diff --git "a/-P2opzekpX8.txt" "b/-P2opzekpX8.txt"
new file mode 100644--- /dev/null
+++ "b/-P2opzekpX8.txt"
@@ -0,0 +1,12307 @@
+align:start position:0%
+ 
+foreign
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+for every symmetry for every continuous
+
+ align:start position:0%
+for every symmetry for every continuous
+ 
+
+ align:start position:0%
+for every symmetry for every continuous
+symmetry there's a conserved current
+
+ align:start position:0%
+symmetry there's a conserved current
+ 
+
+ align:start position:0%
+symmetry there's a conserved current
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+and then we also started talking about
+
+ align:start position:0%
+and then we also started talking about
+ 
+
+ align:start position:0%
+and then we also started talking about
+uh relativistic quantum mechanics
+
+ align:start position:0%
+uh relativistic quantum mechanics
+ 
+
+ align:start position:0%
+uh relativistic quantum mechanics
+uh um how we want to unify
+
+ align:start position:0%
+uh um how we want to unify
+ 
+
+ align:start position:0%
+uh um how we want to unify
+special relativity and quantum mechanics
+
+ align:start position:0%
+special relativity and quantum mechanics
+ 
+
+ align:start position:0%
+special relativity and quantum mechanics
+okay so the most immediate idea for that
+
+ align:start position:0%
+okay so the most immediate idea for that
+ 
+
+ align:start position:0%
+okay so the most immediate idea for that
+is uh what's called the right basic
+
+ align:start position:0%
+is uh what's called the right basic
+ 
+
+ align:start position:0%
+is uh what's called the right basic
+quantum mechanics and the most immediate
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+generalization of the Schrodinger
+
+ align:start position:0%
+generalization of the Schrodinger
+ 
+
+ align:start position:0%
+generalization of the Schrodinger
+equation
+
+ align:start position:0%
+equation
+ 
+
+ align:start position:0%
+equation
+so if you have
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so at the end of last lecture we talked
+
+ align:start position:0%
+so at the end of last lecture we talked
+ 
+
+ align:start position:0%
+so at the end of last lecture we talked
+about say the most immediate realization
+
+ align:start position:0%
+about say the most immediate realization
+ 
+
+ align:start position:0%
+about say the most immediate realization
+of the Schrodinger equation
+
+ align:start position:0%
+of the Schrodinger equation
+ 
+
+ align:start position:0%
+of the Schrodinger equation
+which uh so the
+
+ align:start position:0%
+which uh so the
+ 
+
+ align:start position:0%
+which uh so the
+so if you have e square e equal to say P
+
+ align:start position:0%
+so if you have e square e equal to say P
+ 
+
+ align:start position:0%
+so if you have e square e equal to say P
+Square divided by 2m
+
+ align:start position:0%
+Square divided by 2m
+ 
+
+ align:start position:0%
+Square divided by 2m
+and then you go to non-registic
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+quantum mechanics shielding the equation
+
+ align:start position:0%
+quantum mechanics shielding the equation
+ 
+
+ align:start position:0%
+quantum mechanics shielding the equation
+okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and now if you have e squared equal to p
+
+ align:start position:0%
+and now if you have e squared equal to p
+ 
+
+ align:start position:0%
+and now if you have e squared equal to p
+squared plus M Square
+
+ align:start position:0%
+squared plus M Square
+ 
+
+ align:start position:0%
+squared plus M Square
+for relativistic particle
+
+ align:start position:0%
+for relativistic particle
+ 
+
+ align:start position:0%
+for relativistic particle
+and then you get What's called the
+
+ align:start position:0%
+and then you get What's called the
+ 
+
+ align:start position:0%
+and then you get What's called the
+client golden equation
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and again this PSI has the imputation of
+
+ align:start position:0%
+and again this PSI has the imputation of
+ 
+
+ align:start position:0%
+and again this PSI has the imputation of
+the uh of the wave function so this
+
+ align:start position:0%
+the uh of the wave function so this
+ 
+
+ align:start position:0%
+the uh of the wave function so this
+describes and then this
+
+ align:start position:0%
+describes and then this
+ 
+
+ align:start position:0%
+describes and then this
+so if you as a generalization of this
+
+ align:start position:0%
+so if you as a generalization of this
+ 
+
+ align:start position:0%
+so if you as a generalization of this
+then this means to describe the
+
+ align:start position:0%
+then this means to describe the
+ 
+
+ align:start position:0%
+then this means to describe the
+um
+
+ align:start position:0%
+um
+ 
+
+ align:start position:0%
+um
+the quantum mechanics of a relativistic
+
+ align:start position:0%
+the quantum mechanics of a relativistic
+ 
+
+ align:start position:0%
+the quantum mechanics of a relativistic
+free particle say of mass m okay of mass
+
+ align:start position:0%
+free particle say of mass m okay of mass
+ 
+
+ align:start position:0%
+free particle say of mass m okay of mass
+m
+
+ align:start position:0%
+m
+ 
+
+ align:start position:0%
+m
+so here the PSI
+
+ align:start position:0%
+so here the PSI
+ 
+
+ align:start position:0%
+so here the PSI
+TX
+
+ align:start position:0%
+TX
+ 
+
+ align:start position:0%
+TX
+is the wave function
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+of a relativistic particle
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+relativistic particle
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+of mass n
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and we also notice that this equation
+
+ align:start position:0%
+and we also notice that this equation
+ 
+
+ align:start position:0%
+and we also notice that this equation
+actually is the same
+
+ align:start position:0%
+actually is the same
+ 
+
+ align:start position:0%
+actually is the same
+as the simplest field series equation so
+
+ align:start position:0%
+as the simplest field series equation so
+ 
+
+ align:start position:0%
+as the simplest field series equation so
+we also talked about
+
+ align:start position:0%
+we also talked about
+ 
+
+ align:start position:0%
+we also talked about
+a simplest
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+scalar field Theory
+
+ align:start position:0%
+scalar field Theory
+ 
+
+ align:start position:0%
+scalar field Theory
+classical so here is a a simple simple a
+
+ align:start position:0%
+classical so here is a a simple simple a
+ 
+
+ align:start position:0%
+classical so here is a a simple simple a
+classic simple scalar field Theory
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this series you can write down an
+
+ align:start position:0%
+so this series you can write down an
+ 
+
+ align:start position:0%
+so this series you can write down an
+action
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+of the form
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this is the simplest
+
+ align:start position:0%
+so this is the simplest
+ 
+
+ align:start position:0%
+so this is the simplest
+see we can write down
+
+ align:start position:0%
+see we can write down
+ 
+
+ align:start position:0%
+see we can write down
+and then a relativistic Environ Theory
+
+ align:start position:0%
+and then a relativistic Environ Theory
+ 
+
+ align:start position:0%
+and then a relativistic Environ Theory
+and then equation of motion of this so
+
+ align:start position:0%
+and then equation of motion of this so
+ 
+
+ align:start position:0%
+and then equation of motion of this so
+this is a you view this as a classical
+
+ align:start position:0%
+this is a you view this as a classical
+ 
+
+ align:start position:0%
+this is a you view this as a classical
+field and again this has the equation
+
+ align:start position:0%
+field and again this has the equation
+ 
+
+ align:start position:0%
+field and again this has the equation
+motion you have to see exact the same
+
+ align:start position:0%
+motion you have to see exact the same
+ 
+
+ align:start position:0%
+motion you have to see exact the same
+form as this equation
+
+ align:start position:0%
+form as this equation
+ 
+
+ align:start position:0%
+form as this equation
+so but now here Phi
+
+ align:start position:0%
+so but now here Phi
+ 
+
+ align:start position:0%
+so but now here Phi
+again is the function of TX
+
+ align:start position:0%
+again is the function of TX
+ 
+
+ align:start position:0%
+again is the function of TX
+now has a completely different physical
+
+ align:start position:0%
+now has a completely different physical
+ 
+
+ align:start position:0%
+now has a completely different physical
+interpretation so here is the uh uh this
+
+ align:start position:0%
+interpretation so here is the uh uh this
+ 
+
+ align:start position:0%
+interpretation so here is the uh uh this
+is a classical field
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this is a classical field
+
+ align:start position:0%
+so this is a classical field
+ 
+
+ align:start position:0%
+so this is a classical field
+okay so in this case
+
+ align:start position:0%
+okay so in this case
+ 
+
+ align:start position:0%
+okay so in this case
+the interpretation of the X in here
+
+ align:start position:0%
+the interpretation of the X in here
+ 
+
+ align:start position:0%
+the interpretation of the X in here
+and in here is very different okay so so
+
+ align:start position:0%
+and in here is very different okay so so
+ 
+
+ align:start position:0%
+and in here is very different okay so so
+not only Phi and beside the physical
+
+ align:start position:0%
+not only Phi and beside the physical
+ 
+
+ align:start position:0%
+not only Phi and beside the physical
+interpretation are different the
+
+ align:start position:0%
+interpretation are different the
+ 
+
+ align:start position:0%
+interpretation are different the
+physical interpretation of X also here
+
+ align:start position:0%
+physical interpretation of X also here
+ 
+
+ align:start position:0%
+physical interpretation of X also here
+are different here x is just a label
+
+ align:start position:0%
+are different here x is just a label
+ 
+
+ align:start position:0%
+are different here x is just a label
+is a label for the for the location in
+
+ align:start position:0%
+is a label for the for the location in
+ 
+
+ align:start position:0%
+is a label for the for the location in
+the space which which we Define this
+
+ align:start position:0%
+the space which which we Define this
+ 
+
+ align:start position:0%
+the space which which we Define this
+field
+
+ align:start position:0%
+field
+ 
+
+ align:start position:0%
+field
+but here the x is the eigenvalue
+
+ align:start position:0%
+but here the x is the eigenvalue
+ 
+
+ align:start position:0%
+but here the x is the eigenvalue
+of the position
+
+ align:start position:0%
+of the position
+ 
+
+ align:start position:0%
+of the position
+operator for this right basic particle
+
+ align:start position:0%
+operator for this right basic particle
+ 
+
+ align:start position:0%
+operator for this right basic particle
+okay and so they have very different
+
+ align:start position:0%
+okay and so they have very different
+ 
+
+ align:start position:0%
+okay and so they have very different
+physical imputation
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and so let me just label this equation
+
+ align:start position:0%
+and so let me just label this equation
+ 
+
+ align:start position:0%
+and so let me just label this equation
+by one
+
+ align:start position:0%
+by one
+ 
+
+ align:start position:0%
+by one
+enable this by two
+
+ align:start position:0%
+enable this by two
+ 
+
+ align:start position:0%
+enable this by two
+and this by 2 Prime
+
+ align:start position:0%
+and this by 2 Prime
+ 
+
+ align:start position:0%
+and this by 2 Prime
+okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so we also mentioned that this one has a
+
+ align:start position:0%
+so we also mentioned that this one has a
+ 
+
+ align:start position:0%
+so we also mentioned that this one has a
+the interpretation of this as the wave
+
+ align:start position:0%
+the interpretation of this as the wave
+ 
+
+ align:start position:0%
+the interpretation of this as the wave
+function for relativistic yeah foreign
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so the first he said
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+as you will show in your PSAT 2
+
+ align:start position:0%
+as you will show in your PSAT 2
+ 
+
+ align:start position:0%
+as you will show in your PSAT 2
+uh there's no sensitive
+
+ align:start position:0%
+uh there's no sensitive
+ 
+
+ align:start position:0%
+uh there's no sensitive
+no sensible way
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+to Define
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+a positive definite
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+probability density okay
+
+ align:start position:0%
+probability density okay
+ 
+
+ align:start position:0%
+probability density okay
+so if you want to interpret
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+this has a wave equation
+
+ align:start position:0%
+this has a wave equation
+ 
+
+ align:start position:0%
+this has a wave equation
+now you must have a way then you must
+
+ align:start position:0%
+now you must have a way then you must
+ 
+
+ align:start position:0%
+now you must have a way then you must
+have a probability density because in
+
+ align:start position:0%
+have a probability density because in
+ 
+
+ align:start position:0%
+have a probability density because in
+quantum mechanics probability should be
+
+ align:start position:0%
+quantum mechanics probability should be
+ 
+
+ align:start position:0%
+quantum mechanics probability should be
+conserved okay
+
+ align:start position:0%
+conserved okay
+ 
+
+ align:start position:0%
+conserved okay
+and the second difficulty
+
+ align:start position:0%
+and the second difficulty
+ 
+
+ align:start position:0%
+and the second difficulty
+is that the selective energy state
+
+ align:start position:0%
+is that the selective energy state
+ 
+
+ align:start position:0%
+is that the selective energy state
+because of the square because when you
+
+ align:start position:0%
+because of the square because when you
+ 
+
+ align:start position:0%
+because of the square because when you
+take the square roots then you get the
+
+ align:start position:0%
+take the square roots then you get the
+ 
+
+ align:start position:0%
+take the square roots then you get the
+minus sign and then this negative energy
+
+ align:start position:0%
+minus sign and then this negative energy
+ 
+
+ align:start position:0%
+minus sign and then this negative energy
+states
+
+ align:start position:0%
+states
+ 
+
+ align:start position:0%
+states
+which you cannot avoid in quantum
+
+ align:start position:0%
+which you cannot avoid in quantum
+ 
+
+ align:start position:0%
+which you cannot avoid in quantum
+mechanics
+
+ align:start position:0%
+mechanics
+ 
+
+ align:start position:0%
+mechanics
+even though classically you can just
+
+ align:start position:0%
+even though classically you can just
+ 
+
+ align:start position:0%
+even though classically you can just
+throw them away perhaps okay
+
+ align:start position:0%
+throw them away perhaps okay
+ 
+
+ align:start position:0%
+throw them away perhaps okay
+and the third thing we mentioned at the
+
+ align:start position:0%
+and the third thing we mentioned at the
+ 
+
+ align:start position:0%
+and the third thing we mentioned at the
+end
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+he said for relativistic
+
+ align:start position:0%
+he said for relativistic
+ 
+
+ align:start position:0%
+he said for relativistic
+wave equation
+
+ align:start position:0%
+wave equation
+ 
+
+ align:start position:0%
+wave equation
+you can describe
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+fixed number of particles
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so the particle number cannot change
+
+ align:start position:0%
+so the particle number cannot change
+ 
+
+ align:start position:0%
+so the particle number cannot change
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+so so this way we so this equation if
+
+ align:start position:0%
+so so this way we so this equation if
+ 
+
+ align:start position:0%
+so so this way we so this equation if
+you describe a single particle
+
+ align:start position:0%
+you describe a single particle
+ 
+
+ align:start position:0%
+you describe a single particle
+and if you want to describe two
+
+ align:start position:0%
+and if you want to describe two
+ 
+
+ align:start position:0%
+and if you want to describe two
+particles
+
+ align:start position:0%
+particles
+ 
+
+ align:start position:0%
+particles
+then you leave the two write down a
+
+ align:start position:0%
+then you leave the two write down a
+ 
+
+ align:start position:0%
+then you leave the two write down a
+separate equation
+
+ align:start position:0%
+separate equation
+ 
+
+ align:start position:0%
+separate equation
+for different wave function
+
+ align:start position:0%
+for different wave function
+ 
+
+ align:start position:0%
+for different wave function
+so this is for the two particle wave
+
+ align:start position:0%
+so this is for the two particle wave
+ 
+
+ align:start position:0%
+so this is for the two particle wave
+function will be like this okay and Etc
+
+ align:start position:0%
+function will be like this okay and Etc
+ 
+
+ align:start position:0%
+function will be like this okay and Etc
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+but this does not really make sense
+
+ align:start position:0%
+but this does not really make sense
+ 
+
+ align:start position:0%
+but this does not really make sense
+in a relativistic system
+
+ align:start position:0%
+in a relativistic system
+ 
+
+ align:start position:0%
+in a relativistic system
+because we know that in the relative is
+
+ align:start position:0%
+because we know that in the relative is
+ 
+
+ align:start position:0%
+because we know that in the relative is
+existing e equal to m c Square
+
+ align:start position:0%
+existing e equal to m c Square
+ 
+
+ align:start position:0%
+existing e equal to m c Square
+in any case you have enough energy then
+
+ align:start position:0%
+in any case you have enough energy then
+ 
+
+ align:start position:0%
+in any case you have enough energy then
+you should be able to create particles
+
+ align:start position:0%
+you should be able to create particles
+ 
+
+ align:start position:0%
+you should be able to create particles
+and then that means the lumbar particles
+
+ align:start position:0%
+and then that means the lumbar particles
+ 
+
+ align:start position:0%
+and then that means the lumbar particles
+in the given process is not conserved
+
+ align:start position:0%
+in the given process is not conserved
+ 
+
+ align:start position:0%
+in the given process is not conserved
+okay so if you want to use your quantum
+
+ align:start position:0%
+okay so if you want to use your quantum
+ 
+
+ align:start position:0%
+okay so if you want to use your quantum
+mechanics describe a process and then
+
+ align:start position:0%
+mechanics describe a process and then
+ 
+
+ align:start position:0%
+mechanics describe a process and then
+that's you cannot have a formalism which
+
+ align:start position:0%
+that's you cannot have a formalism which
+ 
+
+ align:start position:0%
+that's you cannot have a formalism which
+the number of particle is fixed which
+
+ align:start position:0%
+the number of particle is fixed which
+ 
+
+ align:start position:0%
+the number of particle is fixed which
+you cannot change
+
+ align:start position:0%
+you cannot change
+ 
+
+ align:start position:0%
+you cannot change
+and so so this is actually the most
+
+ align:start position:0%
+and so so this is actually the most
+ 
+
+ align:start position:0%
+and so so this is actually the most
+fundamental difficulty
+
+ align:start position:0%
+fundamental difficulty
+ 
+
+ align:start position:0%
+fundamental difficulty
+okay is that you cannot change the
+
+ align:start position:0%
+okay is that you cannot change the
+ 
+
+ align:start position:0%
+okay is that you cannot change the
+number of particles
+
+ align:start position:0%
+number of particles
+ 
+
+ align:start position:0%
+number of particles
+and related to this difficulty
+
+ align:start position:0%
+and related to this difficulty
+ 
+
+ align:start position:0%
+and related to this difficulty
+is this interpretation
+
+ align:start position:0%
+is this interpretation
+ 
+
+ align:start position:0%
+is this interpretation
+here we say now if you wanted to
+
+ align:start position:0%
+here we say now if you wanted to
+ 
+
+ align:start position:0%
+here we say now if you wanted to
+we say in here
+
+ align:start position:0%
+we say in here
+ 
+
+ align:start position:0%
+we say in here
+there's a fundamental asymmetry between
+
+ align:start position:0%
+there's a fundamental asymmetry between
+ 
+
+ align:start position:0%
+there's a fundamental asymmetry between
+the T and X okay also yeah maybe let me
+
+ align:start position:0%
+the T and X okay also yeah maybe let me
+ 
+
+ align:start position:0%
+the T and X okay also yeah maybe let me
+put it as four which is also fundamental
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+no additional difficulty there's a
+
+ align:start position:0%
+no additional difficulty there's a
+ 
+
+ align:start position:0%
+no additional difficulty there's a
+fundamental asymmetry
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+between
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+T and X
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so so here in the wave equation T is
+
+ align:start position:0%
+so so here in the wave equation T is
+ 
+
+ align:start position:0%
+so so here in the wave equation T is
+just a parameter
+
+ align:start position:0%
+just a parameter
+ 
+
+ align:start position:0%
+just a parameter
+which we describe the evolution
+
+ align:start position:0%
+which we describe the evolution
+ 
+
+ align:start position:0%
+which we describe the evolution
+but the x is the eigenvalue
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+over Quantum operators
+
+ align:start position:0%
+over Quantum operators
+ 
+
+ align:start position:0%
+over Quantum operators
+eigenvalues of quantum operators
+
+ align:start position:0%
+eigenvalues of quantum operators
+ 
+
+ align:start position:0%
+eigenvalues of quantum operators
+say a corresponding to say hats by
+
+ align:start position:0%
+say a corresponding to say hats by
+ 
+
+ align:start position:0%
+say a corresponding to say hats by
+putting ahead with the load the
+
+ align:start position:0%
+putting ahead with the load the
+ 
+
+ align:start position:0%
+putting ahead with the load the
+corresponding a Quantum operator yeah so
+
+ align:start position:0%
+corresponding a Quantum operator yeah so
+ 
+
+ align:start position:0%
+corresponding a Quantum operator yeah so
+so this is diagonal value of position
+
+ align:start position:0%
+so this is diagonal value of position
+ 
+
+ align:start position:0%
+so this is diagonal value of position
+operators
+
+ align:start position:0%
+operators
+ 
+
+ align:start position:0%
+operators
+and this AC major become even more
+
+ align:start position:0%
+and this AC major become even more
+ 
+
+ align:start position:0%
+and this AC major become even more
+prolonged so if you can see the two
+
+ align:start position:0%
+prolonged so if you can see the two
+ 
+
+ align:start position:0%
+prolonged so if you can see the two
+particles
+
+ align:start position:0%
+particles
+ 
+
+ align:start position:0%
+particles
+okay you have two x here but there's
+
+ align:start position:0%
+okay you have two x here but there's
+ 
+
+ align:start position:0%
+okay you have two x here but there's
+only one t okay
+
+ align:start position:0%
+only one t okay
+ 
+
+ align:start position:0%
+only one t okay
+but
+
+ align:start position:0%
+but
+ 
+
+ align:start position:0%
+but
+so uh so those because of those
+
+ align:start position:0%
+so uh so those because of those
+ 
+
+ align:start position:0%
+so uh so those because of those
+fundamental difficulties
+
+ align:start position:0%
+fundamental difficulties
+ 
+
+ align:start position:0%
+fundamental difficulties
+okay so if you
+
+ align:start position:0%
+okay so if you
+ 
+
+ align:start position:0%
+okay so if you
+connect to this one to four
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so we can we conclude
+
+ align:start position:0%
+so we can we conclude
+ 
+
+ align:start position:0%
+so we can we conclude
+that the
+
+ align:start position:0%
+that the
+ 
+
+ align:start position:0%
+that the
+um
+
+ align:start position:0%
+um
+ 
+
+ align:start position:0%
+um
+relativistic
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+quantum mechanics defined in the sense
+
+ align:start position:0%
+quantum mechanics defined in the sense
+ 
+
+ align:start position:0%
+quantum mechanics defined in the sense
+that you write down a wave equation
+
+ align:start position:0%
+that you write down a wave equation
+ 
+
+ align:start position:0%
+that you write down a wave equation
+and for for wave function don't even it
+
+ align:start position:0%
+and for for wave function don't even it
+ 
+
+ align:start position:0%
+and for for wave function don't even it
+does not be a
+
+ align:start position:0%
+does not be a
+ 
+
+ align:start position:0%
+does not be a
+that's not
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+cannot be
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+a fundamental discussion okay
+
+ align:start position:0%
+a fundamental discussion okay
+ 
+
+ align:start position:0%
+a fundamental discussion okay
+but yeah but right this Quantum kind of
+
+ align:start position:0%
+but yeah but right this Quantum kind of
+ 
+
+ align:start position:0%
+but yeah but right this Quantum kind of
+just refers to this kind of wave
+
+ align:start position:0%
+just refers to this kind of wave
+ 
+
+ align:start position:0%
+just refers to this kind of wave
+equation okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so at most
+
+ align:start position:0%
+so at most
+ 
+
+ align:start position:0%
+so at most
+this can be approximate approximation
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+at the most
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+this is approximate description
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+in situations
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+say there's no
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+there's no particle Creation with a
+
+ align:start position:0%
+there's no particle Creation with a
+ 
+
+ align:start position:0%
+there's no particle Creation with a
+violation
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so so in case which is your
+
+ align:start position:0%
+so so in case which is your
+ 
+
+ align:start position:0%
+so so in case which is your
+party Columbo is fixed
+
+ align:start position:0%
+party Columbo is fixed
+ 
+
+ align:start position:0%
+party Columbo is fixed
+and then the then you can use this as
+
+ align:start position:0%
+and then the then you can use this as
+ 
+
+ align:start position:0%
+and then the then you can use this as
+approximation okay but it cannot be a
+
+ align:start position:0%
+approximation okay but it cannot be a
+ 
+
+ align:start position:0%
+approximation okay but it cannot be a
+fundamental description
+
+ align:start position:0%
+fundamental description
+ 
+
+ align:start position:0%
+fundamental description
+for example later we will talk about the
+
+ align:start position:0%
+for example later we will talk about the
+ 
+
+ align:start position:0%
+for example later we will talk about the
+fumionic version of this wave equation
+
+ align:start position:0%
+fumionic version of this wave equation
+ 
+
+ align:start position:0%
+fumionic version of this wave equation
+so this will describe a particle without
+
+ align:start position:0%
+so this will describe a particle without
+ 
+
+ align:start position:0%
+so this will describe a particle without
+spin so later we will describe the the
+
+ align:start position:0%
+spin so later we will describe the the
+ 
+
+ align:start position:0%
+spin so later we will describe the the
+linear equation for electrons for spin
+
+ align:start position:0%
+linear equation for electrons for spin
+ 
+
+ align:start position:0%
+linear equation for electrons for spin
+half and then then that can indeed be
+
+ align:start position:0%
+half and then then that can indeed be
+ 
+
+ align:start position:0%
+half and then then that can indeed be
+used to describe electron in the
+
+ align:start position:0%
+used to describe electron in the
+ 
+
+ align:start position:0%
+used to describe electron in the
+hydrogen atom as how as far as you don't
+
+ align:start position:0%
+hydrogen atom as how as far as you don't
+ 
+
+ align:start position:0%
+hydrogen atom as how as far as you don't
+create new electrons Etc
+
+ align:start position:0%
+create new electrons Etc
+ 
+
+ align:start position:0%
+create new electrons Etc
+anyway so so so write this Quantum
+
+ align:start position:0%
+anyway so so so write this Quantum
+ 
+
+ align:start position:0%
+anyway so so so write this Quantum
+connect only be described as some kind
+
+ align:start position:0%
+connect only be described as some kind
+ 
+
+ align:start position:0%
+connect only be described as some kind
+of consider as approximate description
+
+ align:start position:0%
+of consider as approximate description
+ 
+
+ align:start position:0%
+of consider as approximate description
+okay but now if you want to you unify
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+special relativity and quantum mechanics
+
+ align:start position:0%
+special relativity and quantum mechanics
+ 
+
+ align:start position:0%
+special relativity and quantum mechanics
+together
+
+ align:start position:0%
+together
+ 
+
+ align:start position:0%
+together
+it turns out that the right formulation
+
+ align:start position:0%
+it turns out that the right formulation
+ 
+
+ align:start position:0%
+it turns out that the right formulation
+is just Quantum field Theory okay
+
+ align:start position:0%
+is just Quantum field Theory okay
+ 
+
+ align:start position:0%
+is just Quantum field Theory okay
+so it turns out that the quantum field
+
+ align:start position:0%
+so it turns out that the quantum field
+ 
+
+ align:start position:0%
+so it turns out that the quantum field
+Theory
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+a corner of your theory
+
+ align:start position:0%
+a corner of your theory
+ 
+
+ align:start position:0%
+a corner of your theory
+addresses these difficulties okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+so it turns out
+
+ align:start position:0%
+so it turns out
+ 
+
+ align:start position:0%
+so it turns out
+it turns out the right way so if we want
+
+ align:start position:0%
+it turns out the right way so if we want
+ 
+
+ align:start position:0%
+it turns out the right way so if we want
+to describe quantum mechanics say of
+
+ align:start position:0%
+to describe quantum mechanics say of
+ 
+
+ align:start position:0%
+to describe quantum mechanics say of
+write a specific particles of mass m
+
+ align:start position:0%
+write a specific particles of mass m
+ 
+
+ align:start position:0%
+write a specific particles of mass m
+okay as we want to do here
+
+ align:start position:0%
+okay as we want to do here
+ 
+
+ align:start position:0%
+okay as we want to do here
+it turns out the proper thing to do
+
+ align:start position:0%
+it turns out the proper thing to do
+ 
+
+ align:start position:0%
+it turns out the proper thing to do
+which is a little bit unintuitive at
+
+ align:start position:0%
+which is a little bit unintuitive at
+ 
+
+ align:start position:0%
+which is a little bit unintuitive at
+first sight
+
+ align:start position:0%
+first sight
+ 
+
+ align:start position:0%
+first sight
+is to start with this field Siri okay
+
+ align:start position:0%
+is to start with this field Siri okay
+ 
+
+ align:start position:0%
+is to start with this field Siri okay
+which seemingly have nothing to do with
+
+ align:start position:0%
+which seemingly have nothing to do with
+ 
+
+ align:start position:0%
+which seemingly have nothing to do with
+write a basic particle but to start with
+
+ align:start position:0%
+write a basic particle but to start with
+ 
+
+ align:start position:0%
+write a basic particle but to start with
+this classical field Theory
+
+ align:start position:0%
+this classical field Theory
+ 
+
+ align:start position:0%
+this classical field Theory
+and then contact it okay it hands out
+
+ align:start position:0%
+and then contact it okay it hands out
+ 
+
+ align:start position:0%
+and then contact it okay it hands out
+turns out once you treat this Theory as
+
+ align:start position:0%
+turns out once you treat this Theory as
+ 
+
+ align:start position:0%
+turns out once you treat this Theory as
+a Quantum field Theory and this becomes
+
+ align:start position:0%
+a Quantum field Theory and this becomes
+ 
+
+ align:start position:0%
+a Quantum field Theory and this becomes
+a theory of arbitrary number of
+
+ align:start position:0%
+a theory of arbitrary number of
+ 
+
+ align:start position:0%
+a theory of arbitrary number of
+relativistic particles of mass m
+
+ align:start position:0%
+relativistic particles of mass m
+ 
+
+ align:start position:0%
+relativistic particles of mass m
+okay and so that's the non-intuitive
+
+ align:start position:0%
+okay and so that's the non-intuitive
+ 
+
+ align:start position:0%
+okay and so that's the non-intuitive
+part and and that's the uh one of the
+
+ align:start position:0%
+part and and that's the uh one of the
+ 
+
+ align:start position:0%
+part and and that's the uh one of the
+miracle say of the field theory is that
+
+ align:start position:0%
+miracle say of the field theory is that
+ 
+
+ align:start position:0%
+miracle say of the field theory is that
+automatically give you a formalism
+
+ align:start position:0%
+automatically give you a formalism
+ 
+
+ align:start position:0%
+automatically give you a formalism
+for for treating arbitrary lumbar
+
+ align:start position:0%
+for for treating arbitrary lumbar
+ 
+
+ align:start position:0%
+for for treating arbitrary lumbar
+particles okay
+
+ align:start position:0%
+particles okay
+ 
+
+ align:start position:0%
+particles okay
+and uh um yeah
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+um and also in field Theory
+
+ align:start position:0%
+um and also in field Theory
+ 
+
+ align:start position:0%
+um and also in field Theory
+so both T and X are parameters okay even
+
+ align:start position:0%
+so both T and X are parameters okay even
+ 
+
+ align:start position:0%
+so both T and X are parameters okay even
+though X only labels your uh uh your
+
+ align:start position:0%
+though X only labels your uh uh your
+ 
+
+ align:start position:0%
+though X only labels your uh uh your
+location so both T and X are parameters
+
+ align:start position:0%
+location so both T and X are parameters
+ 
+
+ align:start position:0%
+location so both T and X are parameters
+and so you can easily to make them to be
+
+ align:start position:0%
+and so you can easily to make them to be
+ 
+
+ align:start position:0%
+and so you can easily to make them to be
+on equal ground to be compatible with
+
+ align:start position:0%
+on equal ground to be compatible with
+ 
+
+ align:start position:0%
+on equal ground to be compatible with
+special relativity
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+um
+
+ align:start position:0%
+um
+ 
+
+ align:start position:0%
+um
+good
+
+ align:start position:0%
+good
+ 
+
+ align:start position:0%
+good
+so any questions on this
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay so we will see that the uh the
+
+ align:start position:0%
+okay so we will see that the uh the
+ 
+
+ align:start position:0%
+okay so we will see that the uh the
+right framework is quantum field three
+
+ align:start position:0%
+right framework is quantum field three
+ 
+
+ align:start position:0%
+right framework is quantum field three
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+so finally it's the last motivation for
+
+ align:start position:0%
+so finally it's the last motivation for
+ 
+
+ align:start position:0%
+so finally it's the last motivation for
+Quantum field Theory
+
+ align:start position:0%
+Quantum field Theory
+ 
+
+ align:start position:0%
+Quantum field Theory
+so we quickly uh uh describe the last uh
+
+ align:start position:0%
+so we quickly uh uh describe the last uh
+ 
+
+ align:start position:0%
+so we quickly uh uh describe the last uh
+so the fields here you can also arise
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+as a limit
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+of discrete systems
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+and this is the uh uh the most relevant
+
+ align:start position:0%
+and this is the uh uh the most relevant
+ 
+
+ align:start position:0%
+and this is the uh uh the most relevant
+for this matter physics for example
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so so let's just consider
+
+ align:start position:0%
+so so let's just consider
+ 
+
+ align:start position:0%
+so so let's just consider
+say uh some yeah let's consider 803
+
+ align:start position:0%
+say uh some yeah let's consider 803
+ 
+
+ align:start position:0%
+say uh some yeah let's consider 803
+example okay
+
+ align:start position:0%
+example okay
+ 
+
+ align:start position:0%
+example okay
+so let's imagine you have
+
+ align:start position:0%
+so let's imagine you have
+ 
+
+ align:start position:0%
+so let's imagine you have
+just number of particles a number of the
+
+ align:start position:0%
+just number of particles a number of the
+ 
+
+ align:start position:0%
+just number of particles a number of the
+atoms say on the on the Chain okay and
+
+ align:start position:0%
+atoms say on the on the Chain okay and
+ 
+
+ align:start position:0%
+atoms say on the on the Chain okay and
+then they're connected by some screens
+
+ align:start position:0%
+then they're connected by some screens
+ 
+
+ align:start position:0%
+then they're connected by some screens
+between them
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay so so this is the uh uh the pro
+
+ align:start position:0%
+okay so so this is the uh uh the pro
+ 
+
+ align:start position:0%
+okay so so this is the uh uh the pro
+yeah consider this to be infinite okay
+
+ align:start position:0%
+yeah consider this to be infinite okay
+ 
+
+ align:start position:0%
+yeah consider this to be infinite okay
+and the spacing between them say is a
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+item I fixed on some matches points
+
+ align:start position:0%
+item I fixed on some matches points
+ 
+
+ align:start position:0%
+item I fixed on some matches points
+and the lattice facing is a okay
+
+ align:start position:0%
+and the lattice facing is a okay
+ 
+
+ align:start position:0%
+and the lattice facing is a okay
+so
+
+ align:start position:0%
+so
+ 
+
+ align:start position:0%
+so
+yeah so we can label the other say
+
+ align:start position:0%
+yeah so we can label the other say
+ 
+
+ align:start position:0%
+yeah so we can label the other say
+each particle by their position for
+
+ align:start position:0%
+each particle by their position for
+ 
+
+ align:start position:0%
+each particle by their position for
+example this is x0 this is X1 this is X2
+
+ align:start position:0%
+example this is x0 this is X1 this is X2
+ 
+
+ align:start position:0%
+example this is x0 this is X1 this is X2
+Etc okay and the typical particle is x n
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+at the location of nth particle is x n
+
+ align:start position:0%
+at the location of nth particle is x n
+ 
+
+ align:start position:0%
+at the location of nth particle is x n
+and so we can also introduce the
+
+ align:start position:0%
+and so we can also introduce the
+ 
+
+ align:start position:0%
+and so we can also introduce the
+deviation
+
+ align:start position:0%
+deviation
+ 
+
+ align:start position:0%
+deviation
+between the equivalent position of each
+
+ align:start position:0%
+between the equivalent position of each
+ 
+
+ align:start position:0%
+between the equivalent position of each
+particle so let's call it n
+
+ align:start position:0%
+particle so let's call it n
+ 
+
+ align:start position:0%
+particle so let's call it n
+okay so now let's consider the Dynamics
+
+ align:start position:0%
+okay so now let's consider the Dynamics
+ 
+
+ align:start position:0%
+okay so now let's consider the Dynamics
+of ether M for this Theory
+
+ align:start position:0%
+of ether M for this Theory
+ 
+
+ align:start position:0%
+of ether M for this Theory
+and so this is the just deviation of the
+
+ align:start position:0%
+and so this is the just deviation of the
+ 
+
+ align:start position:0%
+and so this is the just deviation of the
+nth particle from its equivalent
+
+ align:start position:0%
+nth particle from its equivalent
+ 
+
+ align:start position:0%
+nth particle from its equivalent
+position okay so X and zero is its
+
+ align:start position:0%
+position okay so X and zero is its
+ 
+
+ align:start position:0%
+position okay so X and zero is its
+equivalent position
+
+ align:start position:0%
+equivalent position
+ 
+
+ align:start position:0%
+equivalent position
+so now so now if you write down the
+
+ align:start position:0%
+so now so now if you write down the
+ 
+
+ align:start position:0%
+so now so now if you write down the
+lagrangian for this system
+
+ align:start position:0%
+lagrangian for this system
+ 
+
+ align:start position:0%
+lagrangian for this system
+that we can easily do you just write T
+
+ align:start position:0%
+that we can easily do you just write T
+ 
+
+ align:start position:0%
+that we can easily do you just write T
+minus V the t is the kinetic energy and
+
+ align:start position:0%
+minus V the t is the kinetic energy and
+ 
+
+ align:start position:0%
+minus V the t is the kinetic energy and
+V is the uh um the potential energy so
+
+ align:start position:0%
+V is the uh um the potential energy so
+ 
+
+ align:start position:0%
+V is the uh um the potential energy so
+we can just write it as sum over n
+
+ align:start position:0%
+we can just write it as sum over n
+ 
+
+ align:start position:0%
+we can just write it as sum over n
+over o
+
+ align:start position:0%
+over o
+ 
+
+ align:start position:0%
+over o
+particles and then let's assume they
+
+ align:start position:0%
+particles and then let's assume they
+ 
+
+ align:start position:0%
+particles and then let's assume they
+have the same mass let's write mu
+
+ align:start position:0%
+have the same mass let's write mu
+ 
+
+ align:start position:0%
+have the same mass let's write mu
+ETA n dot Square okay so this is a
+
+ align:start position:0%
+ETA n dot Square okay so this is a
+ 
+
+ align:start position:0%
+ETA n dot Square okay so this is a
+kinetic term so so mu is the mass for
+
+ align:start position:0%
+kinetic term so so mu is the mass for
+ 
+
+ align:start position:0%
+kinetic term so so mu is the mass for
+each particle
+
+ align:start position:0%
+each particle
+ 
+
+ align:start position:0%
+each particle
+and then their their potential yeah
+
+ align:start position:0%
+and then their their potential yeah
+ 
+
+ align:start position:0%
+and then their their potential yeah
+let's assume at each point there is also
+
+ align:start position:0%
+let's assume at each point there is also
+ 
+
+ align:start position:0%
+let's assume at each point there is also
+a yeah let's just yeah then there's some
+
+ align:start position:0%
+a yeah let's just yeah then there's some
+ 
+
+ align:start position:0%
+a yeah let's just yeah then there's some
+uh uh uh interaction because each
+
+ align:start position:0%
+uh uh uh interaction because each
+ 
+
+ align:start position:0%
+uh uh uh interaction because each
+particle are connected by the spring and
+
+ align:start position:0%
+particle are connected by the spring and
+ 
+
+ align:start position:0%
+particle are connected by the spring and
+so they're a harmonic Force
+
+ align:start position:0%
+so they're a harmonic Force
+ 
+
+ align:start position:0%
+so they're a harmonic Force
+between
+
+ align:start position:0%
+between
+ 
+
+ align:start position:0%
+between
+neighboring particles okay
+
+ align:start position:0%
+neighboring particles okay
+ 
+
+ align:start position:0%
+neighboring particles okay
+and now let's imagine also there's a
+
+ align:start position:0%
+and now let's imagine also there's a
+ 
+
+ align:start position:0%
+and now let's imagine also there's a
+harmonic potential which trapped this
+
+ align:start position:0%
+harmonic potential which trapped this
+ 
+
+ align:start position:0%
+harmonic potential which trapped this
+particle itself
+
+ align:start position:0%
+particle itself
+ 
+
+ align:start position:0%
+particle itself
+at each location
+
+ align:start position:0%
+at each location
+ 
+
+ align:start position:0%
+at each location
+okay so this is a very simple uh spring
+
+ align:start position:0%
+okay so this is a very simple uh spring
+ 
+
+ align:start position:0%
+okay so this is a very simple uh spring
+and the particle problem which you
+
+ align:start position:0%
+and the particle problem which you
+ 
+
+ align:start position:0%
+and the particle problem which you
+encounter say in 803
+
+ align:start position:0%
+encounter say in 803
+ 
+
+ align:start position:0%
+encounter say in 803
+okay is this problem clear
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay I I assume most of you have seen
+
+ align:start position:0%
+okay I I assume most of you have seen
+ 
+
+ align:start position:0%
+okay I I assume most of you have seen
+this problem before
+
+ align:start position:0%
+this problem before
+ 
+
+ align:start position:0%
+this problem before
+and the and you're tasking 803 is
+
+ align:start position:0%
+and the and you're tasking 803 is
+ 
+
+ align:start position:0%
+and the and you're tasking 803 is
+actually to uh to find the lower modes
+
+ align:start position:0%
+actually to uh to find the lower modes
+ 
+
+ align:start position:0%
+actually to uh to find the lower modes
+say of this system okay
+
+ align:start position:0%
+say of this system okay
+ 
+
+ align:start position:0%
+say of this system okay
+and in 803 you also describe that we can
+
+ align:start position:0%
+and in 803 you also describe that we can
+ 
+
+ align:start position:0%
+and in 803 you also describe that we can
+in a go to zero limit
+
+ align:start position:0%
+in a go to zero limit
+ 
+
+ align:start position:0%
+in a go to zero limit
+so if the left is spacing is very small
+
+ align:start position:0%
+so if the left is spacing is very small
+ 
+
+ align:start position:0%
+so if the left is spacing is very small
+and if you're only interested in the
+
+ align:start position:0%
+and if you're only interested in the
+ 
+
+ align:start position:0%
+and if you're only interested in the
+behavior of the system at a very large
+
+ align:start position:0%
+behavior of the system at a very large
+ 
+
+ align:start position:0%
+behavior of the system at a very large
+distance say the distance much larger
+
+ align:start position:0%
+distance say the distance much larger
+ 
+
+ align:start position:0%
+distance say the distance much larger
+than a equal imagine lighter than a
+
+ align:start position:0%
+than a equal imagine lighter than a
+ 
+
+ align:start position:0%
+than a equal imagine lighter than a
+then you can essentially choose this
+
+ align:start position:0%
+then you can essentially choose this
+ 
+
+ align:start position:0%
+then you can essentially choose this
+season as a Continuum okay you don't
+
+ align:start position:0%
+season as a Continuum okay you don't
+ 
+
+ align:start position:0%
+season as a Continuum okay you don't
+have to resolve individual particles
+
+ align:start position:0%
+have to resolve individual particles
+ 
+
+ align:start position:0%
+have to resolve individual particles
+and so we can just enable the newer
+
+ align:start position:0%
+and so we can just enable the newer
+ 
+
+ align:start position:0%
+and so we can just enable the newer
+limits so you can choose the chain
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+of particles
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+and uh so you
+
+ align:start position:0%
+and uh so you
+ 
+
+ align:start position:0%
+and uh so you
+so each
+
+ align:start position:0%
+so each
+ 
+
+ align:start position:0%
+so each
+and T you replace it
+
+ align:start position:0%
+and T you replace it
+ 
+
+ align:start position:0%
+and T you replace it
+by ETA x t
+
+ align:start position:0%
+by ETA x t
+ 
+
+ align:start position:0%
+by ETA x t
+so X label is position
+
+ align:start position:0%
+so X label is position
+ 
+
+ align:start position:0%
+so X label is position
+okay the X label is position and T
+
+ align:start position:0%
+okay the X label is position and T
+ 
+
+ align:start position:0%
+okay the X label is position and T
+describes the Dynamics okay so so ETA is
+
+ align:start position:0%
+describes the Dynamics okay so so ETA is
+ 
+
+ align:start position:0%
+describes the Dynamics okay so so ETA is
+the deviation
+
+ align:start position:0%
+the deviation
+ 
+
+ align:start position:0%
+the deviation
+at the location X and it's depend on T
+
+ align:start position:0%
+at the location X and it's depend on T
+ 
+
+ align:start position:0%
+at the location X and it's depend on T
+okay so this is the oscillator
+
+ align:start position:0%
+okay so this is the oscillator
+ 
+
+ align:start position:0%
+okay so this is the oscillator
+and then sum over n
+
+ align:start position:0%
+and then sum over n
+ 
+
+ align:start position:0%
+and then sum over n
+in the lagrangian
+
+ align:start position:0%
+in the lagrangian
+ 
+
+ align:start position:0%
+in the lagrangian
+then we can replace it by integral
+
+ align:start position:0%
+then we can replace it by integral
+ 
+
+ align:start position:0%
+then we can replace it by integral
+over DX
+
+ align:start position:0%
+over DX
+ 
+
+ align:start position:0%
+over DX
+okay and now you just choose this as a
+
+ align:start position:0%
+okay and now you just choose this as a
+ 
+
+ align:start position:0%
+okay and now you just choose this as a
+one-dimensional continuum see some
+
+ align:start position:0%
+one-dimensional continuum see some
+ 
+
+ align:start position:0%
+one-dimensional continuum see some
+integration of the X but of course here
+
+ align:start position:0%
+integration of the X but of course here
+ 
+
+ align:start position:0%
+integration of the X but of course here
+there's a label a nothing spacing so so
+
+ align:start position:0%
+there's a label a nothing spacing so so
+ 
+
+ align:start position:0%
+there's a label a nothing spacing so so
+the uh so the implementational uh here
+
+ align:start position:0%
+the uh so the implementational uh here
+ 
+
+ align:start position:0%
+the uh so the implementational uh here
+the element is a so a times
+
+ align:start position:0%
+the element is a so a times
+ 
+
+ align:start position:0%
+the element is a so a times
+the sum Over N you can replace it by DX
+
+ align:start position:0%
+the sum Over N you can replace it by DX
+ 
+
+ align:start position:0%
+the sum Over N you can replace it by DX
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+is the lattice spacing
+
+ align:start position:0%
+is the lattice spacing
+ 
+
+ align:start position:0%
+is the lattice spacing
+and now you can just write
+
+ align:start position:0%
+and now you can just write
+ 
+
+ align:start position:0%
+and now you can just write
+this Lagrange in terms of Continuum
+
+ align:start position:0%
+this Lagrange in terms of Continuum
+ 
+
+ align:start position:0%
+this Lagrange in terms of Continuum
+Theory
+
+ align:start position:0%
+Theory
+ 
+
+ align:start position:0%
+Theory
+okay now you can write this in the
+
+ align:start position:0%
+okay now you can write this in the
+ 
+
+ align:start position:0%
+okay now you can write this in the
+ground in a Continuum Theory and then
+
+ align:start position:0%
+ground in a Continuum Theory and then
+ 
+
+ align:start position:0%
+ground in a Continuum Theory and then
+let's just do it
+
+ align:start position:0%
+let's just do it
+ 
+
+ align:start position:0%
+let's just do it
+so um so we can write it yeah let me
+
+ align:start position:0%
+so um so we can write it yeah let me
+ 
+
+ align:start position:0%
+so um so we can write it yeah let me
+just write one more step
+
+ align:start position:0%
+just write one more step
+ 
+
+ align:start position:0%
+just write one more step
+so you can write it as sum over a we
+
+ align:start position:0%
+so you can write it as sum over a we
+ 
+
+ align:start position:0%
+so you can write it as sum over a we
+take the a factor out because of the a
+
+ align:start position:0%
+take the a factor out because of the a
+ 
+
+ align:start position:0%
+take the a factor out because of the a
+fact R have to be changed into
+
+ align:start position:0%
+fact R have to be changed into
+ 
+
+ align:start position:0%
+fact R have to be changed into
+integration
+
+ align:start position:0%
+integration
+ 
+
+ align:start position:0%
+integration
+and then you have one half mu divided by
+
+ align:start position:0%
+and then you have one half mu divided by
+ 
+
+ align:start position:0%
+and then you have one half mu divided by
+a
+
+ align:start position:0%
+a
+ 
+
+ align:start position:0%
+a
+ETA N squared
+
+ align:start position:0%
+ETA N squared
+ 
+
+ align:start position:0%
+ETA N squared
+minus one half Lambda a
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so so I just slightly rewrite
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+it's lagranging
+
+ align:start position:0%
+it's lagranging
+ 
+
+ align:start position:0%
+it's lagranging
+so that it is easy to take the Continuum
+
+ align:start position:0%
+so that it is easy to take the Continuum
+ 
+
+ align:start position:0%
+so that it is easy to take the Continuum
+limit
+
+ align:start position:0%
+limit
+ 
+
+ align:start position:0%
+limit
+so we have taken the factor of a out
+
+ align:start position:0%
+so we have taken the factor of a out
+ 
+
+ align:start position:0%
+so we have taken the factor of a out
+but for this term
+
+ align:start position:0%
+but for this term
+ 
+
+ align:start position:0%
+but for this term
+because this contains the difference
+
+ align:start position:0%
+because this contains the difference
+ 
+
+ align:start position:0%
+because this contains the difference
+between the two and we also divided by a
+
+ align:start position:0%
+between the two and we also divided by a
+ 
+
+ align:start position:0%
+between the two and we also divided by a
+in the downstairs and then we need to
+
+ align:start position:0%
+in the downstairs and then we need to
+ 
+
+ align:start position:0%
+in the downstairs and then we need to
+multiply a upstairs and then there's a
+
+ align:start position:0%
+multiply a upstairs and then there's a
+ 
+
+ align:start position:0%
+multiply a upstairs and then there's a
+in the front okay
+
+ align:start position:0%
+in the front okay
+ 
+
+ align:start position:0%
+in the front okay
+another Continuum limit
+
+ align:start position:0%
+another Continuum limit
+ 
+
+ align:start position:0%
+another Continuum limit
+you can just replace this by integral
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and now I just
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+you can just write it as one half
+
+ align:start position:0%
+you can just write it as one half
+ 
+
+ align:start position:0%
+you can just write it as one half
+you tilde
+
+ align:start position:0%
+you tilde
+ 
+
+ align:start position:0%
+you tilde
+is a start Square so now e to n just
+
+ align:start position:0%
+is a start Square so now e to n just
+ 
+
+ align:start position:0%
+is a start Square so now e to n just
+replace it by E to x t
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and here let me call it lamina tilde
+
+ align:start position:0%
+and here let me call it lamina tilde
+ 
+
+ align:start position:0%
+and here let me call it lamina tilde
+partial X is a square
+
+ align:start position:0%
+partial X is a square
+ 
+
+ align:start position:0%
+partial X is a square
+and this term we can just replace it by
+
+ align:start position:0%
+and this term we can just replace it by
+ 
+
+ align:start position:0%
+and this term we can just replace it by
+the derivative of ETA and then this is
+
+ align:start position:0%
+the derivative of ETA and then this is
+ 
+
+ align:start position:0%
+the derivative of ETA and then this is
+just become one half Sigma tilde is a
+
+ align:start position:0%
+just become one half Sigma tilde is a
+ 
+
+ align:start position:0%
+just become one half Sigma tilde is a
+square
+
+ align:start position:0%
+square
+ 
+
+ align:start position:0%
+square
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+and the uh the MU tilde of course
+
+ align:start position:0%
+and the uh the MU tilde of course
+ 
+
+ align:start position:0%
+and the uh the MU tilde of course
+is Mu divided by a
+
+ align:start position:0%
+is Mu divided by a
+ 
+
+ align:start position:0%
+is Mu divided by a
+number is Lambda times a and sigma tilde
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+this is the sigma divided by a
+
+ align:start position:0%
+this is the sigma divided by a
+ 
+
+ align:start position:0%
+this is the sigma divided by a
+okay so the Continuum limit is that
+
+ align:start position:0%
+okay so the Continuum limit is that
+ 
+
+ align:start position:0%
+okay so the Continuum limit is that
+those quantity has to be fixed okay the
+
+ align:start position:0%
+those quantity has to be fixed okay the
+ 
+
+ align:start position:0%
+those quantity has to be fixed okay the
+tier the quantity has to be fixed and
+
+ align:start position:0%
+tier the quantity has to be fixed and
+ 
+
+ align:start position:0%
+tier the quantity has to be fixed and
+then we have a continuum lagrangea
+
+ align:start position:0%
+then we have a continuum lagrangea
+ 
+
+ align:start position:0%
+then we have a continuum lagrangea
+okay and then we have a classical field
+
+ align:start position:0%
+okay and then we have a classical field
+ 
+
+ align:start position:0%
+okay and then we have a classical field
+Theory
+
+ align:start position:0%
+Theory
+ 
+
+ align:start position:0%
+Theory
+and this series is essentially the same
+
+ align:start position:0%
+and this series is essentially the same
+ 
+
+ align:start position:0%
+and this series is essentially the same
+as
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+as that theory
+
+ align:start position:0%
+as that theory
+ 
+
+ align:start position:0%
+as that theory
+okay so if you take this Factor mu tilde
+
+ align:start position:0%
+okay so if you take this Factor mu tilde
+ 
+
+ align:start position:0%
+okay so if you take this Factor mu tilde
+out
+
+ align:start position:0%
+out
+ 
+
+ align:start position:0%
+out
+okay if we take this Factor mu tilde out
+
+ align:start position:0%
+okay if we take this Factor mu tilde out
+ 
+
+ align:start position:0%
+okay if we take this Factor mu tilde out
+okay so let me just take this part of
+
+ align:start position:0%
+okay so let me just take this part of
+ 
+
+ align:start position:0%
+okay so let me just take this part of
+mutilda out in the front just up to
+
+ align:start position:0%
+mutilda out in the front just up to
+ 
+
+ align:start position:0%
+mutilda out in the front just up to
+overall factor and here is Lambda tilde
+
+ align:start position:0%
+overall factor and here is Lambda tilde
+ 
+
+ align:start position:0%
+overall factor and here is Lambda tilde
+by a divided by mu tier that we could
+
+ align:start position:0%
+by a divided by mu tier that we could
+ 
+
+ align:start position:0%
+by a divided by mu tier that we could
+let's call it V Square
+
+ align:start position:0%
+let's call it V Square
+ 
+
+ align:start position:0%
+let's call it V Square
+and this becomes Sigma material divided
+
+ align:start position:0%
+and this becomes Sigma material divided
+ 
+
+ align:start position:0%
+and this becomes Sigma material divided
+by uh by mu till let's call it m Square
+
+ align:start position:0%
+by uh by mu till let's call it m Square
+ 
+
+ align:start position:0%
+by uh by mu till let's call it m Square
+so the V Square
+
+ align:start position:0%
+so the V Square
+ 
+
+ align:start position:0%
+so the V Square
+is equal to Mu tilde
+
+ align:start position:0%
+is equal to Mu tilde
+ 
+
+ align:start position:0%
+is equal to Mu tilde
+is equal to the Lambda tilde
+
+ align:start position:0%
+is equal to the Lambda tilde
+ 
+
+ align:start position:0%
+is equal to the Lambda tilde
+divided by mu tilde
+
+ align:start position:0%
+divided by mu tilde
+ 
+
+ align:start position:0%
+divided by mu tilde
+and the M Square
+
+ align:start position:0%
+and the M Square
+ 
+
+ align:start position:0%
+and the M Square
+is equal to Sigma material
+
+ align:start position:0%
+is equal to Sigma material
+ 
+
+ align:start position:0%
+is equal to Sigma material
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+and then this is just essentially
+
+ align:start position:0%
+and then this is just essentially
+ 
+
+ align:start position:0%
+and then this is just essentially
+identical to that theory
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+when V equal to one okay so when b equal
+
+ align:start position:0%
+when V equal to one okay so when b equal
+ 
+
+ align:start position:0%
+when V equal to one okay so when b equal
+to one
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+become the same
+
+ align:start position:0%
+become the same
+ 
+
+ align:start position:0%
+become the same
+as just two okay equation two
+
+ align:start position:0%
+as just two okay equation two
+ 
+
+ align:start position:0%
+as just two okay equation two
+of course corresponding to relative K is
+
+ align:start position:0%
+of course corresponding to relative K is
+ 
+
+ align:start position:0%
+of course corresponding to relative K is
+a speed of light but in general uh this
+
+ align:start position:0%
+a speed of light but in general uh this
+ 
+
+ align:start position:0%
+a speed of light but in general uh this
+describer can yeah addition I can
+
+ align:start position:0%
+describer can yeah addition I can
+ 
+
+ align:start position:0%
+describer can yeah addition I can
+describe a Contin but in general this is
+
+ align:start position:0%
+describe a Contin but in general this is
+ 
+
+ align:start position:0%
+describe a Contin but in general this is
+a noun in general this can be just a
+
+ align:start position:0%
+a noun in general this can be just a
+ 
+
+ align:start position:0%
+a noun in general this can be just a
+long relativistic field series okay for
+
+ align:start position:0%
+long relativistic field series okay for
+ 
+
+ align:start position:0%
+long relativistic field series okay for
+other values of B okay
+
+ align:start position:0%
+other values of B okay
+ 
+
+ align:start position:0%
+other values of B okay
+foreign
+
+ align:start position:0%
+foreign
+ 
+
+ align:start position:0%
+foreign
+so this is
+
+ align:start position:0%
+so this is
+ 
+
+ align:start position:0%
+so this is
+so even though this example is very
+
+ align:start position:0%
+so even though this example is very
+ 
+
+ align:start position:0%
+so even though this example is very
+simple
+
+ align:start position:0%
+simple
+ 
+
+ align:start position:0%
+simple
+but this is actually a very general way
+
+ align:start position:0%
+but this is actually a very general way
+ 
+
+ align:start position:0%
+but this is actually a very general way
+that we can treat many condensed metal
+
+ align:start position:0%
+that we can treat many condensed metal
+ 
+
+ align:start position:0%
+that we can treat many condensed metal
+systems
+
+ align:start position:0%
+systems
+ 
+
+ align:start position:0%
+systems
+which often in Mobile lattice say
+
+ align:start position:0%
+which often in Mobile lattice say
+ 
+
+ align:start position:0%
+which often in Mobile lattice say
+because because solid you can imagine
+
+ align:start position:0%
+because because solid you can imagine
+ 
+
+ align:start position:0%
+because because solid you can imagine
+all the items on the lattice Etc and if
+
+ align:start position:0%
+all the items on the lattice Etc and if
+ 
+
+ align:start position:0%
+all the items on the lattice Etc and if
+you're only interested in the very
+
+ align:start position:0%
+you're only interested in the very
+ 
+
+ align:start position:0%
+you're only interested in the very
+microscopic Behavior then you can treat
+
+ align:start position:0%
+microscopic Behavior then you can treat
+ 
+
+ align:start position:0%
+microscopic Behavior then you can treat
+solid as a continuum
+
+ align:start position:0%
+solid as a continuum
+ 
+
+ align:start position:0%
+solid as a continuum
+and then and now you can uh now if
+
+ align:start position:0%
+and then and now you can uh now if
+ 
+
+ align:start position:0%
+and then and now you can uh now if
+you're interested in quantum mechanics
+
+ align:start position:0%
+you're interested in quantum mechanics
+ 
+
+ align:start position:0%
+you're interested in quantum mechanics
+of such a system then the quantum field
+
+ align:start position:0%
+of such a system then the quantum field
+ 
+
+ align:start position:0%
+of such a system then the quantum field
+theory that naturally Rises okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay good any questions on this example
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yes
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+sorry so the limit yeah yeah yeah
+
+ align:start position:0%
+sorry so the limit yeah yeah yeah
+ 
+
+ align:start position:0%
+sorry so the limit yeah yeah yeah
+like same windows view yeah yeah so what
+
+ align:start position:0%
+like same windows view yeah yeah so what
+ 
+
+ align:start position:0%
+like same windows view yeah yeah so what
+is that physically
+
+ align:start position:0%
+is that physically
+ 
+
+ align:start position:0%
+is that physically
+strength of yeah yeah it is
+
+ align:start position:0%
+strength of yeah yeah it is
+ 
+
+ align:start position:0%
+strength of yeah yeah it is
+corresponding to the the case that the
+
+ align:start position:0%
+corresponding to the the case that the
+ 
+
+ align:start position:0%
+corresponding to the the case that the
+um uh uh yeah it just tells you that
+
+ align:start position:0%
+um uh uh yeah it just tells you that
+ 
+
+ align:start position:0%
+um uh uh yeah it just tells you that
+relativistic limit is special happens at
+
+ align:start position:0%
+relativistic limit is special happens at
+ 
+
+ align:start position:0%
+relativistic limit is special happens at
+very special points but I guess why why
+
+ align:start position:0%
+very special points but I guess why why
+ 
+
+ align:start position:0%
+very special points but I guess why why
+is that the relative like to me lenders
+
+ align:start position:0%
+is that the relative like to me lenders
+ 
+
+ align:start position:0%
+is that the relative like to me lenders
+like the strength of your spring and
+
+ align:start position:0%
+like the strength of your spring and
+ 
+
+ align:start position:0%
+like the strength of your spring and
+then yeah use your mask right you've got
+
+ align:start position:0%
+then yeah use your mask right you've got
+ 
+
+ align:start position:0%
+then yeah use your mask right you've got
+to be comparable how does that yeah
+
+ align:start position:0%
+to be comparable how does that yeah
+ 
+
+ align:start position:0%
+to be comparable how does that yeah
+there's not much you can read from here
+
+ align:start position:0%
+there's not much you can read from here
+ 
+
+ align:start position:0%
+there's not much you can read from here
+yeah yeah uh it it just like when you
+
+ align:start position:0%
+yeah yeah uh it it just like when you
+ 
+
+ align:start position:0%
+yeah yeah uh it it just like when you
+choose some special parameters then you
+
+ align:start position:0%
+choose some special parameters then you
+ 
+
+ align:start position:0%
+choose some special parameters then you
+can uh have a relativistic limit
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+other questions yes
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yeah so these are all um scalars right
+
+ align:start position:0%
+yeah so these are all um scalars right
+ 
+
+ align:start position:0%
+yeah so these are all um scalars right
+I could you can also have uh you can
+
+ align:start position:0%
+I could you can also have uh you can
+ 
+
+ align:start position:0%
+I could you can also have uh you can
+also you mean uh you can also have
+
+ align:start position:0%
+also you mean uh you can also have
+ 
+
+ align:start position:0%
+also you mean uh you can also have
+tensors or vectors yeah
+
+ align:start position:0%
+tensors or vectors yeah
+ 
+
+ align:start position:0%
+tensors or vectors yeah
+yeah
+
+ align:start position:0%
+yeah
+ 
+
+ align:start position:0%
+yeah
+so like what would you treat with this
+
+ align:start position:0%
+so like what would you treat with this
+ 
+
+ align:start position:0%
+so like what would you treat with this
+pronouns oh you can cheat yeah for
+
+ align:start position:0%
+pronouns oh you can cheat yeah for
+ 
+
+ align:start position:0%
+pronouns oh you can cheat yeah for
+example you can treat four rounds you
+
+ align:start position:0%
+example you can treat four rounds you
+ 
+
+ align:start position:0%
+example you can treat four rounds you
+can also choose spins
+
+ align:start position:0%
+can also choose spins
+ 
+
+ align:start position:0%
+can also choose spins
+and say for example if you have an icing
+
+ align:start position:0%
+and say for example if you have an icing
+ 
+
+ align:start position:0%
+and say for example if you have an icing
+model just consider lattice of spins
+
+ align:start position:0%
+model just consider lattice of spins
+ 
+
+ align:start position:0%
+model just consider lattice of spins
+and then the average Spin and then you
+
+ align:start position:0%
+and then the average Spin and then you
+ 
+
+ align:start position:0%
+and then the average Spin and then you
+can treat it as a scale of field and
+
+ align:start position:0%
+can treat it as a scale of field and
+ 
+
+ align:start position:0%
+can treat it as a scale of field and
+then again you can write down a field
+
+ align:start position:0%
+then again you can write down a field
+ 
+
+ align:start position:0%
+then again you can write down a field
+Theory yeah and actually the the
+
+ align:start position:0%
+Theory yeah and actually the the
+ 
+
+ align:start position:0%
+Theory yeah and actually the the
+Breakthrough
+
+ align:start position:0%
+Breakthrough
+ 
+
+ align:start position:0%
+Breakthrough
+of the phase transition
+
+ align:start position:0%
+of the phase transition
+ 
+
+ align:start position:0%
+of the phase transition
+metaphysics to to understand what phase
+
+ align:start position:0%
+metaphysics to to understand what phase
+ 
+
+ align:start position:0%
+metaphysics to to understand what phase
+transition is really about and describe
+
+ align:start position:0%
+transition is really about and describe
+ 
+
+ align:start position:0%
+transition is really about and describe
+the behavior near the phase transition
+
+ align:start position:0%
+the behavior near the phase transition
+ 
+
+ align:start position:0%
+the behavior near the phase transition
+and precisely coincided with the
+
+ align:start position:0%
+and precisely coincided with the
+ 
+
+ align:start position:0%
+and precisely coincided with the
+development of field Theory and uh yeah
+
+ align:start position:0%
+development of field Theory and uh yeah
+ 
+
+ align:start position:0%
+development of field Theory and uh yeah
+actually increased our understanding of
+
+ align:start position:0%
+actually increased our understanding of
+ 
+
+ align:start position:0%
+actually increased our understanding of
+quantum field Theory yeah
+
+ align:start position:0%
+quantum field Theory yeah
+ 
+
+ align:start position:0%
+quantum field Theory yeah
+foreign
+
+ align:start position:0%
+foreign
+ 
+
+ align:start position:0%
+foreign
+other questions
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay good
+
+ align:start position:0%
+okay good
+ 
+
+ align:start position:0%
+okay good
+just to summarize
+
+ align:start position:0%
+just to summarize
+ 
+
+ align:start position:0%
+just to summarize
+what we have discussed so far
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+or path
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+to
+
+ align:start position:0%
+to
+ 
+
+ align:start position:0%
+to
+qft okay
+
+ align:start position:0%
+qft okay
+ 
+
+ align:start position:0%
+qft okay
+so we have described three parts three
+
+ align:start position:0%
+so we have described three parts three
+ 
+
+ align:start position:0%
+so we have described three parts three
+parts but they're pretty General first
+
+ align:start position:0%
+parts but they're pretty General first
+ 
+
+ align:start position:0%
+parts but they're pretty General first
+you say the quantum that line we offer
+
+ align:start position:0%
+you say the quantum that line we offer
+ 
+
+ align:start position:0%
+you say the quantum that line we offer
+increasing Quantum Dynamics
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+of some classical fields
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+say such as
+
+ align:start position:0%
+say such as
+ 
+
+ align:start position:0%
+say such as
+say Electric magnetic field
+
+ align:start position:0%
+say Electric magnetic field
+ 
+
+ align:start position:0%
+say Electric magnetic field
+or space-time metric if you are
+
+ align:start position:0%
+or space-time metric if you are
+ 
+
+ align:start position:0%
+or space-time metric if you are
+interested in gravity
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+Etc okay so so so in this case we
+
+ align:start position:0%
+Etc okay so so so in this case we
+ 
+
+ align:start position:0%
+Etc okay so so so in this case we
+already have the classical Fields Theory
+
+ align:start position:0%
+already have the classical Fields Theory
+ 
+
+ align:start position:0%
+already have the classical Fields Theory
+but we know the word is quantum and we
+
+ align:start position:0%
+but we know the word is quantum and we
+ 
+
+ align:start position:0%
+but we know the word is quantum and we
+want to understand what's the quantum
+
+ align:start position:0%
+want to understand what's the quantum
+ 
+
+ align:start position:0%
+want to understand what's the quantum
+version of it
+
+ align:start position:0%
+version of it
+ 
+
+ align:start position:0%
+version of it
+and the second is that it unifies
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+special relativity
+
+ align:start position:0%
+special relativity
+ 
+
+ align:start position:0%
+special relativity
+plus quantum mechanics okay so you read
+
+ align:start position:0%
+plus quantum mechanics okay so you read
+ 
+
+ align:start position:0%
+plus quantum mechanics okay so you read
+the field Theory to to unify them and
+
+ align:start position:0%
+the field Theory to to unify them and
+ 
+
+ align:start position:0%
+the field Theory to to unify them and
+the Third Way
+
+ align:start position:0%
+the Third Way
+ 
+
+ align:start position:0%
+the Third Way
+he said is the a large distance
+
+ align:start position:0%
+he said is the a large distance
+ 
+
+ align:start position:0%
+he said is the a large distance
+description
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+of discrete systems
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+so
+
+ align:start position:0%
+so
+ 
+
+ align:start position:0%
+so
+um
+
+ align:start position:0%
+um
+ 
+
+ align:start position:0%
+um
+yeah just combine all three elements
+
+ align:start position:0%
+yeah just combine all three elements
+ 
+
+ align:start position:0%
+yeah just combine all three elements
+together the cover many many areas of
+
+ align:start position:0%
+together the cover many many areas of
+ 
+
+ align:start position:0%
+together the cover many many areas of
+physics okay they cover many many areas
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+good so so now we can just
+
+ align:start position:0%
+good so so now we can just
+ 
+
+ align:start position:0%
+good so so now we can just
+say a little bit about the plan for the
+
+ align:start position:0%
+say a little bit about the plan for the
+ 
+
+ align:start position:0%
+say a little bit about the plan for the
+whole semester
+
+ align:start position:0%
+whole semester
+ 
+
+ align:start position:0%
+whole semester
+okay so here is the plan
+
+ align:start position:0%
+okay so here is the plan
+ 
+
+ align:start position:0%
+okay so here is the plan
+so
+
+ align:start position:0%
+so
+ 
+
+ align:start position:0%
+so
+so this is like just rephrase of the
+
+ align:start position:0%
+so this is like just rephrase of the
+ 
+
+ align:start position:0%
+so this is like just rephrase of the
+outline
+
+ align:start position:0%
+outline
+ 
+
+ align:start position:0%
+outline
+which uh
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so so the first thing we do in chapter
+
+ align:start position:0%
+so so the first thing we do in chapter
+ 
+
+ align:start position:0%
+so so the first thing we do in chapter
+two so here is chapter one chapter two
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+we discussed the simplest field Theory
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+just Express this equation two okay the
+
+ align:start position:0%
+just Express this equation two okay the
+ 
+
+ align:start position:0%
+just Express this equation two okay the
+theory of Q okay 2 and the two prime
+
+ align:start position:0%
+theory of Q okay 2 and the two prime
+ 
+
+ align:start position:0%
+theory of Q okay 2 and the two prime
+yeah a a prime is is equation motion so
+
+ align:start position:0%
+yeah a a prime is is equation motion so
+ 
+
+ align:start position:0%
+yeah a a prime is is equation motion so
+so yeah we in physics we always start
+
+ align:start position:0%
+so yeah we in physics we always start
+ 
+
+ align:start position:0%
+so yeah we in physics we always start
+with a simplistic example okay we always
+
+ align:start position:0%
+with a simplistic example okay we always
+ 
+
+ align:start position:0%
+with a simplistic example okay we always
+start with simplistic example and uh and
+
+ align:start position:0%
+start with simplistic example and uh and
+ 
+
+ align:start position:0%
+start with simplistic example and uh and
+so that's the uh is the one we will
+
+ align:start position:0%
+so that's the uh is the one we will
+ 
+
+ align:start position:0%
+so that's the uh is the one we will
+start with
+
+ align:start position:0%
+start with
+ 
+
+ align:start position:0%
+start with
+so what we will see is that this
+
+ align:start position:0%
+so what we will see is that this
+ 
+
+ align:start position:0%
+so what we will see is that this
+describes
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+that field we describe spin is
+
+ align:start position:0%
+that field we describe spin is
+ 
+
+ align:start position:0%
+that field we describe spin is
+there's no
+
+ align:start position:0%
+there's no
+ 
+
+ align:start position:0%
+there's no
+free
+
+ align:start position:0%
+free
+ 
+
+ align:start position:0%
+free
+massive particles okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay so we will see when we quantize
+
+ align:start position:0%
+okay so we will see when we quantize
+ 
+
+ align:start position:0%
+okay so we will see when we quantize
+that theory two and then we get the
+
+ align:start position:0%
+that theory two and then we get the
+ 
+
+ align:start position:0%
+that theory two and then we get the
+theory of
+
+ align:start position:0%
+theory of
+ 
+
+ align:start position:0%
+theory of
+three Methodist or three spin is massive
+
+ align:start position:0%
+three Methodist or three spin is massive
+ 
+
+ align:start position:0%
+three Methodist or three spin is massive
+particles okay
+
+ align:start position:0%
+particles okay
+ 
+
+ align:start position:0%
+particles okay
+so you say oh that's a little bit boring
+
+ align:start position:0%
+so you say oh that's a little bit boring
+ 
+
+ align:start position:0%
+so you say oh that's a little bit boring
+because in this series three the
+
+ align:start position:0%
+because in this series three the
+ 
+
+ align:start position:0%
+because in this series three the
+particle by three means they don't
+
+ align:start position:0%
+particle by three means they don't
+ 
+
+ align:start position:0%
+particle by three means they don't
+intact okay the particle they just don't
+
+ align:start position:0%
+intact okay the particle they just don't
+ 
+
+ align:start position:0%
+intact okay the particle they just don't
+intact
+
+ align:start position:0%
+intact
+ 
+
+ align:start position:0%
+intact
+and then in chapter 3 we will add
+
+ align:start position:0%
+and then in chapter 3 we will add
+ 
+
+ align:start position:0%
+and then in chapter 3 we will add
+interactions we will describe
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+how to
+
+ align:start position:0%
+how to
+ 
+
+ align:start position:0%
+how to
+treat
+
+ align:start position:0%
+treat
+ 
+
+ align:start position:0%
+treat
+interactions okay
+
+ align:start position:0%
+interactions okay
+ 
+
+ align:start position:0%
+interactions okay
+so we will introduce interactions
+
+ align:start position:0%
+so we will introduce interactions
+ 
+
+ align:start position:0%
+so we will introduce interactions
+and tell you how to treat the
+
+ align:start position:0%
+and tell you how to treat the
+ 
+
+ align:start position:0%
+and tell you how to treat the
+interactions between those particles
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+then in chapter four
+
+ align:start position:0%
+then in chapter four
+ 
+
+ align:start position:0%
+then in chapter four
+we go to the real physics
+
+ align:start position:0%
+we go to the real physics
+ 
+
+ align:start position:0%
+we go to the real physics
+so this scale of fields
+
+ align:start position:0%
+so this scale of fields
+ 
+
+ align:start position:0%
+so this scale of fields
+is also real say for example can be used
+
+ align:start position:0%
+is also real say for example can be used
+ 
+
+ align:start position:0%
+is also real say for example can be used
+to describe the Hicks
+
+ align:start position:0%
+to describe the Hicks
+ 
+
+ align:start position:0%
+to describe the Hicks
+okay but the heat exposure maybe it's a
+
+ align:start position:0%
+okay but the heat exposure maybe it's a
+ 
+
+ align:start position:0%
+okay but the heat exposure maybe it's a
+little bit far from what we uh you
+
+ align:start position:0%
+little bit far from what we uh you
+ 
+
+ align:start position:0%
+little bit far from what we uh you
+normally think about so in chapter four
+
+ align:start position:0%
+normally think about so in chapter four
+ 
+
+ align:start position:0%
+normally think about so in chapter four
+we will go to something which is much
+
+ align:start position:0%
+we will go to something which is much
+ 
+
+ align:start position:0%
+we will go to something which is much
+closer we'll talk about the theory of
+
+ align:start position:0%
+closer we'll talk about the theory of
+ 
+
+ align:start position:0%
+closer we'll talk about the theory of
+electron so this is called the rock
+
+ align:start position:0%
+electron so this is called the rock
+ 
+
+ align:start position:0%
+electron so this is called the rock
+theory
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this Theory describes three
+
+ align:start position:0%
+so this Theory describes three
+ 
+
+ align:start position:0%
+so this Theory describes three
+but spin half
+
+ align:start position:0%
+but spin half
+ 
+
+ align:start position:0%
+but spin half
+particles okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this is a theory of electrons
+
+ align:start position:0%
+so this is a theory of electrons
+ 
+
+ align:start position:0%
+so this is a theory of electrons
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+when we like these interactions okay so
+
+ align:start position:0%
+when we like these interactions okay so
+ 
+
+ align:start position:0%
+when we like these interactions okay so
+this is the three spring hard particles
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and then
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+[Applause]
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+we move on to the maximals here
+
+ align:start position:0%
+we move on to the maximals here
+ 
+
+ align:start position:0%
+we move on to the maximals here
+Maxwell's Theory
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this is the theory of the quantum
+
+ align:start position:0%
+so this is the theory of the quantum
+ 
+
+ align:start position:0%
+so this is the theory of the quantum
+electric and magnetic field okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so we quantize the Maxwell Theory
+
+ align:start position:0%
+so we quantize the Maxwell Theory
+ 
+
+ align:start position:0%
+so we quantize the Maxwell Theory
+say without Source the vacuum box real
+
+ align:start position:0%
+say without Source the vacuum box real
+ 
+
+ align:start position:0%
+say without Source the vacuum box real
+Siri and you find you get free again
+
+ align:start position:0%
+Siri and you find you get free again
+ 
+
+ align:start position:0%
+Siri and you find you get free again
+there's no interaction
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+is
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+particle
+
+ align:start position:0%
+particle
+ 
+
+ align:start position:0%
+particle
+you get a series of
+
+ align:start position:0%
+you get a series of
+ 
+
+ align:start position:0%
+you get a series of
+massive spring one particle
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this is what we call the photon
+
+ align:start position:0%
+so this is what we call the photon
+ 
+
+ align:start position:0%
+so this is what we call the photon
+okay so this is the Quantum for
+
+ align:start position:0%
+okay so this is the Quantum for
+ 
+
+ align:start position:0%
+okay so this is the Quantum for
+electromagnetic field
+
+ align:start position:0%
+electromagnetic field
+ 
+
+ align:start position:0%
+electromagnetic field
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+and then first and then
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+sorry did I so this should be chapter
+
+ align:start position:0%
+sorry did I so this should be chapter
+ 
+
+ align:start position:0%
+sorry did I so this should be chapter
+five now I think I lost my account
+
+ align:start position:0%
+five now I think I lost my account
+ 
+
+ align:start position:0%
+five now I think I lost my account
+so now go to chapter six
+
+ align:start position:0%
+so now go to chapter six
+ 
+
+ align:start position:0%
+so now go to chapter six
+we combine the four and five together
+
+ align:start position:0%
+we combine the four and five together
+ 
+
+ align:start position:0%
+we combine the four and five together
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+combine
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+electrons
+
+ align:start position:0%
+electrons
+ 
+
+ align:start position:0%
+electrons
+so Photon normally we if we don't
+
+ align:start position:0%
+so Photon normally we if we don't
+ 
+
+ align:start position:0%
+so Photon normally we if we don't
+developed by gamma
+
+ align:start position:0%
+developed by gamma
+ 
+
+ align:start position:0%
+developed by gamma
+combine the theory of electron and the
+
+ align:start position:0%
+combine the theory of electron and the
+ 
+
+ align:start position:0%
+combine the theory of electron and the
+photon together
+
+ align:start position:0%
+photon together
+ 
+
+ align:start position:0%
+photon together
+and then plus interactions
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+between them
+
+ align:start position:0%
+between them
+ 
+
+ align:start position:0%
+between them
+and then we get the so-called
+
+ align:start position:0%
+and then we get the so-called
+ 
+
+ align:start position:0%
+and then we get the so-called
+quantum electrodynamics
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this is called QED
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so QD is very general essentially covers
+
+ align:start position:0%
+so QD is very general essentially covers
+ 
+
+ align:start position:0%
+so QD is very general essentially covers
+all the quantum phenomena uh yeah a
+
+ align:start position:0%
+all the quantum phenomena uh yeah a
+ 
+
+ align:start position:0%
+all the quantum phenomena uh yeah a
+microscopic phenomena up to say big
+
+ align:start position:0%
+microscopic phenomena up to say big
+ 
+
+ align:start position:0%
+microscopic phenomena up to say big
+interactions and strong interactions if
+
+ align:start position:0%
+interactions and strong interactions if
+ 
+
+ align:start position:0%
+interactions and strong interactions if
+you could don't go inside the nucleus
+
+ align:start position:0%
+you could don't go inside the nucleus
+ 
+
+ align:start position:0%
+you could don't go inside the nucleus
+and uh uh or don't go to a very high
+
+ align:start position:0%
+and uh uh or don't go to a very high
+ 
+
+ align:start position:0%
+and uh uh or don't go to a very high
+energy and think that's covers
+
+ align:start position:0%
+energy and think that's covers
+ 
+
+ align:start position:0%
+energy and think that's covers
+essentially most of the physics
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yeah yeah and and then the
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+end of this our our course
+
+ align:start position:0%
+end of this our our course
+ 
+
+ align:start position:0%
+end of this our our course
+so do you have any questions on this
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay so this is a road map yes
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+good other questions
+
+ align:start position:0%
+good other questions
+ 
+
+ align:start position:0%
+good other questions
+yes
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+sorry
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yeah it's also Mercedes spin one but
+
+ align:start position:0%
+yeah it's also Mercedes spin one but
+ 
+
+ align:start position:0%
+yeah it's also Mercedes spin one but
+actually the interact with themselves
+
+ align:start position:0%
+actually the interact with themselves
+ 
+
+ align:start position:0%
+actually the interact with themselves
+and so good one is different so so glue
+
+ align:start position:0%
+and so good one is different so so glue
+ 
+
+ align:start position:0%
+and so good one is different so so glue
+on to describe gluons you have to wait
+
+ align:start position:0%
+on to describe gluons you have to wait
+ 
+
+ align:start position:0%
+on to describe gluons you have to wait
+for Quantum field City too
+
+ align:start position:0%
+for Quantum field City too
+ 
+
+ align:start position:0%
+for Quantum field City too
+and uh so so the thing about the photon
+
+ align:start position:0%
+and uh so so the thing about the photon
+ 
+
+ align:start position:0%
+and uh so so the thing about the photon
+is that the photons don't interact with
+
+ align:start position:0%
+is that the photons don't interact with
+ 
+
+ align:start position:0%
+is that the photons don't interact with
+itself but the gluons interact with
+
+ align:start position:0%
+itself but the gluons interact with
+ 
+
+ align:start position:0%
+itself but the gluons interact with
+itself okay yeah so so essentially we
+
+ align:start position:0%
+itself okay yeah so so essentially we
+ 
+
+ align:start position:0%
+itself okay yeah so so essentially we
+treat everything
+
+ align:start position:0%
+treat everything
+ 
+
+ align:start position:0%
+treat everything
+except gloves yeah
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+other questions
+
+ align:start position:0%
+other questions
+ 
+
+ align:start position:0%
+other questions
+other questions
+
+ align:start position:0%
+other questions
+ 
+
+ align:start position:0%
+other questions
+okay good
+
+ align:start position:0%
+okay good
+ 
+
+ align:start position:0%
+okay good
+so now we can just move to chapter two
+
+ align:start position:0%
+so now we can just move to chapter two
+ 
+
+ align:start position:0%
+so now we can just move to chapter two
+now we are talking about this series
+
+ align:start position:0%
+now we are talking about this series
+ 
+
+ align:start position:0%
+now we are talking about this series
+okay so so actually I should not erase
+
+ align:start position:0%
+okay so so actually I should not erase
+ 
+
+ align:start position:0%
+okay so so actually I should not erase
+it
+
+ align:start position:0%
+it
+ 
+
+ align:start position:0%
+it
+so now we talk about this here
+
+ align:start position:0%
+so now we talk about this here
+ 
+
+ align:start position:0%
+so now we talk about this here
+so because because this series describes
+
+ align:start position:0%
+so because because this series describes
+ 
+
+ align:start position:0%
+so because because this series describes
+three particles
+
+ align:start position:0%
+three particles
+ 
+
+ align:start position:0%
+three particles
+so we call it free scalar field CV
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay so so so this is the
+
+ align:start position:0%
+okay so so so this is the
+ 
+
+ align:start position:0%
+okay so so so this is the
+Theory we are interested in so so now we
+
+ align:start position:0%
+Theory we are interested in so so now we
+ 
+
+ align:start position:0%
+Theory we are interested in so so now we
+will describe how to quantize this
+
+ align:start position:0%
+will describe how to quantize this
+ 
+
+ align:start position:0%
+will describe how to quantize this
+series okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+good
+
+ align:start position:0%
+good
+ 
+
+ align:start position:0%
+good
+so so first
+
+ align:start position:0%
+so so first
+ 
+
+ align:start position:0%
+so so first
+we will quickly
+
+ align:start position:0%
+we will quickly
+ 
+
+ align:start position:0%
+we will quickly
+go through
+
+ align:start position:0%
+go through
+ 
+
+ align:start position:0%
+go through
+the the quantitation of harmonic
+
+ align:start position:0%
+the the quantitation of harmonic
+ 
+
+ align:start position:0%
+the the quantitation of harmonic
+oscillator
+
+ align:start position:0%
+oscillator
+ 
+
+ align:start position:0%
+oscillator
+which you should already have done in
+
+ align:start position:0%
+which you should already have done in
+ 
+
+ align:start position:0%
+which you should already have done in
+your in your preset and so uh so we can
+
+ align:start position:0%
+your in your preset and so uh so we can
+ 
+
+ align:start position:0%
+your in your preset and so uh so we can
+do it relatively fast
+
+ align:start position:0%
+do it relatively fast
+ 
+
+ align:start position:0%
+do it relatively fast
+so
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+the organization
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+of pythonic oscillator
+
+ align:start position:0%
+of pythonic oscillator
+ 
+
+ align:start position:0%
+of pythonic oscillator
+in the Heisenberg picture
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so we will see that once we understand
+
+ align:start position:0%
+so we will see that once we understand
+ 
+
+ align:start position:0%
+so we will see that once we understand
+this example
+
+ align:start position:0%
+this example
+ 
+
+ align:start position:0%
+this example
+in the right way
+
+ align:start position:0%
+in the right way
+ 
+
+ align:start position:0%
+in the right way
+and then contacting this field Theory
+
+ align:start position:0%
+and then contacting this field Theory
+ 
+
+ align:start position:0%
+and then contacting this field Theory
+becomes trivial
+
+ align:start position:0%
+becomes trivial
+ 
+
+ align:start position:0%
+becomes trivial
+okay and the quantize in this field
+
+ align:start position:0%
+okay and the quantize in this field
+ 
+
+ align:start position:0%
+okay and the quantize in this field
+series become trigger
+
+ align:start position:0%
+series become trigger
+ 
+
+ align:start position:0%
+series become trigger
+okay so so let's start with a harmonic
+
+ align:start position:0%
+okay so so let's start with a harmonic
+ 
+
+ align:start position:0%
+okay so so let's start with a harmonic
+oscillator
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+for Simplicity I take the mass to be one
+
+ align:start position:0%
+for Simplicity I take the mass to be one
+ 
+
+ align:start position:0%
+for Simplicity I take the mass to be one
+and take the frequency to be y okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yeah yeah let me put the frequency here
+
+ align:start position:0%
+yeah yeah let me put the frequency here
+ 
+
+ align:start position:0%
+yeah yeah let me put the frequency here
+it won't fit
+
+ align:start position:0%
+it won't fit
+ 
+
+ align:start position:0%
+it won't fit
+okay let's take the mass TDY y okay
+
+ align:start position:0%
+okay let's take the mass TDY y okay
+ 
+
+ align:start position:0%
+okay let's take the mass TDY y okay
+and so so for this series so so this is
+
+ align:start position:0%
+and so so for this series so so this is
+ 
+
+ align:start position:0%
+and so so for this series so so this is
+a simple harmonic oscillator which you
+
+ align:start position:0%
+a simple harmonic oscillator which you
+ 
+
+ align:start position:0%
+a simple harmonic oscillator which you
+have seen it
+
+ align:start position:0%
+have seen it
+ 
+
+ align:start position:0%
+have seen it
+uh uh uh maybe for the most of your
+
+ align:start position:0%
+uh uh uh maybe for the most of your
+ 
+
+ align:start position:0%
+uh uh uh maybe for the most of your
+intellectual life
+
+ align:start position:0%
+intellectual life
+ 
+
+ align:start position:0%
+intellectual life
+and the P will be x dot it's a momentum
+
+ align:start position:0%
+and the P will be x dot it's a momentum
+ 
+
+ align:start position:0%
+and the P will be x dot it's a momentum
+the conjugate momentum is x dot and so
+
+ align:start position:0%
+the conjugate momentum is x dot and so
+ 
+
+ align:start position:0%
+the conjugate momentum is x dot and so
+the hamiltonian is the P-Square divided
+
+ align:start position:0%
+the hamiltonian is the P-Square divided
+ 
+
+ align:start position:0%
+the hamiltonian is the P-Square divided
+by two
+
+ align:start position:0%
+by two
+ 
+
+ align:start position:0%
+by two
+of one half Omega Square x squared okay
+
+ align:start position:0%
+of one half Omega Square x squared okay
+ 
+
+ align:start position:0%
+of one half Omega Square x squared okay
+an equation motion
+
+ align:start position:0%
+an equation motion
+ 
+
+ align:start position:0%
+an equation motion
+is X dots double dot equal to x three x
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so
+
+ align:start position:0%
+so
+ 
+
+ align:start position:0%
+so
+so let's first look at this Theory uh
+
+ align:start position:0%
+so let's first look at this Theory uh
+ 
+
+ align:start position:0%
+so let's first look at this Theory uh
+look at harmonic considered as a
+
+ align:start position:0%
+look at harmonic considered as a
+ 
+
+ align:start position:0%
+look at harmonic considered as a
+classical Theory
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so for classical Theory
+
+ align:start position:0%
+so for classical Theory
+ 
+
+ align:start position:0%
+so for classical Theory
+we know how to solve this equation we
+
+ align:start position:0%
+we know how to solve this equation we
+ 
+
+ align:start position:0%
+we know how to solve this equation we
+just need to solve this equation
+
+ align:start position:0%
+just need to solve this equation
+ 
+
+ align:start position:0%
+just need to solve this equation
+so classical solution
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+just given by x t
+
+ align:start position:0%
+just given by x t
+ 
+
+ align:start position:0%
+just given by x t
+equal to a
+
+ align:start position:0%
+equal to a
+ 
+
+ align:start position:0%
+equal to a
+cosine Omega t
+
+ align:start position:0%
+cosine Omega t
+ 
+
+ align:start position:0%
+cosine Omega t
+plus b sine of Omega t
+
+ align:start position:0%
+plus b sine of Omega t
+ 
+
+ align:start position:0%
+plus b sine of Omega t
+and a and a and b just some integration
+
+ align:start position:0%
+and a and a and b just some integration
+ 
+
+ align:start position:0%
+and a and a and b just some integration
+constant
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and for convenience I can also write it
+
+ align:start position:0%
+and for convenience I can also write it
+ 
+
+ align:start position:0%
+and for convenience I can also write it
+in the complex form
+
+ align:start position:0%
+in the complex form
+ 
+
+ align:start position:0%
+in the complex form
+bracket as following
+
+ align:start position:0%
+bracket as following
+ 
+
+ align:start position:0%
+bracket as following
+equal to a
+
+ align:start position:0%
+equal to a
+ 
+
+ align:start position:0%
+equal to a
+expansion minus sign Omega t
+
+ align:start position:0%
+expansion minus sign Omega t
+ 
+
+ align:start position:0%
+expansion minus sign Omega t
+plus a star
+
+ align:start position:0%
+plus a star
+ 
+
+ align:start position:0%
+plus a star
+expression I Omega T and A is some
+
+ align:start position:0%
+expression I Omega T and A is some
+ 
+
+ align:start position:0%
+expression I Omega T and A is some
+complex constant
+
+ align:start position:0%
+complex constant
+ 
+
+ align:start position:0%
+complex constant
+and again it's a integration constant I
+
+ align:start position:0%
+and again it's a integration constant I
+ 
+
+ align:start position:0%
+and again it's a integration constant I
+just rewrite the integration constant
+
+ align:start position:0%
+just rewrite the integration constant
+ 
+
+ align:start position:0%
+just rewrite the integration constant
+slightly differently
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+now these are just integration constants
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so now
+
+ align:start position:0%
+so now
+ 
+
+ align:start position:0%
+so now
+yeah so this is a complete solution of
+
+ align:start position:0%
+yeah so this is a complete solution of
+ 
+
+ align:start position:0%
+yeah so this is a complete solution of
+the problem
+
+ align:start position:0%
+the problem
+ 
+
+ align:start position:0%
+the problem
+so now let's go to Quantum
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so when we go to Quantum
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and then then we replace
+
+ align:start position:0%
+and then then we replace
+ 
+
+ align:start position:0%
+and then then we replace
+then this classical dynamical variable
+
+ align:start position:0%
+then this classical dynamical variable
+ 
+
+ align:start position:0%
+then this classical dynamical variable
+then become
+
+ align:start position:0%
+then become
+ 
+
+ align:start position:0%
+then become
+the the Heisenberg operator
+
+ align:start position:0%
+the the Heisenberg operator
+ 
+
+ align:start position:0%
+the the Heisenberg operator
+becomes the quantum operator in
+
+ align:start position:0%
+becomes the quantum operator in
+ 
+
+ align:start position:0%
+becomes the quantum operator in
+particular in the in the Heisenberg
+
+ align:start position:0%
+particular in the in the Heisenberg
+ 
+
+ align:start position:0%
+particular in the in the Heisenberg
+picture and then then this operator will
+
+ align:start position:0%
+picture and then then this operator will
+ 
+
+ align:start position:0%
+picture and then then this operator will
+depend on time
+
+ align:start position:0%
+depend on time
+ 
+
+ align:start position:0%
+depend on time
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+and now this equation
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+become operating equation okay so now
+
+ align:start position:0%
+become operating equation okay so now
+ 
+
+ align:start position:0%
+become operating equation okay so now
+let's
+
+ align:start position:0%
+let's
+ 
+
+ align:start position:0%
+let's
+maybe I should label my equation
+
+ align:start position:0%
+maybe I should label my equation
+ 
+
+ align:start position:0%
+maybe I should label my equation
+so now this star
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+become an operating equation
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+now star is the operating equation
+
+ align:start position:0%
+now star is the operating equation
+ 
+
+ align:start position:0%
+now star is the operating equation
+for X hat
+
+ align:start position:0%
+for X hat
+ 
+
+ align:start position:0%
+for X hat
+so you have exact the same equation as a
+
+ align:start position:0%
+so you have exact the same equation as a
+ 
+
+ align:start position:0%
+so you have exact the same equation as a
+classical equation but not the
+
+ align:start position:0%
+classical equation but not the
+ 
+
+ align:start position:0%
+classical equation but not the
+interpretation is different
+
+ align:start position:0%
+interpretation is different
+ 
+
+ align:start position:0%
+interpretation is different
+and now now the X hat becomes the uh
+
+ align:start position:0%
+and now now the X hat becomes the uh
+ 
+
+ align:start position:0%
+and now now the X hat becomes the uh
+another X become the operating equation
+
+ align:start position:0%
+another X become the operating equation
+ 
+
+ align:start position:0%
+another X become the operating equation
+foreign
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so now the solution
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+now let me call this star star
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this still solves that equation
+
+ align:start position:0%
+so this still solves that equation
+ 
+
+ align:start position:0%
+so this still solves that equation
+okay so this still solves that equation
+
+ align:start position:0%
+okay so this still solves that equation
+ 
+
+ align:start position:0%
+okay so this still solves that equation
+except
+
+ align:start position:0%
+except
+ 
+
+ align:start position:0%
+except
+so these are just C numbers because
+
+ align:start position:0%
+so these are just C numbers because
+ 
+
+ align:start position:0%
+so these are just C numbers because
+because this is a function of T these
+
+ align:start position:0%
+because this is a function of T these
+ 
+
+ align:start position:0%
+because this is a function of T these
+are C numbers
+
+ align:start position:0%
+are C numbers
+ 
+
+ align:start position:0%
+are C numbers
+but now X becomes the uh so now Quantum
+
+ align:start position:0%
+but now X becomes the uh so now Quantum
+ 
+
+ align:start position:0%
+but now X becomes the uh so now Quantum
+mechanically
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this becomes now become the quantum
+
+ align:start position:0%
+so this becomes now become the quantum
+ 
+
+ align:start position:0%
+so this becomes now become the quantum
+solution
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+so now I've had so this still solves the
+
+ align:start position:0%
+so now I've had so this still solves the
+ 
+
+ align:start position:0%
+so now I've had so this still solves the
+equations so mechanically this becomes
+
+ align:start position:0%
+equations so mechanically this becomes
+ 
+
+ align:start position:0%
+equations so mechanically this becomes
+hat
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+and so this still solves that equation
+
+ align:start position:0%
+and so this still solves that equation
+ 
+
+ align:start position:0%
+and so this still solves that equation
+but these are C numbers
+
+ align:start position:0%
+but these are C numbers
+ 
+
+ align:start position:0%
+but these are C numbers
+the left hand side is the operator
+
+ align:start position:0%
+the left hand side is the operator
+ 
+
+ align:start position:0%
+the left hand side is the operator
+and it can only be that a hat and B hat
+
+ align:start position:0%
+and it can only be that a hat and B hat
+ 
+
+ align:start position:0%
+and it can only be that a hat and B hat
+are operators and also a must be
+
+ align:start position:0%
+are operators and also a must be
+ 
+
+ align:start position:0%
+are operators and also a must be
+operators and the star will replace it
+
+ align:start position:0%
+operators and the star will replace it
+ 
+
+ align:start position:0%
+operators and the star will replace it
+by diagram
+
+ align:start position:0%
+by diagram
+ 
+
+ align:start position:0%
+by diagram
+okay so now
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+say
+
+ align:start position:0%
+say
+ 
+
+ align:start position:0%
+say
+now a you just go to a hat
+
+ align:start position:0%
+now a you just go to a hat
+ 
+
+ align:start position:0%
+now a you just go to a hat
+and a star goes to a dagger it has
+
+ align:start position:0%
+and a star goes to a dagger it has
+ 
+
+ align:start position:0%
+and a star goes to a dagger it has
+dagger okay now these are
+
+ align:start position:0%
+dagger okay now these are
+ 
+
+ align:start position:0%
+dagger okay now these are
+these are integration constants for the
+
+ align:start position:0%
+these are integration constants for the
+ 
+
+ align:start position:0%
+these are integration constants for the
+operating equations
+
+ align:start position:0%
+operating equations
+ 
+
+ align:start position:0%
+operating equations
+so they are just count now they become
+
+ align:start position:0%
+so they are just count now they become
+ 
+
+ align:start position:0%
+so they are just count now they become
+constant operators
+
+ align:start position:0%
+constant operators
+ 
+
+ align:start position:0%
+constant operators
+okay so so so so so they're just
+
+ align:start position:0%
+okay so so so so so they're just
+ 
+
+ align:start position:0%
+okay so so so so so they're just
+constant
+
+ align:start position:0%
+constant
+ 
+
+ align:start position:0%
+constant
+Quantum operators
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay they're just constant operators
+
+ align:start position:0%
+okay they're just constant operators
+ 
+
+ align:start position:0%
+okay they're just constant operators
+so they are integration constants for
+
+ align:start position:0%
+so they are integration constants for
+ 
+
+ align:start position:0%
+so they are integration constants for
+your for your Quantum operating
+
+ align:start position:0%
+your for your Quantum operating
+ 
+
+ align:start position:0%
+your for your Quantum operating
+equations
+
+ align:start position:0%
+equations
+ 
+
+ align:start position:0%
+equations
+so now the solution
+
+ align:start position:0%
+so now the solution
+ 
+
+ align:start position:0%
+so now the solution
+so the as another uh uh yeah so now this
+
+ align:start position:0%
+so the as another uh uh yeah so now this
+ 
+
+ align:start position:0%
+so the as another uh uh yeah so now this
+is your Quantum solution okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this is the form we will often use
+
+ align:start position:0%
+so this is the form we will often use
+ 
+
+ align:start position:0%
+so this is the form we will often use
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+you can also use that form but the
+
+ align:start position:0%
+you can also use that form but the
+ 
+
+ align:start position:0%
+you can also use that form but the
+equivalent but this is the form we often
+
+ align:start position:0%
+equivalent but this is the form we often
+ 
+
+ align:start position:0%
+equivalent but this is the form we often
+use okay
+
+ align:start position:0%
+use okay
+ 
+
+ align:start position:0%
+use okay
+you can also from here you take the
+
+ align:start position:0%
+you can also from here you take the
+ 
+
+ align:start position:0%
+you can also from here you take the
+derivative you can find the P so again
+
+ align:start position:0%
+derivative you can find the P so again
+ 
+
+ align:start position:0%
+derivative you can find the P so again
+this is become an operating equation you
+
+ align:start position:0%
+this is become an operating equation you
+ 
+
+ align:start position:0%
+this is become an operating equation you
+take the direction of X hat X and maybe
+
+ align:start position:0%
+take the direction of X hat X and maybe
+ 
+
+ align:start position:0%
+take the direction of X hat X and maybe
+you find P Etc
+
+ align:start position:0%
+you find P Etc
+ 
+
+ align:start position:0%
+you find P Etc
+okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yeah so you
+
+ align:start position:0%
+yeah so you
+ 
+
+ align:start position:0%
+yeah so you
+the P hats t
+
+ align:start position:0%
+the P hats t
+ 
+
+ align:start position:0%
+the P hats t
+just take a derivative
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay and then you can just
+
+ align:start position:0%
+okay and then you can just
+ 
+
+ align:start position:0%
+okay and then you can just
+walk it out it's very easy
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so now
+
+ align:start position:0%
+so now
+ 
+
+ align:start position:0%
+so now
+so this equation just
+
+ align:start position:0%
+so this equation just
+ 
+
+ align:start position:0%
+so this equation just
+so we already solved the quantum Tobin
+
+ align:start position:0%
+so we already solved the quantum Tobin
+ 
+
+ align:start position:0%
+so we already solved the quantum Tobin
+okay so so because we find the full
+
+ align:start position:0%
+okay so so because we find the full
+ 
+
+ align:start position:0%
+okay so so because we find the full
+evolution full solution to the quantum
+
+ align:start position:0%
+evolution full solution to the quantum
+ 
+
+ align:start position:0%
+evolution full solution to the quantum
+operator equation
+
+ align:start position:0%
+operator equation
+ 
+
+ align:start position:0%
+operator equation
+except that we still leads to
+
+ align:start position:0%
+except that we still leads to
+ 
+
+ align:start position:0%
+except that we still leads to
+impose the canonical condition
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this is just equal to I
+
+ align:start position:0%
+so this is just equal to I
+ 
+
+ align:start position:0%
+so this is just equal to I
+okay so so the standard so if you plug
+
+ align:start position:0%
+okay so so the standard so if you plug
+ 
+
+ align:start position:0%
+okay so so the standard so if you plug
+in the expression for x
+
+ align:start position:0%
+in the expression for x
+ 
+
+ align:start position:0%
+in the expression for x
+and the T and the P into here
+
+ align:start position:0%
+and the T and the P into here
+ 
+
+ align:start position:0%
+and the T and the P into here
+and then you find
+
+ align:start position:0%
+and then you find
+ 
+
+ align:start position:0%
+and then you find
+that a and a dagger
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+accommodate is equal to one okay so this
+
+ align:start position:0%
+accommodate is equal to one okay so this
+ 
+
+ align:start position:0%
+accommodate is equal to one okay so this
+is your familiar
+
+ align:start position:0%
+is your familiar
+ 
+
+ align:start position:0%
+is your familiar
+a creation Annihilation operator for for
+
+ align:start position:0%
+a creation Annihilation operator for for
+ 
+
+ align:start position:0%
+a creation Annihilation operator for for
+harmonic oscillator
+
+ align:start position:0%
+harmonic oscillator
+ 
+
+ align:start position:0%
+harmonic oscillator
+okay for harmonic oscillator
+
+ align:start position:0%
+okay for harmonic oscillator
+ 
+
+ align:start position:0%
+okay for harmonic oscillator
+and uh
+
+ align:start position:0%
+and uh
+ 
+
+ align:start position:0%
+and uh
+and now we can also use the a to build
+
+ align:start position:0%
+and now we can also use the a to build
+ 
+
+ align:start position:0%
+and now we can also use the a to build
+the Hilbert space so because a are the
+
+ align:start position:0%
+the Hilbert space so because a are the
+ 
+
+ align:start position:0%
+the Hilbert space so because a are the
+yeah because all you operate now
+
+ align:start position:0%
+yeah because all you operate now
+ 
+
+ align:start position:0%
+yeah because all you operate now
+or because X and the T
+
+ align:start position:0%
+or because X and the T
+ 
+
+ align:start position:0%
+or because X and the T
+X and P are expressed in terms of a and
+
+ align:start position:0%
+X and P are expressed in terms of a and
+ 
+
+ align:start position:0%
+X and P are expressed in terms of a and
+a dagger so essentially any operator of
+
+ align:start position:0%
+a dagger so essentially any operator of
+ 
+
+ align:start position:0%
+a dagger so essentially any operator of
+this Theory can all be expressed in
+
+ align:start position:0%
+this Theory can all be expressed in
+ 
+
+ align:start position:0%
+this Theory can all be expressed in
+terms of being a diagram okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and then you can just use a and a gagger
+
+ align:start position:0%
+and then you can just use a and a gagger
+ 
+
+ align:start position:0%
+and then you can just use a and a gagger
+because A and negative essentially they
+
+ align:start position:0%
+because A and negative essentially they
+ 
+
+ align:start position:0%
+because A and negative essentially they
+they are fundamental building block of
+
+ align:start position:0%
+they are fundamental building block of
+ 
+
+ align:start position:0%
+they are fundamental building block of
+your full quantum theory and then we can
+
+ align:start position:0%
+your full quantum theory and then we can
+ 
+
+ align:start position:0%
+your full quantum theory and then we can
+also use that to build the helper space
+
+ align:start position:0%
+also use that to build the helper space
+ 
+
+ align:start position:0%
+also use that to build the helper space
+so the healable space
+
+ align:start position:0%
+so the healable space
+ 
+
+ align:start position:0%
+so the healable space
+is defined by the lowest state is
+
+ align:start position:0%
+is defined by the lowest state is
+ 
+
+ align:start position:0%
+is defined by the lowest state is
+evaluated by a
+
+ align:start position:0%
+evaluated by a
+ 
+
+ align:start position:0%
+evaluated by a
+and then and then the in the higher
+
+ align:start position:0%
+and then and then the in the higher
+ 
+
+ align:start position:0%
+and then and then the in the higher
+state
+
+ align:start position:0%
+state
+ 
+
+ align:start position:0%
+state
+uh
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+obtained by acting a dagger
+
+ align:start position:0%
+obtained by acting a dagger
+ 
+
+ align:start position:0%
+obtained by acting a dagger
+on the on the on the ground state okay
+
+ align:start position:0%
+on the on the on the ground state okay
+ 
+
+ align:start position:0%
+on the on the on the ground state okay
+so so so this is your full series okay
+
+ align:start position:0%
+so so so this is your full series okay
+ 
+
+ align:start position:0%
+so so so this is your full series okay
+so this is your 4C
+
+ align:start position:0%
+so this is your 4C
+ 
+
+ align:start position:0%
+so this is your 4C
+and so now
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+uh you can compute anything in this
+
+ align:start position:0%
+uh you can compute anything in this
+ 
+
+ align:start position:0%
+uh you can compute anything in this
+series just miss those knowledge okay
+
+ align:start position:0%
+series just miss those knowledge okay
+ 
+
+ align:start position:0%
+series just miss those knowledge okay
+just with lots of knowledge
+
+ align:start position:0%
+just with lots of knowledge
+ 
+
+ align:start position:0%
+just with lots of knowledge
+so any questions on this regarding the
+
+ align:start position:0%
+so any questions on this regarding the
+ 
+
+ align:start position:0%
+so any questions on this regarding the
+harmonic oscillator
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+good okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so so so let me just summarize
+
+ align:start position:0%
+so so so let me just summarize
+ 
+
+ align:start position:0%
+so so so let me just summarize
+so this is maybe very familiar
+
+ align:start position:0%
+so this is maybe very familiar
+ 
+
+ align:start position:0%
+so this is maybe very familiar
+but let's summarize the rule we have
+
+ align:start position:0%
+but let's summarize the rule we have
+ 
+
+ align:start position:0%
+but let's summarize the rule we have
+been using Okay summarize the steps
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+in context the harmonic oscillator
+
+ align:start position:0%
+in context the harmonic oscillator
+ 
+
+ align:start position:0%
+in context the harmonic oscillator
+and then the and then the same steps can
+
+ align:start position:0%
+and then the and then the same steps can
+ 
+
+ align:start position:0%
+and then the and then the same steps can
+be used to quantize the the field series
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+steps
+
+ align:start position:0%
+steps
+ 
+
+ align:start position:0%
+steps
+of quantization
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so we make it a general
+
+ align:start position:0%
+so we make it a general
+ 
+
+ align:start position:0%
+so we make it a general
+so first
+
+ align:start position:0%
+so first
+ 
+
+ align:start position:0%
+so first
+so the zero step is that the classical
+
+ align:start position:0%
+so the zero step is that the classical
+ 
+
+ align:start position:0%
+so the zero step is that the classical
+equation motion the quantum quantum
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+operator equation
+
+ align:start position:0%
+operator equation
+ 
+
+ align:start position:0%
+operator equation
+okay then the first step
+
+ align:start position:0%
+okay then the first step
+ 
+
+ align:start position:0%
+okay then the first step
+is to find the most General solution
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+to
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+find the most General solution to
+
+ align:start position:0%
+find the most General solution to
+ 
+
+ align:start position:0%
+find the most General solution to
+classical equation motion
+
+ align:start position:0%
+classical equation motion
+ 
+
+ align:start position:0%
+classical equation motion
+yeah just to equation motion
+
+ align:start position:0%
+yeah just to equation motion
+ 
+
+ align:start position:0%
+yeah just to equation motion
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+and then
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+when you've got the quantum
+
+ align:start position:0%
+when you've got the quantum
+ 
+
+ align:start position:0%
+when you've got the quantum
+and then you just promote
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+the integration constants
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+in your classical solution in one
+
+ align:start position:0%
+in your classical solution in one
+ 
+
+ align:start position:0%
+in your classical solution in one
+in the step one
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+to
+
+ align:start position:0%
+to
+ 
+
+ align:start position:0%
+to
+constant operators
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+constant Quantum operators okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this gives the uh then then you have
+
+ align:start position:0%
+so this gives the uh then then you have
+ 
+
+ align:start position:0%
+so this gives the uh then then you have
+the full-time evolution
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+at the quantum level okay okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+now you know the how the quantum
+
+ align:start position:0%
+now you know the how the quantum
+ 
+
+ align:start position:0%
+now you know the how the quantum
+operator evolves
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and then
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+you impose
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+canonical condition
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+s
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay so that will tell you
+
+ align:start position:0%
+okay so that will tell you
+ 
+
+ align:start position:0%
+okay so that will tell you
+the commentators between those
+
+ align:start position:0%
+the commentators between those
+ 
+
+ align:start position:0%
+the commentators between those
+integration constant operators okay
+
+ align:start position:0%
+integration constant operators okay
+ 
+
+ align:start position:0%
+integration constant operators okay
+just as we do here
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and then
+
+ align:start position:0%
+and then
+ 
+
+ align:start position:0%
+and then
+um
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and then um
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+constant operators
+
+ align:start position:0%
+constant operators
+ 
+
+ align:start position:0%
+constant operators
+in queue
+
+ align:start position:0%
+in queue
+ 
+
+ align:start position:0%
+in queue
+now now you know also load the
+
+ align:start position:0%
+now now you know also load the
+ 
+
+ align:start position:0%
+now now you know also load the
+commutation relation between them among
+
+ align:start position:0%
+commutation relation between them among
+ 
+
+ align:start position:0%
+commutation relation between them among
+them and then now can be used to
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+generate the healable space
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+so this step and one two four are very
+
+ align:start position:0%
+so this step and one two four are very
+ 
+
+ align:start position:0%
+so this step and one two four are very
+general
+
+ align:start position:0%
+general
+ 
+
+ align:start position:0%
+general
+and if you can do it
+
+ align:start position:0%
+and if you can do it
+ 
+
+ align:start position:0%
+and if you can do it
+and then you can then you can
+
+ align:start position:0%
+and then you can then you can
+ 
+
+ align:start position:0%
+and then you can then you can
+essentially uh do it applied to gen any
+
+ align:start position:0%
+essentially uh do it applied to gen any
+ 
+
+ align:start position:0%
+essentially uh do it applied to gen any
+system
+
+ align:start position:0%
+system
+ 
+
+ align:start position:0%
+system
+say one degree harmonic acid is one
+
+ align:start position:0%
+say one degree harmonic acid is one
+ 
+
+ align:start position:0%
+say one degree harmonic acid is one
+degrees three then you can apply it two
+
+ align:start position:0%
+degrees three then you can apply it two
+ 
+
+ align:start position:0%
+degrees three then you can apply it two
+degrees three degrees freedom and also
+
+ align:start position:0%
+degrees three degrees freedom and also
+ 
+
+ align:start position:0%
+degrees three degrees freedom and also
+to field series with infinite number
+
+ align:start position:0%
+to field series with infinite number
+ 
+
+ align:start position:0%
+to field series with infinite number
+equals freedom
+
+ align:start position:0%
+equals freedom
+ 
+
+ align:start position:0%
+equals freedom
+okay and now we will apply these two
+
+ align:start position:0%
+okay and now we will apply these two
+ 
+
+ align:start position:0%
+okay and now we will apply these two
+field series yes
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+uh for this procedure you cannot
+
+ align:start position:0%
+uh for this procedure you cannot
+ 
+
+ align:start position:0%
+uh for this procedure you cannot
+but but uh but you can get the finite uh
+
+ align:start position:0%
+but but uh but you can get the finite uh
+ 
+
+ align:start position:0%
+but but uh but you can get the finite uh
+Dimension yeah
+
+ align:start position:0%
+Dimension yeah
+ 
+
+ align:start position:0%
+Dimension yeah
+because the finite here will space don't
+
+ align:start position:0%
+because the finite here will space don't
+ 
+
+ align:start position:0%
+because the finite here will space don't
+have the classical analog so here we
+
+ align:start position:0%
+have the classical analog so here we
+ 
+
+ align:start position:0%
+have the classical analog so here we
+start with the classical system and then
+
+ align:start position:0%
+start with the classical system and then
+ 
+
+ align:start position:0%
+start with the classical system and then
+we contact it and there's a the the the
+
+ align:start position:0%
+we contact it and there's a the the the
+ 
+
+ align:start position:0%
+we contact it and there's a the the the
+the quantum system with a finite
+
+ align:start position:0%
+the quantum system with a finite
+ 
+
+ align:start position:0%
+the quantum system with a finite
+Dimension healable space they
+
+ align:start position:0%
+Dimension healable space they
+ 
+
+ align:start position:0%
+Dimension healable space they
+essentially intrinsically quantum and uh
+
+ align:start position:0%
+essentially intrinsically quantum and uh
+ 
+
+ align:start position:0%
+essentially intrinsically quantum and uh
+yeah and like spin spin is an intrinsic
+
+ align:start position:0%
+yeah and like spin spin is an intrinsic
+ 
+
+ align:start position:0%
+yeah and like spin spin is an intrinsic
+Quantum scene yeah
+
+ align:start position:0%
+Quantum scene yeah
+ 
+
+ align:start position:0%
+Quantum scene yeah
+yes
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yeah yeah yeah it's just because they
+
+ align:start position:0%
+yeah yeah yeah it's just because they
+ 
+
+ align:start position:0%
+yeah yeah yeah it's just because they
+don't have yeah yeah the reason is they
+
+ align:start position:0%
+don't have yeah yeah the reason is they
+ 
+
+ align:start position:0%
+don't have yeah yeah the reason is they
+don't have classical counterpart yeah
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yes um is it always true
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yeah because because if you think about
+
+ align:start position:0%
+yeah because because if you think about
+ 
+
+ align:start position:0%
+yeah because because if you think about
+this way yeah that's a very good
+
+ align:start position:0%
+this way yeah that's a very good
+ 
+
+ align:start position:0%
+this way yeah that's a very good
+question because
+
+ align:start position:0%
+question because
+ 
+
+ align:start position:0%
+question because
+because let's just look at this harmonic
+
+ align:start position:0%
+because let's just look at this harmonic
+ 
+
+ align:start position:0%
+because let's just look at this harmonic
+oscillator and then you can try to think
+
+ align:start position:0%
+oscillator and then you can try to think
+ 
+
+ align:start position:0%
+oscillator and then you can try to think
+generalize it
+
+ align:start position:0%
+generalize it
+ 
+
+ align:start position:0%
+generalize it
+because
+
+ align:start position:0%
+because
+ 
+
+ align:start position:0%
+because
+they are integrating constant of the X
+
+ align:start position:0%
+they are integrating constant of the X
+ 
+
+ align:start position:0%
+they are integrating constant of the X
+and the p
+
+ align:start position:0%
+and the p
+ 
+
+ align:start position:0%
+and the p
+that any operator in your theory can all
+
+ align:start position:0%
+that any operator in your theory can all
+ 
+
+ align:start position:0%
+that any operator in your theory can all
+be expressed in terms of a and a dagger
+
+ align:start position:0%
+be expressed in terms of a and a dagger
+ 
+
+ align:start position:0%
+be expressed in terms of a and a dagger
+and then and then you'll hear the space
+
+ align:start position:0%
+and then and then you'll hear the space
+ 
+
+ align:start position:0%
+and then and then you'll hear the space
+must begin you must be able to generate
+
+ align:start position:0%
+must begin you must be able to generate
+ 
+
+ align:start position:0%
+must begin you must be able to generate
+the herbal space using them yeah yeah
+
+ align:start position:0%
+the herbal space using them yeah yeah
+ 
+
+ align:start position:0%
+the herbal space using them yeah yeah
+because they are the building product of
+
+ align:start position:0%
+because they are the building product of
+ 
+
+ align:start position:0%
+because they are the building product of
+your of your whole operators yeah
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yeah yeah the working State here is
+
+ align:start position:0%
+yeah yeah the working State here is
+ 
+
+ align:start position:0%
+yeah yeah the working State here is
+based on uh it's come from the uh uh uh
+
+ align:start position:0%
+based on uh it's come from the uh uh uh
+ 
+
+ align:start position:0%
+based on uh it's come from the uh uh uh
+uh from the energy right so so once we
+
+ align:start position:0%
+uh from the energy right so so once we
+ 
+
+ align:start position:0%
+uh from the energy right so so once we
+solve x and P
+
+ align:start position:0%
+solve x and P
+ 
+
+ align:start position:0%
+solve x and P
+and then you can write to the
+
+ align:start position:0%
+and then you can write to the
+ 
+
+ align:start position:0%
+and then you can write to the
+hamiltonian in terms of X and P and then
+
+ align:start position:0%
+hamiltonian in terms of X and P and then
+ 
+
+ align:start position:0%
+hamiltonian in terms of X and P and then
+you just look for the lowest energy
+
+ align:start position:0%
+you just look for the lowest energy
+ 
+
+ align:start position:0%
+you just look for the lowest energy
+State and then you find the lowest
+
+ align:start position:0%
+State and then you find the lowest
+ 
+
+ align:start position:0%
+State and then you find the lowest
+energy State just satisfy this equation
+
+ align:start position:0%
+energy State just satisfy this equation
+ 
+
+ align:start position:0%
+energy State just satisfy this equation
+yeah and then from there you can find
+
+ align:start position:0%
+yeah and then from there you can find
+ 
+
+ align:start position:0%
+yeah and then from there you can find
+other states
+
+ align:start position:0%
+other states
+ 
+
+ align:start position:0%
+other states
+yeah the same thing we are going to yeah
+
+ align:start position:0%
+yeah the same thing we are going to yeah
+ 
+
+ align:start position:0%
+yeah the same thing we are going to yeah
+the same strategy we are going to use
+
+ align:start position:0%
+the same strategy we are going to use
+ 
+
+ align:start position:0%
+the same strategy we are going to use
+for the uh of a Quantum velocity
+
+ align:start position:0%
+for the uh of a Quantum velocity
+ 
+
+ align:start position:0%
+for the uh of a Quantum velocity
+okay good
+
+ align:start position:0%
+okay good
+ 
+
+ align:start position:0%
+okay good
+okay good so so now become a mechanical
+
+ align:start position:0%
+okay good so so now become a mechanical
+ 
+
+ align:start position:0%
+okay good so so now become a mechanical
+we can just uh apply this to to this
+
+ align:start position:0%
+we can just uh apply this to to this
+ 
+
+ align:start position:0%
+we can just uh apply this to to this
+Theory
+
+ align:start position:0%
+Theory
+ 
+
+ align:start position:0%
+Theory
+okay we can just apply these two three
+
+ align:start position:0%
+okay we can just apply these two three
+ 
+
+ align:start position:0%
+okay we can just apply these two three
+Theory and now let me add here so here
+
+ align:start position:0%
+Theory and now let me add here so here
+ 
+
+ align:start position:0%
+Theory and now let me add here so here
+the canonical momentum density conjugate
+
+ align:start position:0%
+the canonical momentum density conjugate
+ 
+
+ align:start position:0%
+the canonical momentum density conjugate
+to Phi is called the pi before it's just
+
+ align:start position:0%
+to Phi is called the pi before it's just
+ 
+
+ align:start position:0%
+to Phi is called the pi before it's just
+the atom derivative of Phi
+
+ align:start position:0%
+the atom derivative of Phi
+ 
+
+ align:start position:0%
+the atom derivative of Phi
+and the hamiltonian density you can find
+
+ align:start position:0%
+and the hamiltonian density you can find
+ 
+
+ align:start position:0%
+and the hamiltonian density you can find
+it explicitly
+
+ align:start position:0%
+it explicitly
+ 
+
+ align:start position:0%
+it explicitly
+is pi square plus
+
+ align:start position:0%
+is pi square plus
+ 
+
+ align:start position:0%
+is pi square plus
+one half
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+and then this is a classical equation
+
+ align:start position:0%
+and then this is a classical equation
+ 
+
+ align:start position:0%
+and then this is a classical equation
+motion okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+now let's just solve this classical
+
+ align:start position:0%
+now let's just solve this classical
+ 
+
+ align:start position:0%
+now let's just solve this classical
+equation motion
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this equation can be
+
+ align:start position:0%
+so this equation can be
+ 
+
+ align:start position:0%
+so this equation can be
+so this equation is easy to solve
+
+ align:start position:0%
+so this equation is easy to solve
+ 
+
+ align:start position:0%
+so this equation is easy to solve
+because the because the translation
+
+ align:start position:0%
+because the because the translation
+ 
+
+ align:start position:0%
+because the because the translation
+symmetry
+
+ align:start position:0%
+symmetry
+ 
+
+ align:start position:0%
+symmetry
+okay you can just do a free transform
+
+ align:start position:0%
+okay you can just do a free transform
+ 
+
+ align:start position:0%
+okay you can just do a free transform
+okay so we can fully transform
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so so now let's do the um
+
+ align:start position:0%
+so so now let's do the um
+ 
+
+ align:start position:0%
+so so now let's do the um
+you can freely transform
+
+ align:start position:0%
+you can freely transform
+ 
+
+ align:start position:0%
+you can freely transform
+so 2 Prime
+
+ align:start position:0%
+so 2 Prime
+ 
+
+ align:start position:0%
+so 2 Prime
+to be solved
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+so we can just write 5X
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+equal to expenditure
+
+ align:start position:0%
+equal to expenditure
+ 
+
+ align:start position:0%
+equal to expenditure
+minus IET
+
+ align:start position:0%
+minus IET
+ 
+
+ align:start position:0%
+minus IET
+plus I
+
+ align:start position:0%
+plus I
+ 
+
+ align:start position:0%
+plus I
+K dot X
+
+ align:start position:0%
+K dot X
+ 
+
+ align:start position:0%
+K dot X
+okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay so and then you you can see that
+
+ align:start position:0%
+okay so and then you you can see that
+ 
+
+ align:start position:0%
+okay so and then you you can see that
+this
+
+ align:start position:0%
+this
+ 
+
+ align:start position:0%
+this
+just solves the
+
+ align:start position:0%
+just solves the
+ 
+
+ align:start position:0%
+just solves the
+so this is provide
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+a basis
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+of solutions
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+22 prime okay
+
+ align:start position:0%
+22 prime okay
+ 
+
+ align:start position:0%
+22 prime okay
+it's just a plain wave okay just plain
+
+ align:start position:0%
+it's just a plain wave okay just plain
+ 
+
+ align:start position:0%
+it's just a plain wave okay just plain
+weight
+
+ align:start position:0%
+weight
+ 
+
+ align:start position:0%
+weight
+oh
+
+ align:start position:0%
+oh
+ 
+
+ align:start position:0%
+oh
+so now we plug this into there
+
+ align:start position:0%
+so now we plug this into there
+ 
+
+ align:start position:0%
+so now we plug this into there
+then you just get the dispersion
+
+ align:start position:0%
+then you just get the dispersion
+ 
+
+ align:start position:0%
+then you just get the dispersion
+relation
+
+ align:start position:0%
+relation
+ 
+
+ align:start position:0%
+relation
+e Square
+
+ align:start position:0%
+e Square
+ 
+
+ align:start position:0%
+e Square
+should be M squared plus K Square
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay so so we'll delote this as
+
+ align:start position:0%
+okay so so we'll delote this as
+ 
+
+ align:start position:0%
+okay so so we'll delote this as
+Omega K Square
+
+ align:start position:0%
+Omega K Square
+ 
+
+ align:start position:0%
+Omega K Square
+so Omega k
+
+ align:start position:0%
+so Omega k
+ 
+
+ align:start position:0%
+so Omega k
+defines to be
+
+ align:start position:0%
+defines to be
+ 
+
+ align:start position:0%
+defines to be
+just the K Square
+
+ align:start position:0%
+just the K Square
+ 
+
+ align:start position:0%
+just the K Square
+plus M squared
+
+ align:start position:0%
+plus M squared
+ 
+
+ align:start position:0%
+plus M squared
+okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so ee when you take the square root of e
+
+ align:start position:0%
+so ee when you take the square root of e
+ 
+
+ align:start position:0%
+so ee when you take the square root of e
+so you can take a plus minus Omega k
+
+ align:start position:0%
+so you can take a plus minus Omega k
+ 
+
+ align:start position:0%
+so you can take a plus minus Omega k
+okay can be plus minus Omega K so we
+
+ align:start position:0%
+okay can be plus minus Omega K so we
+ 
+
+ align:start position:0%
+okay can be plus minus Omega K so we
+normally call the solution
+
+ align:start position:0%
+normally call the solution
+ 
+
+ align:start position:0%
+normally call the solution
+we normally separate
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so for historical reasons okay
+
+ align:start position:0%
+so for historical reasons okay
+ 
+
+ align:start position:0%
+so for historical reasons okay
+we normally quote
+
+ align:start position:0%
+we normally quote
+ 
+
+ align:start position:0%
+we normally quote
+Define UK
+
+ align:start position:0%
+Define UK
+ 
+
+ align:start position:0%
+Define UK
+X to be
+
+ align:start position:0%
+X to be
+ 
+
+ align:start position:0%
+X to be
+experiential matters I Omega k
+
+ align:start position:0%
+experiential matters I Omega k
+ 
+
+ align:start position:0%
+experiential matters I Omega k
+t
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+X so okay now we have inserted the
+
+ align:start position:0%
+X so okay now we have inserted the
+ 
+
+ align:start position:0%
+X so okay now we have inserted the
+positive root of E
+
+ align:start position:0%
+positive root of E
+ 
+
+ align:start position:0%
+positive root of E
+so this is normally called the positive
+
+ align:start position:0%
+so this is normally called the positive
+ 
+
+ align:start position:0%
+so this is normally called the positive
+Energy Solution
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+even though this
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+even though this name is Elizabeth
+
+ align:start position:0%
+even though this name is Elizabeth
+ 
+
+ align:start position:0%
+even though this name is Elizabeth
+misleading okay so so uh actually this
+
+ align:start position:0%
+misleading okay so so uh actually this
+ 
+
+ align:start position:0%
+misleading okay so so uh actually this
+we don't Define the energy actually uh
+
+ align:start position:0%
+we don't Define the energy actually uh
+ 
+
+ align:start position:0%
+we don't Define the energy actually uh
+yeah later we will see this is not
+
+ align:start position:0%
+yeah later we will see this is not
+ 
+
+ align:start position:0%
+yeah later we will see this is not
+really the energy of a particle and so
+
+ align:start position:0%
+really the energy of a particle and so
+ 
+
+ align:start position:0%
+really the energy of a particle and so
+so this is just a traditional name okay
+
+ align:start position:0%
+so this is just a traditional name okay
+ 
+
+ align:start position:0%
+so this is just a traditional name okay
+this is just a traditional name
+
+ align:start position:0%
+this is just a traditional name
+ 
+
+ align:start position:0%
+this is just a traditional name
+conventional name
+
+ align:start position:0%
+conventional name
+ 
+
+ align:start position:0%
+conventional name
+and then you can Define the compressed
+
+ align:start position:0%
+and then you can Define the compressed
+ 
+
+ align:start position:0%
+and then you can Define the compressed
+conjugate of K
+
+ align:start position:0%
+conjugate of K
+ 
+
+ align:start position:0%
+conjugate of K
+now you have
+
+ align:start position:0%
+now you have
+ 
+
+ align:start position:0%
+now you have
+then corresponding to you have
+
+ align:start position:0%
+then corresponding to you have
+ 
+
+ align:start position:0%
+then corresponding to you have
+minus
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+Omega K in there
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so we
+
+ align:start position:0%
+so we
+ 
+
+ align:start position:0%
+so we
+yeah we take this conjugate
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and so this is called the reactive
+
+ align:start position:0%
+and so this is called the reactive
+ 
+
+ align:start position:0%
+and so this is called the reactive
+Energy Solution
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+so all together
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+therefore the former company side of
+
+ align:start position:0%
+therefore the former company side of
+ 
+
+ align:start position:0%
+therefore the former company side of
+solutions okay so complete side of
+
+ align:start position:0%
+solutions okay so complete side of
+ 
+
+ align:start position:0%
+solutions okay so complete side of
+solutions
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so
+
+ align:start position:0%
+so
+ 
+
+ align:start position:0%
+so
+complete the basis the complete side of
+
+ align:start position:0%
+complete the basis the complete side of
+ 
+
+ align:start position:0%
+complete the basis the complete side of
+yeah complete
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+phases
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+is formed by
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+UK
+
+ align:start position:0%
+UK
+ 
+
+ align:start position:0%
+UK
+and the UK UK star
+
+ align:start position:0%
+and the UK UK star
+ 
+
+ align:start position:0%
+and the UK UK star
+for okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so these are the uh when you these are
+
+ align:start position:0%
+so these are the uh when you these are
+ 
+
+ align:start position:0%
+so these are the uh when you these are
+the complete set of solutions to that
+
+ align:start position:0%
+the complete set of solutions to that
+ 
+
+ align:start position:0%
+the complete set of solutions to that
+wave equation okay so that's just the uh
+
+ align:start position:0%
+wave equation okay so that's just the uh
+ 
+
+ align:start position:0%
+wave equation okay so that's just the uh
+um yeah the uh
+
+ align:start position:0%
+um yeah the uh
+ 
+
+ align:start position:0%
+um yeah the uh
+any questions on this
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this is just like a classically this
+
+ align:start position:0%
+so this is just like a classically this
+ 
+
+ align:start position:0%
+so this is just like a classically this
+is like a wave okay just like a plane
+
+ align:start position:0%
+is like a wave okay just like a plane
+ 
+
+ align:start position:0%
+is like a wave okay just like a plane
+wave which you should also have seen in
+
+ align:start position:0%
+wave which you should also have seen in
+ 
+
+ align:start position:0%
+wave which you should also have seen in
+803 you know
+
+ align:start position:0%
+803 you know
+ 
+
+ align:start position:0%
+803 you know
+or
+
+ align:start position:0%
+or
+ 
+
+ align:start position:0%
+or
+good
+
+ align:start position:0%
+good
+ 
+
+ align:start position:0%
+good
+so now we can find that so now we can
+
+ align:start position:0%
+so now we can find that so now we can
+ 
+
+ align:start position:0%
+so now we can find that so now we can
+write down the most General
+
+ align:start position:0%
+write down the most General
+ 
+
+ align:start position:0%
+write down the most General
+so this is a basis
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so these are the counterparts of the
+
+ align:start position:0%
+so these are the counterparts of the
+ 
+
+ align:start position:0%
+so these are the counterparts of the
+exponential plus minus the Omega T here
+
+ align:start position:0%
+exponential plus minus the Omega T here
+ 
+
+ align:start position:0%
+exponential plus minus the Omega T here
+okay so now we can write down the most
+
+ align:start position:0%
+okay so now we can write down the most
+ 
+
+ align:start position:0%
+okay so now we can write down the most
+General solutions by just putting the
+
+ align:start position:0%
+General solutions by just putting the
+ 
+
+ align:start position:0%
+General solutions by just putting the
+integration constant
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+to the most General
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+classical Solutions
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so you can just write Phi X
+
+ align:start position:0%
+so you can just write Phi X
+ 
+
+ align:start position:0%
+so you can just write Phi X
+equal to
+
+ align:start position:0%
+equal to
+ 
+
+ align:start position:0%
+equal to
+integrate of all possible value
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+of K because this is for all K so we
+
+ align:start position:0%
+of K because this is for all K so we
+ 
+
+ align:start position:0%
+of K because this is for all K so we
+just sum up all of them
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and uh so we so
+
+ align:start position:0%
+and uh so we so
+ 
+
+ align:start position:0%
+and uh so we so
+so this factor is for for just for
+
+ align:start position:0%
+so this factor is for for just for
+ 
+
+ align:start position:0%
+so this factor is for for just for
+convenience okay it's just a convention
+
+ align:start position:0%
+convenience okay it's just a convention
+ 
+
+ align:start position:0%
+convenience okay it's just a convention
+you don't have to put it here it's just
+
+ align:start position:0%
+you don't have to put it here it's just
+ 
+
+ align:start position:0%
+you don't have to put it here it's just
+a convention
+
+ align:start position:0%
+a convention
+ 
+
+ align:start position:0%
+a convention
+and then and then we have
+
+ align:start position:0%
+and then and then we have
+ 
+
+ align:start position:0%
+and then and then we have
+a k
+
+ align:start position:0%
+a k
+ 
+
+ align:start position:0%
+a k
+u k
+
+ align:start position:0%
+u k
+ 
+
+ align:start position:0%
+u k
+plus a k star
+
+ align:start position:0%
+plus a k star
+ 
+
+ align:start position:0%
+plus a k star
+UK stop
+
+ align:start position:0%
+UK stop
+ 
+
+ align:start position:0%
+UK stop
+okay so this is just the most General
+
+ align:start position:0%
+okay so this is just the most General
+ 
+
+ align:start position:0%
+okay so this is just the most General
+set of solutions
+
+ align:start position:0%
+set of solutions
+ 
+
+ align:start position:0%
+set of solutions
+with AK and the AK Star as integration
+
+ align:start position:0%
+with AK and the AK Star as integration
+ 
+
+ align:start position:0%
+with AK and the AK Star as integration
+constant
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this is a full set of the integration
+
+ align:start position:0%
+so this is a full set of the integration
+ 
+
+ align:start position:0%
+so this is a full set of the integration
+constant okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+good
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so now when you go to Quantum level
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so now we can just follow the rule
+
+ align:start position:0%
+so now we can just follow the rule
+ 
+
+ align:start position:0%
+so now we can just follow the rule
+okay we find the the most General
+
+ align:start position:0%
+okay we find the the most General
+ 
+
+ align:start position:0%
+okay we find the the most General
+classical solution
+
+ align:start position:0%
+classical solution
+ 
+
+ align:start position:0%
+classical solution
+and in the column level we just promote
+
+ align:start position:0%
+and in the column level we just promote
+ 
+
+ align:start position:0%
+and in the column level we just promote
+the distribute operator
+
+ align:start position:0%
+the distribute operator
+ 
+
+ align:start position:0%
+the distribute operator
+you just put a hat there
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and change this to dagger
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay so now this becomes
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+your basis of quantum operators
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+so these are the cons the full set of
+
+ align:start position:0%
+so these are the cons the full set of
+ 
+
+ align:start position:0%
+so these are the cons the full set of
+constant Quantum operators
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and this solves your Theory on
+
+ align:start position:0%
+and this solves your Theory on
+ 
+
+ align:start position:0%
+and this solves your Theory on
+okay so uh so this solves the the
+
+ align:start position:0%
+okay so uh so this solves the the
+ 
+
+ align:start position:0%
+okay so uh so this solves the the
+operator equation
+
+ align:start position:0%
+operator equation
+ 
+
+ align:start position:0%
+operator equation
+and dissolves the operating equation
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so now the Legacy
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+is to impose the canonical
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+commutation relation
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so firstly we have to uh now we have to
+
+ align:start position:0%
+so firstly we have to uh now we have to
+ 
+
+ align:start position:0%
+so firstly we have to uh now we have to
+do a little bit thinking okay
+
+ align:start position:0%
+do a little bit thinking okay
+ 
+
+ align:start position:0%
+do a little bit thinking okay
+so so far you just uh straightforward
+
+ align:start position:0%
+so so far you just uh straightforward
+ 
+
+ align:start position:0%
+so so far you just uh straightforward
+but now we have to do a little bit
+
+ align:start position:0%
+but now we have to do a little bit
+ 
+
+ align:start position:0%
+but now we have to do a little bit
+thinking
+
+ align:start position:0%
+thinking
+ 
+
+ align:start position:0%
+thinking
+so so for
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+for finite
+
+ align:start position:0%
+for finite
+ 
+
+ align:start position:0%
+for finite
+for harmonic oscillator or for
+
+ align:start position:0%
+for harmonic oscillator or for
+ 
+
+ align:start position:0%
+for harmonic oscillator or for
+Quantum system of a single variable
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+you just have X you you just have this
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+you just have these and now we need to
+
+ align:start position:0%
+you just have these and now we need to
+ 
+
+ align:start position:0%
+you just have these and now we need to
+come up
+
+ align:start position:0%
+come up
+ 
+
+ align:start position:0%
+come up
+with the generalization of these two
+
+ align:start position:0%
+with the generalization of these two
+ 
+
+ align:start position:0%
+with the generalization of these two
+field Theory
+
+ align:start position:0%
+field Theory
+ 
+
+ align:start position:0%
+field Theory
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+so so we need to come up with the
+
+ align:start position:0%
+so so we need to come up with the
+ 
+
+ align:start position:0%
+so so we need to come up with the
+generation of that to field City with
+
+ align:start position:0%
+generation of that to field City with
+ 
+
+ align:start position:0%
+generation of that to field City with
+corresponding to
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+by X
+
+ align:start position:0%
+by X
+ 
+
+ align:start position:0%
+by X
+so now let me
+
+ align:start position:0%
+so now let me
+ 
+
+ align:start position:0%
+so now let me
+make it the time and the special
+
+ align:start position:0%
+make it the time and the special
+ 
+
+ align:start position:0%
+make it the time and the special
+coordinate separate
+
+ align:start position:0%
+coordinate separate
+ 
+
+ align:start position:0%
+coordinate separate
+and these conjugate the momentum is Phi
+
+ align:start position:0%
+and these conjugate the momentum is Phi
+ 
+
+ align:start position:0%
+and these conjugate the momentum is Phi
+it is pi okay
+
+ align:start position:0%
+it is pi okay
+ 
+
+ align:start position:0%
+it is pi okay
+conjugate the momentum density
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so we should do them at the same time
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+remember
+
+ align:start position:0%
+remember
+ 
+
+ align:start position:0%
+remember
+T is the is the same Evolution operator
+
+ align:start position:0%
+T is the is the same Evolution operator
+ 
+
+ align:start position:0%
+T is the is the same Evolution operator
+so they have to be valued at the same
+
+ align:start position:0%
+so they have to be valued at the same
+ 
+
+ align:start position:0%
+so they have to be valued at the same
+time as it's a equal time a canonical of
+
+ align:start position:0%
+time as it's a equal time a canonical of
+ 
+
+ align:start position:0%
+time as it's a equal time a canonical of
+quantitative condition is always at the
+
+ align:start position:0%
+quantitative condition is always at the
+ 
+
+ align:start position:0%
+quantitative condition is always at the
+equal time
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+but the x is a label of operators so so
+
+ align:start position:0%
+but the x is a label of operators so so
+ 
+
+ align:start position:0%
+but the x is a label of operators so so
+actually they don't have to be the same
+
+ align:start position:0%
+actually they don't have to be the same
+ 
+
+ align:start position:0%
+actually they don't have to be the same
+okay so here can be X here can be X
+
+ align:start position:0%
+okay so here can be X here can be X
+ 
+
+ align:start position:0%
+okay so here can be X here can be X
+Prime okay so now we have to come up
+
+ align:start position:0%
+Prime okay so now we have to come up
+ 
+
+ align:start position:0%
+Prime okay so now we have to come up
+with the generation of what is this
+
+ align:start position:0%
+with the generation of what is this
+ 
+
+ align:start position:0%
+with the generation of what is this
+quantity for filter okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so so now we just need to do a little
+
+ align:start position:0%
+so so now we just need to do a little
+ 
+
+ align:start position:0%
+so so now we just need to do a little
+bit guesswork okay you can easily guess
+
+ align:start position:0%
+bit guesswork okay you can easily guess
+ 
+
+ align:start position:0%
+bit guesswork okay you can easily guess
+it
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so before we do that you have any
+
+ align:start position:0%
+so before we do that you have any
+ 
+
+ align:start position:0%
+so before we do that you have any
+questions on this yes
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yeah yeah X is always so X is always
+
+ align:start position:0%
+yeah yeah X is always so X is always
+ 
+
+ align:start position:0%
+yeah yeah X is always so X is always
+here it's always just the label of the
+
+ align:start position:0%
+here it's always just the label of the
+ 
+
+ align:start position:0%
+here it's always just the label of the
+spatial location right yes yeah it's a
+
+ align:start position:0%
+spatial location right yes yeah it's a
+ 
+
+ align:start position:0%
+spatial location right yes yeah it's a
+label for the yeah yeah it's your field
+
+ align:start position:0%
+label for the yeah yeah it's your field
+ 
+
+ align:start position:0%
+label for the yeah yeah it's your field
+Theory enable yes so
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+operators has been constant in time is
+
+ align:start position:0%
+operators has been constant in time is
+ 
+
+ align:start position:0%
+operators has been constant in time is
+there any way that you can get them
+
+ align:start position:0%
+there any way that you can get them
+ 
+
+ align:start position:0%
+there any way that you can get them
+where it's like the evolution is more
+
+ align:start position:0%
+where it's like the evolution is more
+ 
+
+ align:start position:0%
+where it's like the evolution is more
+complex rather than just a constant
+
+ align:start position:0%
+complex rather than just a constant
+ 
+
+ align:start position:0%
+complex rather than just a constant
+operator and it
+
+ align:start position:0%
+operator and it
+ 
+
+ align:start position:0%
+operator and it
+yeah so so normally if you have second
+
+ align:start position:0%
+yeah so so normally if you have second
+ 
+
+ align:start position:0%
+yeah so so normally if you have second
+order differential equation
+
+ align:start position:0%
+order differential equation
+ 
+
+ align:start position:0%
+order differential equation
+you always have some integration
+
+ align:start position:0%
+you always have some integration
+ 
+
+ align:start position:0%
+you always have some integration
+constant yeah that's it yeah
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yeah
+
+ align:start position:0%
+yeah
+ 
+
+ align:start position:0%
+yeah
+yeah
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+other questions
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yes
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+directly into the operator
+
+ align:start position:0%
+directly into the operator
+ 
+
+ align:start position:0%
+directly into the operator
+right yeah yeah so that's a very good
+
+ align:start position:0%
+right yeah yeah so that's a very good
+ 
+
+ align:start position:0%
+right yeah yeah so that's a very good
+the uh that's a very good uh
+
+ align:start position:0%
+the uh that's a very good uh
+ 
+
+ align:start position:0%
+the uh that's a very good uh
+[Music]
+
+ align:start position:0%
+[Music]
+ 
+
+ align:start position:0%
+[Music]
+questions
+
+ align:start position:0%
+questions
+ 
+
+ align:start position:0%
+questions
+so that's just extension
+
+ align:start position:0%
+so that's just extension
+ 
+
+ align:start position:0%
+so that's just extension
+of our usual procedure for the quantum
+
+ align:start position:0%
+of our usual procedure for the quantum
+ 
+
+ align:start position:0%
+of our usual procedure for the quantum
+mechanics
+
+ align:start position:0%
+mechanics
+ 
+
+ align:start position:0%
+mechanics
+so so the usual procedure when you even
+
+ align:start position:0%
+so so the usual procedure when you even
+ 
+
+ align:start position:0%
+so so the usual procedure when you even
+just for harmonic also the for single
+
+ align:start position:0%
+just for harmonic also the for single
+ 
+
+ align:start position:0%
+just for harmonic also the for single
+variable system you have this
+
+ align:start position:0%
+variable system you have this
+ 
+
+ align:start position:0%
+variable system you have this
+correspondence between the classical
+
+ align:start position:0%
+correspondence between the classical
+ 
+
+ align:start position:0%
+correspondence between the classical
+system the quantum system when you
+
+ align:start position:0%
+system the quantum system when you
+ 
+
+ align:start position:0%
+system the quantum system when you
+quantize the classical system then the
+
+ align:start position:0%
+quantize the classical system then the
+ 
+
+ align:start position:0%
+quantize the classical system then the
+classical equation working becomes a
+
+ align:start position:0%
+classical equation working becomes a
+ 
+
+ align:start position:0%
+classical equation working becomes a
+Quantum operating equation uh here we
+
+ align:start position:0%
+Quantum operating equation uh here we
+ 
+
+ align:start position:0%
+Quantum operating equation uh here we
+just use the same rule because Quantum
+
+ align:start position:0%
+just use the same rule because Quantum
+ 
+
+ align:start position:0%
+just use the same rule because Quantum
+field is just the theory of infinite
+
+ align:start position:0%
+field is just the theory of infinite
+ 
+
+ align:start position:0%
+field is just the theory of infinite
+number of degrees freedoms we are not
+
+ align:start position:0%
+number of degrees freedoms we are not
+ 
+
+ align:start position:0%
+number of degrees freedoms we are not
+changing the rule of quantum mechanics
+
+ align:start position:0%
+changing the rule of quantum mechanics
+ 
+
+ align:start position:0%
+changing the rule of quantum mechanics
+and so that's why we just again just
+
+ align:start position:0%
+and so that's why we just again just
+ 
+
+ align:start position:0%
+and so that's why we just again just
+promote the classical equation into the
+
+ align:start position:0%
+promote the classical equation into the
+ 
+
+ align:start position:0%
+promote the classical equation into the
+operating equation
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+other questions
+
+ align:start position:0%
+other questions
+ 
+
+ align:start position:0%
+other questions
+yes
+
+ align:start position:0%
+yes
+ 
+
+ align:start position:0%
+yes
+one way of understanding like the
+
+ align:start position:0%
+one way of understanding like the
+ 
+
+ align:start position:0%
+one way of understanding like the
+Heisenberg equations for the
+
+ align:start position:0%
+Heisenberg equations for the
+ 
+
+ align:start position:0%
+Heisenberg equations for the
+Quantum mechanic system brackets for
+
+ align:start position:0%
+Quantum mechanic system brackets for
+ 
+
+ align:start position:0%
+Quantum mechanic system brackets for
+like the classroom is there something
+
+ align:start position:0%
+like the classroom is there something
+ 
+
+ align:start position:0%
+like the classroom is there something
+like that for field Theory as well
+
+ align:start position:0%
+like that for field Theory as well
+ 
+
+ align:start position:0%
+like that for field Theory as well
+um yeah yeah yeah there is yeah so so
+
+ align:start position:0%
+um yeah yeah yeah there is yeah so so
+ 
+
+ align:start position:0%
+um yeah yeah yeah there is yeah so so
+classically you can Define the possum
+
+ align:start position:0%
+classically you can Define the possum
+ 
+
+ align:start position:0%
+classically you can Define the possum
+bracket
+
+ align:start position:0%
+bracket
+ 
+
+ align:start position:0%
+bracket
+between the uh between the classical
+
+ align:start position:0%
+between the uh between the classical
+ 
+
+ align:start position:0%
+between the uh between the classical
+field variables and then and then
+
+ align:start position:0%
+field variables and then and then
+ 
+
+ align:start position:0%
+field variables and then and then
+Quantum mechanically just become commit
+
+ align:start position:0%
+Quantum mechanically just become commit
+ 
+
+ align:start position:0%
+Quantum mechanically just become commit
+uh uh uh quantum commutators
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yeah yeah you can also do that that's
+
+ align:start position:0%
+yeah yeah you can also do that that's
+ 
+
+ align:start position:0%
+yeah yeah you can also do that that's
+right yeah so so one way to come to this
+
+ align:start position:0%
+right yeah so so one way to come to this
+ 
+
+ align:start position:0%
+right yeah so so one way to come to this
+is you first describe first you need to
+
+ align:start position:0%
+is you first describe first you need to
+ 
+
+ align:start position:0%
+is you first describe first you need to
+generalize
+
+ align:start position:0%
+generalize
+ 
+
+ align:start position:0%
+generalize
+your standards possible brackets
+
+ align:start position:0%
+your standards possible brackets
+ 
+
+ align:start position:0%
+your standards possible brackets
+for final number because freedom to
+
+ align:start position:0%
+for final number because freedom to
+ 
+
+ align:start position:0%
+for final number because freedom to
+classical field Theory and then you can
+
+ align:start position:0%
+classical field Theory and then you can
+ 
+
+ align:start position:0%
+classical field Theory and then you can
+just generalize that to the uh to
+
+ align:start position:0%
+just generalize that to the uh to
+ 
+
+ align:start position:0%
+just generalize that to the uh to
+Quantum yeah indeed uh that's one route
+
+ align:start position:0%
+Quantum yeah indeed uh that's one route
+ 
+
+ align:start position:0%
+Quantum yeah indeed uh that's one route
+of doing it yeah
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay other questions
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+good so we'll be discussed the answer
+
+ align:start position:0%
+good so we'll be discussed the answer
+ 
+
+ align:start position:0%
+good so we'll be discussed the answer
+okay the answer is very easy to guess
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so so remember
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so um
+
+ align:start position:0%
+so um
+ 
+
+ align:start position:0%
+so um
+if you have a single X and P
+
+ align:start position:0%
+if you have a single X and P
+ 
+
+ align:start position:0%
+if you have a single X and P
+that's what you have okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+but if you have more than one particles
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+if you have more than one particles
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+say just hint
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+say you have multiple particles
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+system in quantum mechanics and then you
+
+ align:start position:0%
+system in quantum mechanics and then you
+ 
+
+ align:start position:0%
+system in quantum mechanics and then you
+have x a
+
+ align:start position:0%
+have x a
+ 
+
+ align:start position:0%
+have x a
+and the PA as your dynamical variable
+
+ align:start position:0%
+and the PA as your dynamical variable
+ 
+
+ align:start position:0%
+and the PA as your dynamical variable
+so a equal to 1 to say ends it says the
+
+ align:start position:0%
+so a equal to 1 to say ends it says the
+ 
+
+ align:start position:0%
+so a equal to 1 to say ends it says the
+lumbar particles okay
+
+ align:start position:0%
+lumbar particles okay
+ 
+
+ align:start position:0%
+lumbar particles okay
+and then your canonical quantization
+
+ align:start position:0%
+and then your canonical quantization
+ 
+
+ align:start position:0%
+and then your canonical quantization
+condition just become x a
+
+ align:start position:0%
+condition just become x a
+ 
+
+ align:start position:0%
+condition just become x a
+t
+
+ align:start position:0%
+t
+ 
+
+ align:start position:0%
+t
+p b t
+
+ align:start position:0%
+p b t
+ 
+
+ align:start position:0%
+p b t
+equal to I Delta ad
+
+ align:start position:0%
+equal to I Delta ad
+ 
+
+ align:start position:0%
+equal to I Delta ad
+and the different XA is commute
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+and different P commute
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay given the P commutes okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so
+
+ align:start position:0%
+so
+ 
+
+ align:start position:0%
+so
+so now this A and B
+
+ align:start position:0%
+so now this A and B
+ 
+
+ align:start position:0%
+so now this A and B
+are essentially just replaced by x and x
+
+ align:start position:0%
+are essentially just replaced by x and x
+ 
+
+ align:start position:0%
+are essentially just replaced by x and x
+Prime
+
+ align:start position:0%
+Prime
+ 
+
+ align:start position:0%
+Prime
+so x and x Prime are just continual of
+
+ align:start position:0%
+so x and x Prime are just continual of
+ 
+
+ align:start position:0%
+so x and x Prime are just continual of
+those A and B okay remember we kind of
+
+ align:start position:0%
+those A and B okay remember we kind of
+ 
+
+ align:start position:0%
+those A and B okay remember we kind of
+emphasize the X and the X Prime are the
+
+ align:start position:0%
+emphasize the X and the X Prime are the
+ 
+
+ align:start position:0%
+emphasize the X and the X Prime are the
+labels of your degrees freedom
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so now you can just guess
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay so we must have
+
+ align:start position:0%
+okay so we must have
+ 
+
+ align:start position:0%
+okay so we must have
+the following scene
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so so from here we must have
+
+ align:start position:0%
+so so from here we must have
+ 
+
+ align:start position:0%
+so so from here we must have
+Phi TX
+
+ align:start position:0%
+Phi TX
+ 
+
+ align:start position:0%
+Phi TX
+Phi TX Prime
+
+ align:start position:0%
+Phi TX Prime
+ 
+
+ align:start position:0%
+Phi TX Prime
+must be zero
+
+ align:start position:0%
+must be zero
+ 
+
+ align:start position:0%
+must be zero
+and the pi TX so Pi is the analog of P
+
+ align:start position:0%
+and the pi TX so Pi is the analog of P
+ 
+
+ align:start position:0%
+and the pi TX so Pi is the analog of P
+here
+
+ align:start position:0%
+here
+ 
+
+ align:start position:0%
+here
+yeah so those are operators
+
+ align:start position:0%
+yeah so those are operators
+ 
+
+ align:start position:0%
+yeah so those are operators
+TX Prime
+
+ align:start position:0%
+TX Prime
+ 
+
+ align:start position:0%
+TX Prime
+must be zero
+
+ align:start position:0%
+must be zero
+ 
+
+ align:start position:0%
+must be zero
+and then Phi
+
+ align:start position:0%
+and then Phi
+ 
+
+ align:start position:0%
+and then Phi
+TX
+
+ align:start position:0%
+TX
+ 
+
+ align:start position:0%
+TX
+which pi
+
+ align:start position:0%
+which pi
+ 
+
+ align:start position:0%
+which pi
+TX Prime
+
+ align:start position:0%
+TX Prime
+ 
+
+ align:start position:0%
+TX Prime
+should be something can only be zero but
+
+ align:start position:0%
+should be something can only be zero but
+ 
+
+ align:start position:0%
+should be something can only be zero but
+when X is not equal to X Prime
+
+ align:start position:0%
+when X is not equal to X Prime
+ 
+
+ align:start position:0%
+when X is not equal to X Prime
+can only be long zero when x equal to X
+
+ align:start position:0%
+can only be long zero when x equal to X
+ 
+
+ align:start position:0%
+can only be long zero when x equal to X
+Prime okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay as a generation of this
+
+ align:start position:0%
+okay as a generation of this
+ 
+
+ align:start position:0%
+okay as a generation of this
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+and uh so so you can now you can guess
+
+ align:start position:0%
+and uh so so you can now you can guess
+ 
+
+ align:start position:0%
+and uh so so you can now you can guess
+so what should this be
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so what
+
+ align:start position:0%
+so what
+ 
+
+ align:start position:0%
+so what
+yeah
+
+ align:start position:0%
+yeah
+ 
+
+ align:start position:0%
+yeah
+yeah I just used to be just directed huh
+
+ align:start position:0%
+yeah I just used to be just directed huh
+ 
+
+ align:start position:0%
+yeah I just used to be just directed huh
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+but now you answer the question why has
+
+ align:start position:0%
+but now you answer the question why has
+ 
+
+ align:start position:0%
+but now you answer the question why has
+to be direct Delta or maybe y should not
+
+ align:start position:0%
+to be direct Delta or maybe y should not
+ 
+
+ align:start position:0%
+to be direct Delta or maybe y should not
+be
+
+ align:start position:0%
+be
+ 
+
+ align:start position:0%
+be
+say the derivatives of direct data okay
+
+ align:start position:0%
+say the derivatives of direct data okay
+ 
+
+ align:start position:0%
+say the derivatives of direct data okay
+say y should not be say 100th derivative
+
+ align:start position:0%
+say y should not be say 100th derivative
+ 
+
+ align:start position:0%
+say y should not be say 100th derivative
+of direct Delta
+
+ align:start position:0%
+of direct Delta
+ 
+
+ align:start position:0%
+of direct Delta
+and that question can be addressed just
+
+ align:start position:0%
+and that question can be addressed just
+ 
+
+ align:start position:0%
+and that question can be addressed just
+by a form dimensional analysis
+
+ align:start position:0%
+by a form dimensional analysis
+ 
+
+ align:start position:0%
+by a form dimensional analysis
+so so here we know it's somehow this
+
+ align:start position:0%
+so so here we know it's somehow this
+ 
+
+ align:start position:0%
+so so here we know it's somehow this
+must be really direct Delta and now
+
+ align:start position:0%
+must be really direct Delta and now
+ 
+
+ align:start position:0%
+must be really direct Delta and now
+let's decide
+
+ align:start position:0%
+let's decide
+ 
+
+ align:start position:0%
+let's decide
+so now you can do the dimension do a
+
+ align:start position:0%
+so now you can do the dimension do a
+ 
+
+ align:start position:0%
+so now you can do the dimension do a
+little bit dimensional analysis
+
+ align:start position:0%
+little bit dimensional analysis
+ 
+
+ align:start position:0%
+little bit dimensional analysis
+so so if you just write down the action
+
+ align:start position:0%
+so so if you just write down the action
+ 
+
+ align:start position:0%
+so so if you just write down the action
+yeah the action I have just erased sorry
+
+ align:start position:0%
+yeah the action I have just erased sorry
+ 
+
+ align:start position:0%
+yeah the action I have just erased sorry
+so so if you look back on the action let
+
+ align:start position:0%
+so so if you look back on the action let
+ 
+
+ align:start position:0%
+so so if you look back on the action let
+me just alternate the idea because I'm
+
+ align:start position:0%
+me just alternate the idea because I'm
+ 
+
+ align:start position:0%
+me just alternate the idea because I'm
+sure you can do dimensional analysis
+
+ align:start position:0%
+sure you can do dimensional analysis
+ 
+
+ align:start position:0%
+sure you can do dimensional analysis
+yourself
+
+ align:start position:0%
+yourself
+ 
+
+ align:start position:0%
+yourself
+so if you look at the action
+
+ align:start position:0%
+so if you look at the action
+ 
+
+ align:start position:0%
+so if you look at the action
+so the action is dimension is in the
+
+ align:start position:0%
+so the action is dimension is in the
+ 
+
+ align:start position:0%
+so the action is dimension is in the
+lateral unit we are using
+
+ align:start position:0%
+lateral unit we are using
+ 
+
+ align:start position:0%
+lateral unit we are using
+so so if uh so from that you can deduce
+
+ align:start position:0%
+so so if uh so from that you can deduce
+ 
+
+ align:start position:0%
+so so if uh so from that you can deduce
+the dimension of Phi
+
+ align:start position:0%
+the dimension of Phi
+ 
+
+ align:start position:0%
+the dimension of Phi
+should be 1 over l so one over the lens
+
+ align:start position:0%
+should be 1 over l so one over the lens
+ 
+
+ align:start position:0%
+should be 1 over l so one over the lens
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+and from the fact that the pi
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+where is pi or maybe how also erase is
+
+ align:start position:0%
+where is pi or maybe how also erase is
+ 
+
+ align:start position:0%
+where is pi or maybe how also erase is
+equal to Five Dot
+
+ align:start position:0%
+equal to Five Dot
+ 
+
+ align:start position:0%
+equal to Five Dot
+means Pi should be Dimension 1 over L
+
+ align:start position:0%
+means Pi should be Dimension 1 over L
+ 
+
+ align:start position:0%
+means Pi should be Dimension 1 over L
+Square
+
+ align:start position:0%
+Square
+ 
+
+ align:start position:0%
+Square
+okay because you take the derivative one
+
+ align:start position:0%
+okay because you take the derivative one
+ 
+
+ align:start position:0%
+okay because you take the derivative one
+time and then there's a lot of factor of
+
+ align:start position:0%
+time and then there's a lot of factor of
+ 
+
+ align:start position:0%
+time and then there's a lot of factor of
+L
+
+ align:start position:0%
+L
+ 
+
+ align:start position:0%
+L
+then that means on the right hand side
+
+ align:start position:0%
+then that means on the right hand side
+ 
+
+ align:start position:0%
+then that means on the right hand side
+here
+
+ align:start position:0%
+here
+ 
+
+ align:start position:0%
+here
+must be something 1 over L to a cube
+
+ align:start position:0%
+must be something 1 over L to a cube
+ 
+
+ align:start position:0%
+must be something 1 over L to a cube
+okay because there's no other parameters
+
+ align:start position:0%
+okay because there's no other parameters
+ 
+
+ align:start position:0%
+okay because there's no other parameters
+here okay yeah because uh here there
+
+ align:start position:0%
+here okay yeah because uh here there
+ 
+
+ align:start position:0%
+here okay yeah because uh here there
+should be a I okay and if it is a
+
+ align:start position:0%
+should be a I okay and if it is a
+ 
+
+ align:start position:0%
+should be a I okay and if it is a
+dimension a one over Cube then can only
+
+ align:start position:0%
+dimension a one over Cube then can only
+ 
+
+ align:start position:0%
+dimension a one over Cube then can only
+be the data function not 100 derivative
+
+ align:start position:0%
+be the data function not 100 derivative
+ 
+
+ align:start position:0%
+be the data function not 100 derivative
+data function so so so so this thing
+
+ align:start position:0%
+data function so so so so this thing
+ 
+
+ align:start position:0%
+data function so so so so this thing
+should be just a data function okay
+
+ align:start position:0%
+should be just a data function okay
+ 
+
+ align:start position:0%
+should be just a data function okay
+okay so so um
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this you know the convention that
+
+ align:start position:0%
+so this you know the convention that
+ 
+
+ align:start position:0%
+so this you know the convention that
+there should be I
+
+ align:start position:0%
+there should be I
+ 
+
+ align:start position:0%
+there should be I
+and then it should be just the three
+
+ align:start position:0%
+and then it should be just the three
+ 
+
+ align:start position:0%
+and then it should be just the three
+data function
+
+ align:start position:0%
+data function
+ 
+
+ align:start position:0%
+data function
+and this indeed have the dimension one
+
+ align:start position:0%
+and this indeed have the dimension one
+ 
+
+ align:start position:0%
+and this indeed have the dimension one
+of uh L Cube okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+good
+
+ align:start position:0%
+good
+ 
+
+ align:start position:0%
+good
+so now you can just plug
+
+ align:start position:0%
+so now you can just plug
+ 
+
+ align:start position:0%
+so now you can just plug
+so you have the expression for X for Phi
+
+ align:start position:0%
+so you have the expression for X for Phi
+ 
+
+ align:start position:0%
+so you have the expression for X for Phi
+you take the time derivative of this
+
+ align:start position:0%
+you take the time derivative of this
+ 
+
+ align:start position:0%
+you take the time derivative of this
+you get the expression for for pi
+
+ align:start position:0%
+you get the expression for for pi
+ 
+
+ align:start position:0%
+you get the expression for for pi
+and now you can just plug them into here
+
+ align:start position:0%
+and now you can just plug them into here
+ 
+
+ align:start position:0%
+and now you can just plug them into here
+you can just plug them into here okay
+
+ align:start position:0%
+you can just plug them into here okay
+ 
+
+ align:start position:0%
+you can just plug them into here okay
+and then you can find the commutation
+
+ align:start position:0%
+and then you can find the commutation
+ 
+
+ align:start position:0%
+and then you can find the commutation
+relation between those a case
+
+ align:start position:0%
+relation between those a case
+ 
+
+ align:start position:0%
+relation between those a case
+okay and so this is a slightly TD
+
+ align:start position:0%
+okay and so this is a slightly TD
+ 
+
+ align:start position:0%
+okay and so this is a slightly TD
+calculation
+
+ align:start position:0%
+calculation
+ 
+
+ align:start position:0%
+calculation
+which is
+
+ align:start position:0%
+which is
+ 
+
+ align:start position:0%
+which is
+however a little bit fun
+
+ align:start position:0%
+however a little bit fun
+ 
+
+ align:start position:0%
+however a little bit fun
+which of course I will leave you to do
+
+ align:start position:0%
+which of course I will leave you to do
+ 
+
+ align:start position:0%
+which of course I will leave you to do
+so so if you just plug them in and then
+
+ align:start position:0%
+so so if you just plug them in and then
+ 
+
+ align:start position:0%
+so so if you just plug them in and then
+you can deduce
+
+ align:start position:0%
+you can deduce
+ 
+
+ align:start position:0%
+you can deduce
+at the following commutation relation
+
+ align:start position:0%
+at the following commutation relation
+ 
+
+ align:start position:0%
+at the following commutation relation
+between A's
+
+ align:start position:0%
+between A's
+ 
+
+ align:start position:0%
+between A's
+so so this is the I I think this is in P
+
+ align:start position:0%
+so so this is the I I think this is in P
+ 
+
+ align:start position:0%
+so so this is the I I think this is in P
+said two uh but I can still change my
+
+ align:start position:0%
+said two uh but I can still change my
+ 
+
+ align:start position:0%
+said two uh but I can still change my
+mind yeah I wanted to put in pizza two
+
+ align:start position:0%
+mind yeah I wanted to put in pizza two
+ 
+
+ align:start position:0%
+mind yeah I wanted to put in pizza two
+so you find the the commentator between
+
+ align:start position:0%
+so you find the the commentator between
+ 
+
+ align:start position:0%
+so you find the the commentator between
+a
+
+ align:start position:0%
+a
+ 
+
+ align:start position:0%
+a
+and the commentator between a dagger
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yeah so so now I will suppress the Hat
+
+ align:start position:0%
+yeah so so now I will suppress the Hat
+ 
+
+ align:start position:0%
+yeah so so now I will suppress the Hat
+okay because at the right height I think
+
+ align:start position:0%
+okay because at the right height I think
+ 
+
+ align:start position:0%
+okay because at the right height I think
+over and over I will be too tired
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so these are zero
+
+ align:start position:0%
+so these are zero
+ 
+
+ align:start position:0%
+so these are zero
+okay so the commutation relation between
+
+ align:start position:0%
+okay so the commutation relation between
+ 
+
+ align:start position:0%
+okay so the commutation relation between
+a and zero and a
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+you think
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+between a
+
+ align:start position:0%
+between a
+ 
+
+ align:start position:0%
+between a
+and a dagger
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so this gives you
+
+ align:start position:0%
+so this gives you
+ 
+
+ align:start position:0%
+so this gives you
+two pair Cube
+
+ align:start position:0%
+two pair Cube
+ 
+
+ align:start position:0%
+two pair Cube
+that the function k
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay so this is a three data function
+
+ align:start position:0%
+okay so this is a three data function
+ 
+
+ align:start position:0%
+okay so this is a three data function
+okay okay
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+so again this is a straightforward
+
+ align:start position:0%
+so again this is a straightforward
+ 
+
+ align:start position:0%
+so again this is a straightforward
+generation
+
+ align:start position:0%
+generation
+ 
+
+ align:start position:0%
+generation
+so if you have multiple harmonic
+
+ align:start position:0%
+so if you have multiple harmonic
+ 
+
+ align:start position:0%
+so if you have multiple harmonic
+oscillators
+
+ align:start position:0%
+oscillators
+ 
+
+ align:start position:0%
+oscillators
+so if you have considered the multiple
+
+ align:start position:0%
+so if you have considered the multiple
+ 
+
+ align:start position:0%
+so if you have considered the multiple
+harmonic oscillators before
+
+ align:start position:0%
+harmonic oscillators before
+ 
+
+ align:start position:0%
+harmonic oscillators before
+and then the a between the different
+
+ align:start position:0%
+and then the a between the different
+ 
+
+ align:start position:0%
+and then the a between the different
+harmonicles because K Prime are just
+
+ align:start position:0%
+harmonicles because K Prime are just
+ 
+
+ align:start position:0%
+harmonicles because K Prime are just
+here just corresponding to
+
+ align:start position:0%
+here just corresponding to
+ 
+
+ align:start position:0%
+here just corresponding to
+essentially you have yeah here it just
+
+ align:start position:0%
+essentially you have yeah here it just
+ 
+
+ align:start position:0%
+essentially you have yeah here it just
+is essentially you have infinite number
+
+ align:start position:0%
+is essentially you have infinite number
+ 
+
+ align:start position:0%
+is essentially you have infinite number
+of harmonic oscillators and each one of
+
+ align:start position:0%
+of harmonic oscillators and each one of
+ 
+
+ align:start position:0%
+of harmonic oscillators and each one of
+the enabled by a k okay so this is just
+
+ align:start position:0%
+the enabled by a k okay so this is just
+ 
+
+ align:start position:0%
+the enabled by a k okay so this is just
+like essentially we find yeah let me
+
+ align:start position:0%
+like essentially we find yeah let me
+ 
+
+ align:start position:0%
+like essentially we find yeah let me
+just write it here
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+[Applause]
+
+ align:start position:0%
+[Applause]
+ 
+
+ align:start position:0%
+[Applause]
+so from those commutation relations
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+we conclude
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+conclude
+
+ align:start position:0%
+conclude
+ 
+
+ align:start position:0%
+conclude
+this series will be quantize after we
+
+ align:start position:0%
+this series will be quantize after we
+ 
+
+ align:start position:0%
+this series will be quantize after we
+quantize it
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+become an infinite number
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+independent harmonic oscillators
+
+ align:start position:0%
+independent harmonic oscillators
+ 
+
+ align:start position:0%
+independent harmonic oscillators
+decoupled harmonical status
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+harmonic oscillators labeled by
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+continuous parameter k
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+the K is yeah
+
+ align:start position:0%
+the K is yeah
+ 
+
+ align:start position:0%
+the K is yeah
+okay is the wave Lumber
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+okay
+
+ align:start position:0%
+okay
+ 
+
+ align:start position:0%
+okay
+so for each k
+
+ align:start position:0%
+so for each k
+ 
+
+ align:start position:0%
+so for each k
+there is a
+
+ align:start position:0%
+there is a
+ 
+
+ align:start position:0%
+there is a
+an A and so between between a a
+
+ align:start position:0%
+an A and so between between a a
+ 
+
+ align:start position:0%
+an A and so between between a a
+themselves it's zero between a dag it's
+
+ align:start position:0%
+themselves it's zero between a dag it's
+ 
+
+ align:start position:0%
+themselves it's zero between a dag it's
+zero but a a dagger
+
+ align:start position:0%
+zero but a a dagger
+ 
+
+ align:start position:0%
+zero but a a dagger
+they not equal to zero
+
+ align:start position:0%
+they not equal to zero
+ 
+
+ align:start position:0%
+they not equal to zero
+and uh so this is again the continuing
+
+ align:start position:0%
+and uh so this is again the continuing
+ 
+
+ align:start position:0%
+and uh so this is again the continuing
+generation of one okay this is a
+
+ align:start position:0%
+generation of one okay this is a
+ 
+
+ align:start position:0%
+generation of one okay this is a
+Continuum generation of one uh because
+
+ align:start position:0%
+Continuum generation of one uh because
+ 
+
+ align:start position:0%
+Continuum generation of one uh because
+you have a continuous variables yes
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yeah then you cannot say for sure
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yeah yeah yeah
+
+ align:start position:0%
+yeah yeah yeah
+ 
+
+ align:start position:0%
+yeah yeah yeah
+no but but you see the conservation
+
+ align:start position:0%
+no but but you see the conservation
+ 
+
+ align:start position:0%
+no but but you see the conservation
+condition
+
+ align:start position:0%
+condition
+ 
+
+ align:start position:0%
+condition
+is in quantum mechanics quantum
+
+ align:start position:0%
+is in quantum mechanics quantum
+ 
+
+ align:start position:0%
+is in quantum mechanics quantum
+mechanics T and X are not on the same
+
+ align:start position:0%
+mechanics T and X are not on the same
+ 
+
+ align:start position:0%
+mechanics T and X are not on the same
+protein you can require your action to
+
+ align:start position:0%
+protein you can require your action to
+ 
+
+ align:start position:0%
+protein you can require your action to
+be x and t to be on the same 14 once you
+
+ align:start position:0%
+be x and t to be on the same 14 once you
+ 
+
+ align:start position:0%
+be x and t to be on the same 14 once you
+start the quantize your theory and then
+
+ align:start position:0%
+start the quantize your theory and then
+ 
+
+ align:start position:0%
+start the quantize your theory and then
+T will have a pronounced row
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+because I wanted to so I couldn't write
+
+ align:start position:0%
+because I wanted to so I couldn't write
+ 
+
+ align:start position:0%
+because I wanted to so I couldn't write
+that like the commutation relation as
+
+ align:start position:0%
+that like the commutation relation as
+ 
+
+ align:start position:0%
+that like the commutation relation as
+functions in the four Vector x no Delta
+
+ align:start position:0%
+functions in the four Vector x no Delta
+ 
+
+ align:start position:0%
+functions in the four Vector x no Delta
+x no no no no no no no the canonical
+
+ align:start position:0%
+x no no no no no no no the canonical
+ 
+
+ align:start position:0%
+x no no no no no no no the canonical
+computation variation have to be imposed
+
+ align:start position:0%
+computation variation have to be imposed
+ 
+
+ align:start position:0%
+computation variation have to be imposed
+at equal time
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+other questions
+
+ align:start position:0%
+other questions
+ 
+
+ align:start position:0%
+other questions
+good
+
+ align:start position:0%
+good
+ 
+
+ align:start position:0%
+good
+so um
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+yeah so so so essentially we just get
+
+ align:start position:0%
+yeah so so so essentially we just get
+ 
+
+ align:start position:0%
+yeah so so so essentially we just get
+and now it's just trivial okay so so you
+
+ align:start position:0%
+and now it's just trivial okay so so you
+ 
+
+ align:start position:0%
+and now it's just trivial okay so so you
+can just build up your hillbow space
+
+ align:start position:0%
+can just build up your hillbow space
+ 
+
+ align:start position:0%
+can just build up your hillbow space
+essentially you just have infinite
+
+ align:start position:0%
+essentially you just have infinite
+ 
+
+ align:start position:0%
+essentially you just have infinite
+number harmonical signatures
+
+ align:start position:0%
+number harmonical signatures
+ 
+
+ align:start position:0%
+number harmonical signatures
+okay just have infinite number of
+
+ align:start position:0%
+okay just have infinite number of
+ 
+
+ align:start position:0%
+okay just have infinite number of
+harmonic concentrators
+
+ align:start position:0%
+harmonic concentrators
+ 
+
+ align:start position:0%
+harmonic concentrators
+and there's no surprise you get the
+
+ align:start position:0%
+and there's no surprise you get the
+ 
+
+ align:start position:0%
+and there's no surprise you get the
+infinite number of harmonic oscillators
+
+ align:start position:0%
+infinite number of harmonic oscillators
+ 
+
+ align:start position:0%
+infinite number of harmonic oscillators
+because we mentioned that this field
+
+ align:start position:0%
+because we mentioned that this field
+ 
+
+ align:start position:0%
+because we mentioned that this field
+Theory
+
+ align:start position:0%
+Theory
+ 
+
+ align:start position:0%
+Theory
+can be actually written as a Continuum
+
+ align:start position:0%
+can be actually written as a Continuum
+ 
+
+ align:start position:0%
+can be actually written as a Continuum
+limit
+
+ align:start position:0%
+limit
+ 
+
+ align:start position:0%
+limit
+of these particles on the chain
+
+ align:start position:0%
+of these particles on the chain
+ 
+
+ align:start position:0%
+of these particles on the chain
+which in these eight or three examples
+
+ align:start position:0%
+which in these eight or three examples
+ 
+
+ align:start position:0%
+which in these eight or three examples
+you know that is a harmonic oscillator
+
+ align:start position:0%
+you know that is a harmonic oscillator
+ 
+
+ align:start position:0%
+you know that is a harmonic oscillator
+once you uh find the lower mode they're
+
+ align:start position:0%
+once you uh find the lower mode they're
+ 
+
+ align:start position:0%
+once you uh find the lower mode they're
+all just from a bunch of harmonicles
+
+ align:start position:0%
+all just from a bunch of harmonicles
+ 
+
+ align:start position:0%
+all just from a bunch of harmonicles
+features and this is just a
+
+ align:start position:0%
+features and this is just a
+ 
+
+ align:start position:0%
+features and this is just a
+three-dimensional version of that okay
+
+ align:start position:0%
+three-dimensional version of that okay
+ 
+
+ align:start position:0%
+three-dimensional version of that okay
+and now we will
+
+ align:start position:0%
+and now we will
+ 
+
+ align:start position:0%
+and now we will
+yeah today we are running all the uh
+
+ align:start position:0%
+ 
+ 
+
+ align:start position:0%
+ 
+all the time so next time we will see
+
+ align:start position:0%
+all the time so next time we will see
+ 
+
+ align:start position:0%
+all the time so next time we will see
+that each excitations of the harmonic
+
+ align:start position:0%
+that each excitations of the harmonic
+ 
+
+ align:start position:0%
+that each excitations of the harmonic
+oscillator
+
+ align:start position:0%
+oscillator
+ 
+
+ align:start position:0%
+oscillator
+can be interpreted as a space-time
+
+ align:start position:0%
+can be interpreted as a space-time
+ 
+
+ align:start position:0%
+can be interpreted as a space-time
+particle okay so that's the crossing of
+
+ align:start position:0%
+particle okay so that's the crossing of
+ 
+
+ align:start position:0%
+particle okay so that's the crossing of
+it and now you have this infinite number
+
+ align:start position:0%
+it and now you have this infinite number
+ 
+
+ align:start position:0%
+it and now you have this infinite number
+of harmonic oscillator and now you can
+
+ align:start position:0%
+of harmonic oscillator and now you can
+ 
+
+ align:start position:0%
+of harmonic oscillator and now you can
+act and now you can Define the vacuum
+
+ align:start position:0%
+act and now you can Define the vacuum
+ 
+
+ align:start position:0%
+act and now you can Define the vacuum
+and then actually this creation
+
+ align:start position:0%
+and then actually this creation
+ 
+
+ align:start position:0%
+and then actually this creation
+operators on the vacuum and now you find
+
+ align:start position:0%
+operators on the vacuum and now you find
+ 
+
+ align:start position:0%
+operators on the vacuum and now you find
+each excitation actually equals one into
+
+ align:start position:0%
+each excitation actually equals one into
+ 
+
+ align:start position:0%
+each excitation actually equals one into
+a particle
+
+ align:start position:0%
+a particle
+ 
+
+ align:start position:0%
+a particle
+and and has the uh a corresponding to
+
+ align:start position:0%
+and and has the uh a corresponding to
+ 
+
+ align:start position:0%
+and and has the uh a corresponding to
+relativistic particle and that's how you
+
+ align:start position:0%
+relativistic particle and that's how you
+ 
+
+ align:start position:0%
+relativistic particle and that's how you
+can have actually arbitrary lumbar
+
+ align:start position:0%
+can have actually arbitrary lumbar
+ 
+
+ align:start position:0%
+can have actually arbitrary lumbar
+particles
+
+ align:start position:0%
+particles
+ 
+
+ align:start position:0%
+particles
+in this series and uh yeah because you
+
+ align:start position:0%
+in this series and uh yeah because you
+ 
+
+ align:start position:0%
+in this series and uh yeah because you
+can excite as many times as you want
+
+ align:start position:0%
+can excite as many times as you want
+ 
+
+ align:start position:0%
+can excite as many times as you want
+okay each excitation is a particle
+
+ align:start position:0%
+okay each excitation is a particle
+ 
+
+ align:start position:0%
+okay each excitation is a particle
+good good okay so so I think it's a good
+
+ align:start position:0%
+good good okay so so I think it's a good
+ 
+
+ align:start position:0%
+good good okay so so I think it's a good
+time yeah we are two minutes I think
+
+ align:start position:0%
+time yeah we are two minutes I think
+ 
+
+ align:start position:0%
+time yeah we are two minutes I think
+early but I think it's a very good place
+
+ align:start position:0%
+early but I think it's a very good place
+ 
+
+ align:start position:0%
+early but I think it's a very good place
+to to break
\ No newline at end of file