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Relations between classical and quantum timeevolution of fields 
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#1
Feb2112, 08:21 AM

P: 679

This question is gonna be a bit vague and might lead to nowhere, but still I'll take the risk and try to ask it here.
I know in general how to quantize a field, and from the quantized field one gets the quantized Hamitonian thus the timeevolution operator. However, I wonder what're the precise relations between the classical and quantum time evolution. I can think of one, i.e. [tex]U(t)\langle\alpha\hat{\psi}(x)\alpha\rangle= \langle\alphae^{i\hat{H}t}\hat{\psi}(x)e^{i\hat{H}t}\alpha\rangle[/tex] Where [itex]U(t)[/itex] is the time evolution on the classical field, [itex]\hat{\psi}(x)[/itex] is the field operator, and [itex]\alpha\rangle[/itex] is some quantum state. However if one looks into more details, there is some thing odd(I think) happening: Take a free field for example, the eigenvectors of quantum timeevolution are Fock states, but the field expectation value of Fock states is 0, which is not an eigenvector of the classical timeevolution, since the eigenvetors of classical evolution are planewaves. So what is happening here mathematically? Also I'd like to know more about the relations classical and quantum timeevolution, any reference is also appreciated. 


#2
Feb2112, 06:25 PM

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Consider a classical quantity f, i.e., a function on phase space: f(q,p), where typically both q and p are functions of time. In this context, the time evolution of f is given by [tex] \dot f ~:=~ \frac{df}{dt} ~=~ \left\{f,H \right\}_{PB} [/tex] where the "PB" subscript stands for Poisson Bracket. Typically f is some element constructed from the generators of the dynamical algebra for the (class of) system under study. To quantize that system, one seeks a representation of the same dynamical algebra as operators on a Hilbert space. (If the dynamical algebra involves quadratic and higher products of the generators, then modifications are often needed. E.g., one might have to symmetrize the generators in the product.) But assuming that has been done, f and H now correspond to Hilbert space operators, and the time evolution of f is given by [tex] \dot f ~=~ \frac{i}{\hbar} \, \left[f,H \right] [/tex] [tex] q(t) ~=~ e^{i\omega t}a_0 + e^{i\omega t}a_0^* ~=:~ a(t)+a^*(t) [/tex] where [itex]a_0[/itex] is an arbitrary complex number, and [itex]a^*_0[/itex] its conjugate. The momentum p(t) can be calculated from the time derivative of q(t) and one can easily show that the Poisson bracket relationship between q and p, i.e., [itex]\{q,p\}=1[/itex] remains satisfied for all time. For a quantum harmonic oscillator, the solution is very similar: [tex] Q(t) ~=~ e^{i\omega t} a_0 + e^{i\omega t} a^*_0 [/tex] except that now [itex]a_0, a^*_0[/itex] are operators satisfying [tex] [a_0 , a^*_0] ~=~ \hbar [/tex] and one can show from the above that this also remains satisfied for all time. Summary: the classical and quantum cases are more similar than different. 


#3
Feb2212, 03:30 AM

P: 679

[tex]\langle n_{k_1},n_{k_2}\ldots\psi(x)n_{k_1},n_{k_2} \ldots \rangle=0[/tex]Which means the classical correpondence of Fock state is a trivial classical field (0 field value everywhere) ,because the field operator is linear w.r.t creation and annhilation operators. This seems a bit weird to me, because naively one expects an eigenvector of quantized timeevolution(i.e. eigenvector of Hamiltonian) should correspond to a classical field which is an eigenvetor of the classical time evolution. I'm just wondering what happened during the quantization so that this naive expectation fails. 


#4
Feb2212, 06:18 AM

P: 66

Relations between classical and quantum timeevolution of fields
But even in the above mentioned simplest states, other field observables will have non vanishing expectation values. Try for instance computing the expectation value for the energy or momentum density for a scalar field in a 1quantum state (it's a pretty simple exercise). 


#5
Feb2212, 08:33 AM

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#6
Feb2812, 07:02 AM

P: 679

And I have a follow up question: Is there any relation between the spectrum of classical field equation and quantum Hamiltonian? For free field the spectrum of classical field equation is just the oneparticle spectrum quantum Hamiltonian, so is there any principle about quantization saying similar things in general?



#7
Feb2812, 07:23 AM

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#8
Feb2812, 07:24 AM

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#9
Feb2812, 07:54 AM

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#10
Feb2812, 07:59 AM

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#11
Feb2812, 10:01 AM

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#12
Feb2812, 10:09 AM

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The dispersion relation of a classical linear field theory becomes the 1particle energy operator in the corresponding free quantum field theory. The Hamiltonian itself is the secondquantized form of the 1particle energy operator. A classical nonlinear field theory doesn't have a fixed dispersion relation, and the corresponding quantum field theory has no welldefined singleparticle energy operator, as it is an interacting theory. 


#13
Feb2812, 10:25 AM

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#14
Feb2812, 10:38 AM

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#15
Feb2812, 10:51 AM

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Also, they exhibit the closeness to the classical situation much better than number states, which have no classical analogue. 


#16
Feb2812, 10:57 AM

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#17
Feb2812, 11:14 AM

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#18
Feb2912, 09:04 AM

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