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How do we compute correlation function in the Schrodinger picture?

+ 3 like - 0 dislike
179 views

From concreteness' sake consider $\phi^4$ theory with a real scalar (even though the choice of the theory has nothing to do in principle with what I am going to ask).

Consider thefollowing correlation function
$$
<\Omega|T\{\phi(x_1)\phi(x_2)\}|\Omega>
$$
My question is, do we have to understand the formula above as given in the Heisenberg picture, the Schrodinger picture, or it doesn't matter? What motivates this question is that if we were in the Schrodinger picture i wouldn't know how to take the time ordering of the fields. On the other hand, if it really is meant to be understood in the Heisenberg picture, it is dissatisfying for me not to have a means to compute correlation functions in the Schrodinger picture. So which one is it?

asked Jun 22, 2016 in Theoretical Physics by Dmitry hand me the Kalashnikov (720 points) [ no revision ]

1 Answer

+ 3 like - 0 dislike

The fact that $\phi(x)$ has the time in the operator identifies the formula as one in the Heisenberg picture. In the Schroedinger picture, operators are time independent, unless time occurs explicitly as parameter in the interaction.

Even outside of field theory, time correlations are awkward to represent in the Schroedinger picture, where only a single time is natural. One can put one of the times into the Schroedinger state, and the time difference remains in the operator expression, as a conjugation by exponentiated Hamiltonians.

Moreover, the meaning as a time correlation is apparent only in the Heisenberg picture in the Schroedinger picture it is just an obscure formal expression without ready interpretation. Thus anything using multiple times is natural only in the Heisenberg picture (although formally one can rewrite it also in the Schroedinger picture).

This shows that the Heisenberg picture is the fundamental one, and the Schroedinger picture is derived.

answered Jun 22, 2016 by Arnold Neumaier (12,385 points) [ revision history ]
edited Jun 23, 2016 by Arnold Neumaier

The Schroedinger equation can be safely solved with help of Green's function containing two times, no problem.

If the Heisenberg picture is fundamental, then any other derived picture is as fundamental as the original one. No variable change can make the resulting picture less fundamental.

The Heisenberg picture is also "derived" when the second quantization is considered (introduced).

@VladimirKalitvianski: The Greens function is known only for exactly solvable problems. Most problems don't belong to this class.

Exact (but unknown) Green's function $G(x_1,x_2)$ is expressed via exact (but unknown) solutions $\phi_n(x)$ where you can see the same structure as a "correlation function". There is a perturbation theory for GF similar to that for $\phi_n(x)$, namely, it contains nothing else but the perturbative expansion of each exact $E_n$ and $\psi_n$ in the exact $\phi_n(x)=e^{-iE_n t}\psi_n({\bf{x}}) $.

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