# Infinite degrees of freedom

## Main Question or Discussion Point

Currently, I am working on Thermal Quantum Field Theory.

In the introduction to that, many authors point out that infinitely many degrees of freedom and infinite volume are special.

In one reference that I am reading said "The famous equivalence between the Heisenberg and the Schro ̈dinger picture simply breaks down" when the degrees of freedom are infinite.

Could you explain these statements more concretely?

Thank you

## Answers and Replies

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strangerep
Currently, I am working on Thermal Quantum Field Theory.
In one reference that I am reading
In questions like this you should quote the specific reference. Have you looked at Umezawa's textbook "Thermofield Dynamics & Condensed States"?

said "The famous equivalence between the Heisenberg and the Schro ̈dinger picture simply breaks down" when the degrees of freedom are infinite.

Could you explain these statements more concretely?
Dirac explained the idea "concretely" in these references:

P.A.M. Dirac, "Quantum Electrodynamics with Dead Wood",
Phys. Rev, vol 139, no 3B, (1965), pB684.

and more extensively in his "Lectures on quantum field theory" given at Yeshiva, 1966. (Some university libraries have a copy of this small book.)

Dirac's explanation is "concrete" in the sense that he takes a specific interaction Hamiltonian, and shows that the dynamics is not sensibly solvable in the Schrodinger picture -- since the Hamiltonian is unbounded and even an infinitesimal time evolution causes a divergence.

However, in the Heisenberg picture, the dynamics is easily solvable as a differential equation, yielding analytic expressions for the time-dependent annihilation/creation operators. Dirac then notes that, "in this sense, the Heisenberg picture is better".

If you can't access the above references, I have written it up in some private notes, but I'd have to convert them from standard Latex to PF latex...