Is the Classical Limit of Quantum Field Theory a Valid Inquiry?

[maverick280857]

Hi,

Is it meaningful to inquire about the classical limit of a quantum field theory? Specifically, is it possible to formally recover NRQM and RQM from quantum field theory? I am told this is a wrong/ill-posed question, so I wanted to get a clearer idea about it...after all, in a QM course, the classical limit of Schrödinger's equation is shown as the Hamilton Jacobi equation. Are there any analogues in QFT?

Sorry if this is a wrong question to ask, but in that case, I would appreciate being corrected. :-)

Thanks in advance.

 

[Fredrik]

Hi,

Is it meaningful to inquire about the classical limit of a quantum field theory? Specifically, is it possible to formally recover NRQM and RQM from quantum field theory? I am told this is a wrong/ill-posed question, so I wanted to get a clearer idea about it...after all, in a QM course, the classical limit of Schrödinger's equation is shown as the Hamilton Jacobi equation. Are there any analogues in QFT?

Sorry if this is a wrong question to ask, but in that case, I would appreciate being corrected. :-)

Thanks in advance.

A relativistic quantum field theory is a specific theory of matter in the framework of relativistic quantum mechanics, so you don't need to "recover" relativistic QM. It's already a part of the theory.

The procedure to recover non-relativistic QM from relativistic QM is to replace the Poincaré group with the Galilei group. This can be done by taking the limit c→∞.

You should also be more careful about how you use the word "classical". I recommend that you only use it to mean "non-quantum" and never "non-relativistic". Now I'm confused about what you're asking.

 

[RUTA]

The classical limit is easy to obtain in the path integral formalism (A. Zee, Quantum Field Theory in a Nutshell, Princeton Univ Press, 2003), essentially it's just the Euler-Lagrange field equations for the Lagrangian density in your action (p 19). NRQM is just (0+1)-dimensional QFT (p 18). As Fredrik said, the action of QFT is already Lorentz invariant (p 17).

 

[maverick280857]

Thanks RUTA and Fredrik.

You should also be more careful about how you use the word "classical". I recommend that you only use it to mean "non-quantum" and never "non-relativistic". Now I'm confused about what you're asking.

I'm sorry, I meant "non-relativistic"; that is why I asked if NRQM can be shown to be a special case of QFT.