# Do expectation values of quantum fields behave like classical fields?

A famous result in quantum mechanics is Ehrenfest's theorem which states that the expectation values of observables are governed by the classical equations of motion.

Does a similar statement hold for quantum fields?

In particular, in the context of the Higgs mechanism we usually investigate the potential and its minima by treating the field as classical. Let's say we find that the potential has two minima at $$\pm v$$. The connection to the quantum description is then made by claiming that the vacuum expecatation value $$\langle 0|\phi|0\rangle$$ is equal to the classical minimum: $$\langle 0|\phi|0\rangle = \pm v \, .$$ This can be calculated by taking the classical limit $$\hbar \to 0$$ of the path integral (c.f. page 563 in Schwartz's QFT book) $$\langle 0|\phi|0\rangle = \lim_{\hbar \to 0} \int D\phi e^{\frac{i}{\hbar}\int d^4x\mathcal L [\phi] } \phi = v\, .$$

Can this be understood analogous to the Ehrenfest theorem in quantum mechanics?

• hint: consider what happens inside an interferometer – John Dvorak Dec 14 '19 at 11:22

The QFT version of Ehrenfest's theorem are the Schwinger-Dyson equations, stating that the classical equation of motion (in presence of a source $$J$$) $$\frac{\delta S}{\delta \phi} + J = 0$$ holds as an operator equation $$\langle \frac{\delta S}{\delta \phi} + J \rangle_J = 0$$ in the quantum theory. If you evaluate this at $$J=0$$, you get that the classical equations of motions hold "up to contact terms" inside correlation functions, i.e. $$\langle \frac{\delta S}{\delta \phi(x)} \pi_i\phi(x_i)\rangle = \mathrm{i}\sum_i \langle \phi(x_1)\dots\phi(x_{i_1})\delta(x - x_i)\phi(x_{i+1}\dots \phi(x_n)\rangle$$ where the r.h.s. consists only of correlators in one field less than the r.h.s.