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I've learnt quantum field theory for a semester but I still can't understand the role of classical equation of motion in QFT.

I have looked up for several books. They all discuss classical field theory. And they turn to the quantum part without leaving comments. It seems the following things have nothing to do with it. Then why discuss it?

I can vaguely tell that maybe the solution space of classical of motion has something to do with the Hilbert space (maybe through mode expansion?) but I'm not sure.

I don't work in high energy physics. Maybe it's a trivial question but please help me.

edited: I came up with this idea when learning the Non-Abelian Chern-simons field theory via the quantum hall notes by David Tong. He calculates( in Page 189) the ground space degeneracy of chern simons action by determining the solutions of classical equation of motion. It seems he finds the classical degree of freedom and quantizes them.

So I think if it is the case. Then what the quantum field theory does is find the degree of freedom of classical motion (the solution of classical equation) and quantize them by creating of annihilating operators. But I don't know whether it is compatible with the canonical quantization or not.

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  • $\begingroup$ Related: physics.stackexchange.com/q/276722/59281 $\endgroup$
    – octonion
    Dec 7 '21 at 7:24
  • $\begingroup$ Yes, I know how to do perturbation and renormlization through field integral. But still I don't ware the role classical equation of motion plays. $\endgroup$
    – Taveren Sa
    Dec 7 '21 at 9:19
  • $\begingroup$ It seems there are more constraints it puts to the Hilbert space or (fock space). The path integral quantization seems to make the classical trajectory less unique and thus we should not gain anything strong from the classical equation unless you want do semi classical analysis. $\endgroup$
    – Taveren Sa
    Dec 7 '21 at 9:26
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    $\begingroup$ It seems the edit v3 changed the question significantly. $\endgroup$
    – Qmechanic
    Dec 7 '21 at 10:03
  • $\begingroup$ Haaa, Nevertheless , thanks for answering! $\endgroup$
    – Taveren Sa
    Dec 7 '21 at 10:21
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There is an intimate connection between classical and quantum physics, see e.g. Bohr's correspondence principle, Ehrenfest theorem, the WKB approximation & the Schwinger-Dyson (SD) equations for starters.

One particular case of the SD equations $\langle \frac{\delta S}{\delta\phi}\rangle=0$ shows that the classical EOMs are satisfied in a quantum averaged sense in the quantum world.

Moreover, one may show that the classical paths give the dominant contributions to the quantum path integral.

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For this question, I think two types of equations should be discussed together.

  1. Equations of motion for the fundamental fields in a Lagrangian theory. These are what come up in the path integral formulation which tells you how probable a certain deviation should be.
  2. Conservation equations satisfied by polynomials in those fields that correspond to currents. Even though there could be anomalies, we should generically expect that these equations still have something to say about the quantum theory.

A few issues about quantization have already been brought up. For instance, a free quantum field is often expressed in terms of plane waves $e^{i p x}$ because these solve the Klein-Gordon equation. But exciting each plane wave individually is only something we can do because $\phi(x)$ is a simple sum of them and this is only true because the Klein-Gordon equation is linear. So in this sense, the fact that the Hilbert space is a Fock space is indeed a consequence of the equation of motion. It is therefore understandable that textbooks don't discuss interacting QFTs in the same way. If they did, they would essentially be limited to certain integrable QFTs that can be treated with a similar approach.

However, I think the biggest change in viewpoint comes about because we promote fields to operators. There is no way to measure an operator valued distribution directly. Experimentalists like S-matrix elements or Green's functions where we can plug in a position or a momentum and get a number. So we then have to ask whether equations of motion can be brought "outside the brackets" and taken to act on correlation functions themselves. The answer to this is yes but it comes with subtleties. For instance, the free field propagator is a Green's function of the Klein-Gordon equation so instead of $$ (\partial_x^2 + m^2) \left < \phi(x) \phi(y) \right > = 0 $$ we have $$ (\partial_x^2 + m^2) \left < \phi(x) \phi(y) \right > = \delta(x - y). $$ This is what's known as an operator equation which turns into the classical equation if we only consider it at separated points. This distinction is related to the fact that two fields can be multiplied at will in a classical theory whereas composite operators are singular in a quantum theory and need to be renormalized. This happens for conservation laws too such as $$ \partial_\mu \left < T^{\mu\nu}(x) O_1(x_1) \dots O_n(x_n) \right > = -\sum_i \delta(x - x_i) \partial^\mu_i \left < O_1(x_1) \dots O_n(x_n) \right > $$ which is the operator equation for the stress tensor.

The thing is, once this leap to operator equations is made, they absolutely constrain the quantum theory in powerful ways. Ward identities in gauge theories, which relate different Green's functions, come about because of current conservation. Some scaling dimensions in conformal theories can be computed exactly because equations of motion are equivalent to shortening conditions. I.e. they specify that an operator must transform in a special representation of the conformal group where various descendants are guaranteed to vanish. This analysis is also important when we add interactions that break conformal invariance. We know that interactions lead to anomalous dimensions but this implies that short representations must become long. This again leads to constraints (sometimes powerful enough to obviate Feynman diagrams) because a generic Hilbert space would not provide the matter that the short multiplet needs to "eat". Finally, if one adopts superspace, all these short supermultiplets (with names like "half BPS" and such) can be regarded as solutions to equations of motion as well. And clever uses of these equations of motion is behind some modern methods of studying supersymmetric QFT by means of a "twist" which turns an otherwise complicated observable into something that is topological or holomorphic.

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  • $\begingroup$ Thanks for answering! I think it's great. Does that mean classical equation dominants when the fields are apart? And only symmetry is important when they are getting closer. So we use ward identity to promote classical symmetry to quantum operator. Maybe that's the reason why, when we do OPE , the remaining parts are those when fields coordinates are very close. $\endgroup$
    – Taveren Sa
    Dec 15 '21 at 2:50
  • $\begingroup$ Some correlation functions are designed to be used in an integral which makes it impossible to always keep points separated. But yeah, I think that's right. For an integral involving a current, the EOM is what makes it pick out the regions where the current is approaching the other insertions. And this tells you what the leading term in each OPE has to be. $\endgroup$ Dec 16 '21 at 11:41
  • $\begingroup$ Thank you very much! $\endgroup$
    – Taveren Sa
    Dec 17 '21 at 8:18

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