# A Is there a principle of stationary action for QFT?

#### jordi

Classical mechanics (and classical field theory) has the principle of stationary action (Hamilton's principle) as main principle. The Euler-Lagrange equations are derived from that principle, by using calculus of variations, on functionals (functions of functions).

Is there an equivalent principle of stationary action for QFT? I understand that the Schwinger-Dyson equations are the "equivalent" of the Euler-Lagrange equations, but instead of being differential equations in 1 (or 4) variables, the variable is a function itself (infinite dimensional).

I imagine that if the analogy works out, one should use functions of functionals (functions of functions of functions). Or maybe the "principle for QFT" does not look like a principle of stationary action at all.

So, is there some kind of "principle" that gives the Schwinger-Dyson equations in QFT, in the same way the principle of stationary action gives the Euler-Lagrange equations for classical field theory?

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#### atyy

Science Advisor
Thr fubdamental formulation of quantum mechanics involves quantum states as rays in Hilbert space, unitaty evolution via the Schroedinger equation, observables and the Born rule.

This can often be formulated using the sum over all paths formalism of the path integral. The "saddle point" approximation to the path integral recovers the classical least action principle.

• vanhees71 and jordi

#### jordi

Well, yes. What you say it is true for Quantum Mechanics. I was talking about QFT (of course, QFT is also Quantum Mechanics, but ...).

But my point is different: for sure there are several ways to quantize a theory: canonical quantization, path integral ...

But for example the path integral is very different from the dynamical law of classical mechanics: the path integral gives you the generating functional, and you build the correlation functions by functional differentiation of the generating functional. In perturbation theory, you just use the Feynman diagrams.

But maybe one would like to have a dynamical law, à la Hamilton principle, from which the Schwinger-Dyson equations could be derived. In the same way the Euler-Lagrange equations are derived from the Hamilton principle, using calculus of variations.

It is just more elegant to state the Hamilton principle as our "axiom" than the Euler-Lagrange equations (even though probably all conceivable calculations would be the same).

Analogously, it would be more elegant to have a Hamilton-like principle, from which the Schwinger-Dyson equations could be derived, rather than purely stating the Schwinger-Dyson equations as an axiom.

Science Advisor

#### jordi

This quantum action principle by Schwinger seems quite mysterious (to me, an ignorant). Is there a "for dummies" paper or book?

#### A. Neumaier

Science Advisor
This quantum action principle by Schwinger seems quite mysterious (to me, an ignorant). Is there a "for dummies" paper or book?
It is far from intuitive and out of fashion, was never popular. It has been superseded by the path integral formalism, which turned out to be much more versatile.

• jordi

#### jordi

I understand that, thank you.

I would like that there could be a "principle" guiding to QFT. In the end, in classical physics this is a strong argument (Hamilton's principle leading to the Euler-Lagrange equations, and then to classical mechanics and classical field theory).

Instead, in QFT we "have the solution" (the partition function), but we do not have a principle, and even the equivalent equations to Euler-Lagrange, the Schwinger-Dyson equations, are not very emphasized.

Books usually start with canonical quantization, or with the path integral. For sure, this is for good reasons: path integral is the easiest way to calculate, especially with gauge theories.

But I am surprised about the lack of interest to put QFT into a more "elegant" framework. OK, maybe it cannot be done, or it is not wise to do so.

#### A. Neumaier

Science Advisor
I am surprised about the lack of interest to put QFT into a more "elegant" framework. OK, maybe it cannot be done, or it is not wise to do so.
No one has found an elegant way to do this. One either has to live with the limitations in the state of the art, or improve it. But the latter is hard, since the easier possibilities have all been tried and failed to give an elegant framework.

• vanhees71 and jordi

#### jordi

Your answer makes sense.

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