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A lot of theories in physics can be derived from a variational principle: some action functional S on a space of e.g. field configurations $\phi$ is given, and the equations of the theory follow from the requirement that the action functional be stationary. In physics textbooks the functional derivative $\delta S/\delta \phi$ is then often introduced, and one says that the action is stationary for a given field configuration $\phi_0$ whenever

\begin{align} \frac{\delta S}{\delta \phi}\bigg|_{\phi_0} = 0. \end{align}

Unfortunately a definition of this object, the functional derivative, is almost never given, or we might get a definition like

\begin{align} \frac{\delta \phi(x)}{\delta \phi(y)} = \delta(x-y), \end{align}

where $\delta(x-y)$ represents the Dirac delta function. After that we usually get some rules of manipulating functional derivatives, like the product rule.

This all strucks me as some 'playing with symbols', while having no idea what we are doing. I have been looking for like a year to find books or articles where these things are treated rigorously, but have found next to nothing. Of course in the mathematics textbooks, like Gelfand & Fomin, variational problems are treated rigorously, but the way the variation of a functional is defined in those treatments is very different from how physicists define the functional derivative and use it. And treatments of the Frechet derivative and the Gateaux derivative seem to be disconnected from the physics way even more.

So I would really appreciate it if someone could give me some references (books, articles, ...) where the physicists functional derivative is treated rigorously!

The best thing I found so far is actually the Wikipedia article, which claims that the variation $\delta S$ of some functional can be expressed as \begin{align} \delta S[\delta\phi] = \int \frac{\delta S}{\delta \phi}(x) \delta\phi(x)\text{d}x. \end{align} for some function $\frac{\delta S}{\delta \phi}$, which would then be called the functional derivative. But the thing is, I looked at all the references the wikipedia lists there, and not even one of them proves this statement, or discusses e.g. the proper function spaces one should look at. And google has not brought me any further either...

Thanks!

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  • $\begingroup$ Or maybe, just maybe, a physicist like Feynman understood more of what he was doing than you realize. $\endgroup$ – Paul Sep 21 '17 at 11:03
  • $\begingroup$ Phycicists have no rigorous interacting quantum field theories in four dimensional space time. There are models is two or three dimensions where some of these methods can be made rigorous. This is called "constructive quantum field theory". Google or search mathoverflow. $\endgroup$ – Keith McClary Sep 22 '17 at 16:53

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