Is there any useful results, or even any meaning to integration over a space of functionals? For example, consider $$\int\mathcal{D}S\,e^{-Z[S]},$$ where $Z[S]=\int\mathcal{D}\phi\,e^{-S[\phi]}$ is the partition function of a (Euclidean) quantum field theory with action $S[\phi]$. Such an expression would perhaps carry some fundamental problems like

  1. Would it converge for general $Z[S]$? (I think not)
  2. What does it even means?
  3. Is it at least well-defined?

We could take our wisdom from (ordinary) path integrals and try to remediate problem 1 by taking quotients, like $$\frac{\int\mathcal{D}S\,e^{-Z[S]}S_{\text{Some action of interest}}[\phi]}{\int\mathcal{D}S\,e^{-Z[S]}},$$ but the expression still looks either useless, meaningless, or hopelessly complicated. So, is there any meaning to these "higher order" functional integrals?

Edit: One could go more general and define $\int\mathcal{D}S\,W[S]$, for arbitrary $W[S]$, but I think the above less general expression may be a better entry point in thinking about usefulness, e.g. In some very non-rigorous way it's like a quotient of QFTs.

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    $\begingroup$ if you can define a measure for a functional, you can integrate it $\endgroup$
    – lurscher
    Feb 10, 2018 at 18:07
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    $\begingroup$ @lurscher But justifying one particular measure over another can be a problem. $\endgroup$ Feb 11, 2018 at 0:27

1 Answer 1


Note that even in traditional path integrals over fields, $\mathcal{D}\phi$ is meaningless. There is no Lebesgue measure over infinite-dimensional spaces. So $\mathcal{D}S$ should also be meaningless. One can define (field) path integrals as honest integrals with respect to a probability measure, namely, justify an equation like $$ \frac{\int \mathcal{D}\phi\ F(\phi)e^{-S(\phi)}}{\int \mathcal{D}\phi\ e^{-S(\phi)}}\ =\ \int_{S'(\mathbb{R}^d)} d\mu(\phi) \ F(\phi)\ . $$ Here $\mu$ should be a Borel probability measure on the space of temperate Schwartz distributions on $d$-dimensional spacetime where the fields $\phi$ live. In your case, one would first need a suitable space $W$ of test-functionals on say $S'(\mathbb{R}^d)$, and then you could try to construct your path integrals using Borel probability measures on $W'$. For this you need $W$ to have nice properties like being nuclear and Fréchet. The issue tough is the lack of explicit examples. There are examples in the area of probability theory called "white noise calculus" (see keywords like Kondratiev triple, Hida distribution). But I don't know if the kind of (functional) path integrals you are thinking about arise naturally anywhere.


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