# What does it mean for a field to be defined by a measure?

In Quantum Physics by Glimm and Jaffe they mention on p. 90 that

The Euclidean fields are defined by a probability measure $$d\mu(\phi) = d\mu$$ on the space of real distributions. Here $$d\mu$$ plays the same role as does the Feynman-Kac measure in quantum mechanics

and they go on to introduce a generating functional $$S\{f\} = \int e^{i\phi(f)}d\mu\tag{6.1.4}$$ that is the inverse Fourier transform of a Borel probability measure $$d\mu$$ on the space of test functions. Here $$\phi$$ is the field and $$f$$ is any test function.

I am a little confused on what this all means. I am familiar with measure theory, but not in this context. How does a (probability) measure define a field? What is the significance of this measure?

As a disclaimer, I am unfamiliar with the Feynman-Kac formula.

• ...how did you arrive at page 90 of this book being unfamiliar with the Feynman-Kac formula when chapter 3, which is titled "The Feynman-Kac formula", starts on page 43? Commented Jan 25, 2023 at 23:59
• @ACuriousMind I am not reading through this book cover to cover, I am using it as a supplement to other books. I came across this section and was intrigued. Commented Jan 26, 2023 at 1:11

I believe this is a common issue in bridging the mathematically rigorous analysis of Glimm and Jaffe and the standard Physics discussion in QFT textbooks. But the basic point is this. Imagine you have some field $$\phi(x)$$ whose dynamics is governed by an action $$S[\phi]$$. As an operator on some Hilbert space, you may think that $$\phi(x)$$ is really defined by its insertions into correlation functions $$\langle \phi(x)\cdots \rangle$$. This is a pretty common idea for example in Conformal Field Theory.

Now, the thing is that these kinds of correlation functions can be obtained from a generating functional $$Z[J]$$. Indeed, if you know the correlation functions you can construct $$Z[J]$$ as $$Z[J]=\sum_{n=0}^\infty \dfrac{1}{n!}\int d x_1\cdots dx_n J(x_1)\cdots J(x_n)\langle \phi(x_1)\cdots \phi(x_n)\rangle,$$

and then you may show that $$\langle \phi(x_1)\cdots \phi(x_n)\rangle =\dfrac{\delta^n Z[J]}{\delta J(x_1)\cdots \delta J(x_n)}\bigg|_{J=0}.$$

If you have more fields you then just need to add more sources. Now one may either formally, or rigorously as done in Glimm & Jaffe, consider a functional Fourier transform of $$Z[J]$$. Such functional Fourier transform defines a measure. Indeed, the claim is that in a suitable space of distributions, there exists a measure $$d\mu(\phi)$$ such that $$Z[J]=\int d\mu(\phi) \exp\left[i\int dx \phi(x)J(x)\right].$$

This measure $$d\mu(\phi)$$ is the "path integral measure", which physicists write as "$$\mathfrak{D}\phi e^{-S[\phi]}$$". Let me stress that the translation-invariant measure $$\mathfrak{D}\phi$$ that Physicists employ does not really exist as a rigorous construct in measure theory, because there is no infinite-dimensional Lebesgue measure. Nevertheless, the object "$$\mathfrak{D}\phi e^{-S[\phi]}$$" can be given meaning, and it is this measure $$d\mu(\phi)$$.

Maybe the non-trivial step then is in realizing that $$d\mu(\phi)$$ is the Physicist "$$\mathfrak{D}\phi e^{-S[\phi]}$$". One way of doing this, that I find quite elegant, is by observing that $$Z[J]$$ has to obey a functional differential equation known as the Dyson-Schwinger equation. If one formally solves this equation by a functional Fourier transform, one finds out $$Z[J]$$ in the form above, where $$d\mu(\phi)$$ is really the known "$$\mathfrak{D}\phi e^{-S[\phi]}$$". One may then realize that the rigorous version of the story must really be what Glimm & Jaffe are writing down in a logically organized manner. One reference for this is the book by Rivers, "Path Integral Methods in QFT", but if I'm not mistaken, Schwartz also discusses the solution to the Dyson-Schwinger functional equation by functional Fourier transform.

Now we can summarize it like this: the measure $$d\mu(\phi)$$ is really defined by the action, i.e., the dynamics of the theory. It defines the generating functional $$Z[J]$$ by means of the construction of the functional Fourier transform of a measure, and such generating functional defines, by functional differentiation, insertions of the field $$\phi(x)$$ into correlation functions. Since a field is defined by its insertions into correlators, ultimately the measure defines the field.

Finally, let me stress that really constructing $$d\mu(\phi)$$ is easier said than done. It is known how to be done for free fields and for some interacting cases in lower dimensions. The interacting case in four dimensions is really an open problem. Nevertheless, one can resort to perturbation theory. As soon as $$d\mu_0(\phi)$$ for the free action $$S_0[\phi]$$ is known, one may perturbatively build $$d\mu(\phi)$$ for the interacting action $$S[\phi]=S_0[\phi]+gV[\phi]$$ by really defining the interacting $$Z[J]$$ through integration against the free measure $$d\mu_0(\phi)$$. This is the starting point for Feynman diagrams in QFT.

• Thank you for this wonderful answer. If I have understood correctly, we start with some field $\phi$, which is completely determined by correlation functions. These correlation functions in turn define a generating functional, which can be used to (re)obtain the correlation functions. We then obtain a measure by taking the Fourier transform of this generating functional. By searching for the measure $d\mu$ we are effectively carrying out this process in reverse, where we look for the measure such that the above process holds and thus defines the original field $\phi$. Is this correct? Commented Jan 25, 2023 at 6:40
• You're welcome! Well, if you already knew all correlation functions $Z[J]$ would be of little use. The point of constructing $Z[J]$ is that we may obtain it first and use it to define the correlation functions. In fact, $Z[J]$ can be shown to obey the Dyson-Schwinger equation, so solving it is a way of doing that. In fact, as I said the free measure $d\mu_0(\phi)$ is well-known how to be constructed. It is what we call a Gaussian measure and it is really defined by the associated $Z_0[J]$ which can be defined in analogy with the finite-dimensional version for quadratic actions.
– Gold
Commented Jan 25, 2023 at 12:22
• In practice we really use $Z_0[J]$ to construct $d\mu_0$ and then we use $d\mu_0$ to construct one perturbation series for $Z[J]$ which in turn defines $d\mu$. The point is that if one writes down the formal path integral that would yield $Z_0[J]$ one observes it is a Gaussian integral. The finite-dimensional counterparts can be evaluated quite easily. One then either (1) formally reproduces the steps used in the finite-dimensional case or (2) picks the end result for the finite-dimensional case and proposes the continuum generalization as a definition for $Z_0[J]$.
– Gold
Commented Jan 25, 2023 at 18:35
• Then since we have $Z_0[J]$ we know under some assumptions that it is the Fourier transform of a measure $d\mu_0$. This is then a so-called Gaussian measure. Now the trick is to observe that the formal expression for $Z[J]$ is $$Z[J]=\int \mathfrak{D}\phi e^{-S_0[\phi]-gV[\phi]}\exp i\int dx \phi(x)J(x).$$ Observing that $\mathfrak{D}\phi e^{-S_0[\phi]}$ should really be $d\mu_0(\phi)$ and since we now have a proper construction of such a measure, we can then propose $Z[J]$ as $$Z[J]=\int d\mu_0(\phi) e^{-gV[\phi]}\exp i\int dx \phi(x)J(x)$$ i.e., one integral against $d\mu_0(\phi)$.
– Gold
Commented Jan 25, 2023 at 18:38
• At this point one may then expand in a power series in the coupling, that being the starting point of perturbation theory. Notice that in this line of reasoning there is an interplay between formal manipulations and recognizing the results of these formal manipulations as rigorously defined objects.
– Gold
Commented Jan 25, 2023 at 18:39