# QFT and measures on distributions

Recently I came across the following slogan: ,,constructing quantum field theory on a space $$X$$ means constructing a measure on the space of all (Schwarz?) distributions $$\mathcal{S}'(X)$$''. I would like to understand this slogan in some more detail: here is my attempt. Constructing quantum field theory means constructing the Fock space $$\mathcal{F}(H)$$ where $$H$$ is a single particle state space. This Fock space is naturally isomorphic with the space $$L^2$$ on the space of distributions-the underlying isomorphism is called Segal isomorphism. This functional realization is more convienient to work with thus the main task is to construct a measure on the space of distributions. However as I have understood, the above argument is valid provided we restrict ourselves to Gaussian measures-so this approach does not cover situations where there is some interaction present. I would be grateful if anybody could shed some light on those issues.

• See: A Perspective on Constructive Quantum Field Theory, p.9 . Interacting theories have been constructed in 1+1 dimensional spacetime. Commented Jan 13, 2020 at 19:29
• Thank you for pointing out this amazing article! Commented Jan 14, 2020 at 10:27

The term "slogan" may not be the best choice of words here. This is not about a commercial for toothpaste. Also, taking as a starting point "constructing a QFT means constructing a Fock space", while to some extent true in the free case, is not correct in general. If you are familiar with the Wightman axioms defining a QFT and want to understand what they have to do with probability measures on a space of distributions $$\mathcal{S}'$$, then you can look at Section 6.1 of the book by Glimm and Jaffe "Quantum Physics, A Functional Integral Point of View", 2nd Edition, for a precise set of axioms (going back to J. Fröhlich) on such measures for them to correspond to QFTs. Another useful reference on the matter, is the article "Time-ordered products and Schwinger functions", in CMP 1978, by Eckmann and Epstein.