It is often stated that there is no mathematically rigorous definition of Feynman's functional integral, except for some very specific examples.

I may be very naïve, but to me there is at least one possible definition, one that is perfectly rigorous in mathematical terms. To put it simply, define the $n$-point function $$ G_n(x_1,\dots,x_n)\equiv\sum_{\text{graphs}}F_\mathrm{EG}(x_1,\dots,x_n) $$ where $F_\mathrm{EG}\colon\text{graphs}\to\mathbb C[[g]]$ is the value of the graph under the Feynman rules of the theory, calculated according to the Epstein-Glaser formalism. We work in the ring formal power series in the coupling constant $g$, with coefficients in, say, the space of distributions over $\mathbb R^d$.

With this, $G_n$ is a perfectly well-defined distribution. We may therefore set $$ Z[j]\equiv \sum_{n\in\mathbb N}\int_{\mathbb R^{dn}}G_n(x_1,\dots,x_n)j(x_1)\cdots j(x_n)\ \mathrm dx_1\cdots\mathrm dx_n $$ where $j\in C^\infty_c(\mathbb R^d)$, and integration is understood in the sense of distrubutions.

In "physical terms", $Z[j]$ corresponds to the standard functional integral $$ Z[j]\equiv\int_{C'^\infty_c(\mathbb R^d)}\ \mathrm e^{-S[\varphi]+\varphi\cdot j}\ \mathrm d\varphi $$ but regarded as a formal power series (in both $g$ and $j$).

Why is this definition unreasonable? It appears that it satisfies some nice properties (such as the "fundamental theorem of calculus", in the form of Dyson-Schwinger), but it may obscure some others (such as linearity). It agrees with standard integration in the $d=0$ case, and I believe it also agrees with the cases where the functional integral is well-defined (free theories, $d=1$, etc.). Yet, it is never mentioned in any reference I've read. Is there any reason not to take it seriously?

  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – ACuriousMind Feb 14 '18 at 18:18

The definition you are using in your question is the one that everybody who does rigorous perturbative renormalization uses. The particular choice of method BPHZ vs. Epstein-Glaser, etc. doesn't matter. They both give you the renormalized $n$-point correlation functions as formal power series in either $\hbar$ (a bit more canonical) or the renormalized coupling constant $g_{\rm R}$. Now the issue is that a measuring instrument usually returns numerical values rather than elements of $\mathbb{R}[[g]]$. Moreover, quantum transition probabilities must be positive. How would you express unitarity for a QFT if all you got are formal power series? It is desirable to have a rigorous construction of the $G(x_1,\ldots,x_n)$ as honest distributions rather than formal power series with distributional coefficients. This is the job of constructive quantum field theory.

Edit as per AFT's comment: I don't think it is that easy to define positivity for formal power series, e.g., by imposing it order by order. Although I should say I didn't think hard about the issue, so some might have better insights about it. If I look at the formal power series $$ \sum_{n=0}^{\infty} \frac{(-\hbar)^n}{n!}\ \in\ \mathbb{R}[[\hbar]]\ , $$ I can't really tell that it is positive, unless I sum it up to $e^{-\hbar}$. This is perhaps not even a good example, since at least this series converges. The perturbation series in QFT is expected to have zero radius of convergence and the general term wildly oscillates in sign and magnitude. A better example is zero-dimensional $\phi^4$ theory: $$ Z(g)=\int_{\mathbb{R}} e^{-\phi^2-g\phi^4} d\phi $$ which is perfectly well defined and nonnegative for $g\ge 0$. The corresponding series in $\mathbb{R}[[g]]$ is $$ \sum_{n=0}^{\infty}\frac{(-g)^n}{n!}\int_{\mathbb{R}} \phi^{4n}e^{-\phi^2} d\phi =\sum_{n=0}^{\infty} (-g)^n\ \frac{\Gamma\left(2n+\frac{1}{2}\right)}{\Gamma(n+1)}\ . $$ Moreover, what would one mean by positivity? : P1) positivity for all values of the parameter $g$ or $\hbar$, P2) positivity for one specific value like $\hbar=1.05457\times 10^{-34}$, or P3) positivity for small values? For P1, imposing positivity order by order, e.g., truncating the series at some $n$ is very bad. For $n$ odd, one has a polynomial of odd degree which will take negative values. I think P1 and P2 require a summation procedure, i.e., going from $\mathbb{R}[[g]]$ to $\mathbb{R}$. One could define P3 as just the positivity of the zero-th order term but this seems too coarse.

Finally, note that there has been recent work on violation of unitarity in QFTs in non integer dimension (see this article). I didn't look at it, but I suspect they must have addressed this positivity issue somehow.


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