Is Nelson-Symanzik positivity compatible with fermionic statistics?

Question

Let $\{ S_n \}_{n =0}^\infty$ be a sequence of tempered distributions where $S_n \in \mathcal{S}'(\mathbb{R}^{nd})$ where $d \in \{2,3,4\}$ is fixed. Moreover, we put three additional conditions:

1. $S_0 =1$

$ \bigl \lvert S_n(f_1 \otimes \cdots \otimes f_n) \bigr \rvert \leq n! \prod_{i=1}^n \lVert f_i \rVert$ for $f_1, \cdots, f_n \in \mathcal{S}(\mathbb{R}^d)$ and some continuous seminorm $\lVert \cdot \rVert$ on $\mathcal{S}(\mathbb{R}^d)$.

Let $\{F_n\}_{n=0}^\infty$ be a sequence of Schwartz functions such that $F_n \in \mathcal{S}(\mathbb{R}^{nd})$ and only finitely many of $F_n$'s are nonzero. Then, $$ \sum_{n,m=0}^\infty S_{n+m}\bigl( F_n^* \otimes F_m) \geq 0 $$ where $F^*_n(x_1 \cdots, x_n) = \overline{ F_n(x_n, \cdots, x_1)}$. Here, $x_1, \cdots, x_n$ are elements of $\mathbb{R}^d$.

Then, it is well-known, cf. this MO post that there exists a unique probability measure $\mu$ on $\mathcal{S}'(\mathbb{R}^d)$ satisfying $$ \int_{\mathcal{S}'(\mathbb{R}^d)} \phi(f_1) \cdots \phi(f_n) d\mu(\phi) = S_n (f_1 \otimes \cdots \otimes f_n) \text{ for } f_1, \cdots, f_n \in \mathcal{S}(\mathbb{R}^d) $$

Here, the third condition assumed above is known as Nelson-Symanzik positivity as presented in a paper by Frohlich. And it seems almost exclusively associated with Bosonic fields in the context of Euclidean QFT.

However, there is no mention of Bosonic / Fermionic statistics either in above results or in the cited paper by Frohlich. So, I wonder if Fermionic statistics is compatible with Nelson-Symanzik positivity. More specifically,

Is it possible for the above $\{ S_n \}_{n=0}^\infty$ to also satisfy the following anti-commutation property? $$ S_n (x_{\pi(1)}, \cdots, x_{\pi(n)} ) = sgn(\pi)S_n(x_1, \cdots, x_n) $$ where $\pi$ is a permutation on $\{1,2, \cdots, n\}$.

I heard vaguely (possibly in the context of stochastic quantization) that functional integral approach involving a probability measure is only viable for Bosonic theories and does not work for Fermions. I wonder if this is indeed true. Could anyone please provide any insight?

Answer 1

Since the product of real numbers is commutative, for any permutation $\sigma$, $$ \int_{S'}\phi(f_{\sigma(1)})\cdots\phi(f_{\sigma(n)})\ d\mu(\phi)= \int_{S'}\phi(f_{1})\cdots\phi(f_{n})\ d\mu(\phi) $$ and therefore the moments would satisfy $$ S_n(f_{\sigma(1)}\otimes\cdots\otimes f_{\sigma(n)}) = S_n(f_1\otimes\cdots\otimes f_n) $$ and not a transformation rule like $$ S_n(f_{\sigma(1)}\otimes\cdots\otimes f_{\sigma(n)}) ={\rm sgn}(\sigma)\times S_n(f_1\otimes\cdots\otimes f_n)\ . $$ For the latter one needs Grassmann-Berezin "integrals".

I don't know a good definition of the Fermionic analogue of $\int\cdots dx$ in infinite dimension and on the continuum. One can however define the Fermionic analogue of $\int\cdots e^{-x^2}\ dx$ using the result of the Isserlis-Wick theorem (with signs) as a definition. Over a finite lattice, one can rigorously define the Berezin integral, i.e., the analogue of $\int\cdots dx$.

See the thesis by Andrew Swan for an introduction to Grassmann-Berezin integration https://www.repository.cam.ac.uk/handle/1810/324960

See also the Aisenstadt Lectures by Joel Feldman https://personal.math.ubc.ca/~feldman/papers/aisen-all.pdf

There is also the book "Non-perturbative Renormalization" by Vieri Mastropietro on rigorous renormalization group techniques for Fermionic theories.