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? $$ S_n (x_{\pi(1)}, \cdots, x_{\pi(n)} ) = sign(\pi)S_n(x_1, \cdots, x_n) $$ 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?

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?

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 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 to be 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? $$ S_n (x_{\pi(1)}, \cdots, x_{\pi(n)} ) = sgn(\pi)S_n(x_1, \cdots, x_n) \text{ where } \pi \text{ is a permutation of $n$ elements.} $$

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?

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? $$ S_n (x_{\pi(1)}, \cdots, x_{\pi(n)} ) = sign(\pi)S_n(x_1, \cdots, x_n) $$ 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?