$\theta$-splitting densities and reflection positivity

Abstract

A simple condition is given that is sufficient to determine whether a measure that is absolutely continuous with respect to a Gaußian measure on the space of distributions is reflection positive. It readily generalises conventional lattice results to an abstract setting, enabling the construction of many reflection positive measures that are not supported on lattices.

1 Introduction

Reflection positivity is one of the pillars of Euclidean quantum field theories. It is readily established for wide sets of Gaußian measures but for non-Gaußian measures, the author feels that—with the exception of measures supported on lattices—there is no general framework that can be easily applied. For measures that are absolutely continuous with respect to Gaußian measures, that is fixed in this article by introducing the set of $\theta$-splitting functions, which can work as densities to directly generalise the lattice methods used, e.g. in [1]. The result is very simple: Given a $\theta$-invariant reflection positive Gaußian measure and applying a measurable density to it that is $\theta$-splitting, the outcome is a reflection positive measure.

In general, physically relevant measures in $d \ge 3$ dimensions are typically not absolutely continuous with respect to the Gaußian free field measure. Hence, one still needs to find ways to regularise the models of interest in order to apply the theorems in this work. However, reflection positivity is preserved by the weak convergence of measures (see, e.g. [6]), since it implies the pointwise convergence of corresponding characteristic functions. Hence, any limit point of a sequence of regularised models corresponding to reflection positive measures is reflection positive as well.

2 Preliminaries

A locally convex space is a real topological vector space whose topology is induced by some family of seminorms. The dual of a locally convex space X equipped with the strong dual topology will be denoted by $X^*_\beta$. Inner products denoted with round brackets $(\cdot ,\cdot )$ are taken to be $\mathbb {R}$-bilinear. Throughout this work, $d \in \mathbb {N}$ is fixed. We shall work on the spaces

$$\begin{aligned} \mathcal {D} := \mathcal {D}(\mathbb {R}^{d+1}) \qquad \text {and}\qquad \mathcal {D}_+ := \mathcal {D}(\mathbb {R}_{>0} \times \mathbb {R}^d). \end{aligned}$$

(1)

of real test functions with their canonical LF topologies [2, p. 131–133]. Let us denote the corresponding continuous restriction map by $\pi _+: \mathcal {D}^*_\beta \rightarrow (\mathcal {D}_+)^*_\beta$ (see, e.g. [2, p. 245–246]). $\mathcal {D}$ and $\mathcal {D}_+$ as well as their strong duals $\mathcal {D}^*_\beta$ and $(\mathcal {D}_+)^*_\beta$ are complete [2, Theorem 13.1], barrelled [2, p. 347], nuclear spaces [2, p. 530] (hence, reflexive by [3, p. 147]) that are also Lusin spaces [4, p. 128] and thus in particular Souslin spaces.

Theorem 2.1

[5, Lemma 6.4.2.(ii), Lemma 6.6.4] Let X and Y be Souslin spaces. Then, the Borel $\sigma$-algebra of $X \times Y$ coincides with the $\sigma$-algebra generated by all products of Borel sets in X and Y, respectively.

In this work, a measure is taken to be a countably additive nonnegative function on a $\sigma$-algebra. A Borel measure is thus a measure on a Borel $\sigma$-algebra and a Radon measure is a Borel measure that is inner regular over compact sets. A centred Gaußian measure on a locally convex space X is a Borel probability measure with the property that the pushforward measures by elements of $X^*$ are centred Gaußians or the Dirac delta measure $\delta _0$ at the origin. One can in general consider non-Radon Gaußian measures on locally convex spaces. However, every Borel measure on the spaces $\mathcal {D}^*_\beta , (\mathcal {D}_+)^*_\beta$ and countable products thereof is automatically Radon [5, Theorem 7.4.3].

A subset $A \subseteq X$ is $\varvec{\mu }$-measurable with respect to a measure $\mu$ on some $\sigma$-algebra $\mathcal {A}$ on X if it is in the Lebesgue completion $\mathcal {A}_\mu$ of $\mathcal {A}$ with respect to $\mu$. Similarly, a function $f: X \rightarrow [-\infty , \infty ]$ is $\varvec{\mu }$-measurable if the preimage of every Borel subset of $[-\infty ,\infty ]$ is in $\mathcal {A}_\mu$. Likewise, $f: X \rightarrow [-\infty , \infty ]$ is $\varvec{\mu }$-integrable if f is $\mu$-measurable and $\int |f| \text {d}\mu < \infty$. A subset $A \subseteq X$ is $\varvec{\mu }$-negligible if it is a subset of some $B \in \mathcal {A}$ with $\mu (B) = 0$.

The pushforward of a Borel measure $\mu$ on a Hausdorff space X by a continuous function $f: X \rightarrow Y$ to a Hausdorff space Y will be denoted by $f_* \mu$. It is automatically a Borel measure on Y and if $\mu$ is Radon, so is $f_* \mu$ [5, Theorem 9.1.1.(i)]. The convolution of two Borel measures $\mu$ and $\nu$ on a Souslin locally convex space X is given by $\mu * \nu = s_*(\mu \times \nu )$ where $s: X \times X \rightarrow X, (x,y) \mapsto x+y$. This is well defined by Theorem 2.1.

To every finite Borel measure $\mu$ on a locally convex space X, we associate its characteristic function $\hat{\mu }: X^* \rightarrow \mathbb {C}$ with

$$\begin{aligned} \phi \mapsto \int _X \exp \left[ i \phi \left( x \right) \right] \text {d} \mu \left( x \right) . \end{aligned}$$

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It is well known that two Radon measures on a locally convex space are equal if and only if their characteristic functions are equal [5, Lemma 7.13.5]. Moreover, if $\mu$ is a centred Gaußian measure on X, its characteristic function is given by

$$\begin{aligned} \hat{\mu } \left( \phi \right) = \exp \left[ - \frac{1}{2} \left( \phi , \phi \right) _{L^2(\mu )} \right] \end{aligned}$$

(3)

for all $\phi \in X^*$ [6, Theorem 2.2.4, Corollary 2.2.5].

Theorem 2.2

Let $f: X \rightarrow Y$ be a continuous map from a Souslin space X to a Hausdorff space Y. Then, for every Borel set $B \subseteq X$, f(B) is measurable by any Radon measure on Y.

Proof

Since every Borel subset of a Souslin space is Souslin [4, p. 96 Theorem 3], this follows directly from [6, Theorem A.3.15]. $\square$

Corollary 2.3

Let $p: X \rightarrow Y$ be a continuous map from a Souslin space X to a Hausdorff space Y and $\mu$ a Radon measure on X. Then, every function $f: Y \rightarrow [-\infty ,\infty ]$ with the property that $f \circ p$ is $\mu$-measurable is $(p_*\mu )$-measurable.

Proof

First, note that p(X) is $(p_*\mu )$-measurable by the preceeding theorem. Now, letting $B \subset [-\infty ,\infty ]$ be a Borel set, we have

$$\begin{aligned} p^{-1} \left( f^{-1} \left( B \right) \right) = A \cup N_1 \end{aligned}$$

(4)

for some Borel subset $A \subseteq X$ and some $\mu$-negligible set $N_1 \subseteq X$. For brevity, let $N_2 = Y {\setminus } p(X)$, which is clearly $(p_* \mu )$-negligible. Then,

$$\begin{aligned} \begin{aligned} f^{-1} \left( B \right)&= \left[ f^{-1} \left( B \right) \cap p(X) \right] \cup \left[ f^{-1} \left( B \right) \cap N_2 \right] \\&= p \left( p^{-1} \left( f^{-1} \left( B \right) \right) \right) \cup \left[ f^{-1} \left( B \right) \cap N_2 \right] \\&= p \left( A \right) \cup p \left( N_1 \right) \cup \left[ f^{-1} \left( B \right) \cap N_2 \right] . \end{aligned} \end{aligned}$$

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p(A) is $(p_* \mu )$-measurable by the preceeding theorem and $p (N_1 )$ as well as $f^{-1} ( B ) \cap N_2$ are clearly $(p_* \mu )$-negligible. $\square$

We close this section by a simple lemma on positive semidefinite matrices.

Lemma 2.4

[7, Satz VII] Let $N \in \mathbb {N}$ and A, B be positive semidefinite $N \times N$ matrices with respect to the standard inner product on $\mathbb {C}^N$. Then, the matrix $(A_{m,n} B_{m,n})_{m,n = 1}^N$ given by component-wise multiplication is positive semidefinite.

Proof

Diagonalising B by a unitary matrix U, we obtain

$$\begin{aligned} B_{m,n} = \sum _{a = 1}^N U_{m,a}^* \lambda _a U_{n,a} \end{aligned}$$

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for some nonnegative numbers $\lambda _1, \dots , \lambda _N$. Hence, for any $c \in \mathbb {C}^N$,

$$\begin{aligned} \sum _{m,n,a,b = 1}^N c_m^* A_{m,n} B_{m,n} c_n = \sum _{a = 1}^N \lambda _a \sum _{m,n = 1}^N \left( U_{m,a} c_m \right) ^* A_{m,n} \left( U_{n,a} c_n \right) \ge 0. \end{aligned}$$

(7)

$\square$

3 Reflection positivity

On $\mathbb {R}^{d+1}$, we define the operation of time reflection which we shall denote by $\theta : \mathbb {R}^{d+1} \rightarrow \mathbb {R}^{d+1}, (x_1, \dots , x_{d+1}) \mapsto (-x_1, x_2, \dots , x_{d+1})$. By a slight abuse of notation, $\theta$ extends continuously and linearly to $\mathcal {D}$ and $\mathcal {D}^*_\beta$ in the obvious way.

Definition 3.1

[1, p. 90] Let $\mu$ be a finite Borel measure on $\mathcal {D}^*_\beta$. Then, $\mu$ is reflection positive if for every sequence $(\phi _n)_{n \in \mathbb {N}}$ in $\mathcal {D}_+$, every sequence $(c_n)_{n \in \mathbb {N}}$ of complex numbers and every $N \in \mathbb {N}$,

$$\begin{aligned} \sum _{m,n = 1}^N c_m^* \hat{\mu } \left( \phi _m - \theta \phi _n \right) c_n \ge 0. \end{aligned}$$

(8)

Furthermore, $\mu$ is $\varvec{\theta }$-invariant if $\theta _* \mu = \mu$.

To begin with, let us recapitulate two of the most important (in the author’s opinion) theorems on reflection positive measures along with their proofs.

Theorem 3.2

[1, Theorem 6.2.3] Let $\mu$ be a finite, reflection positive Borel measure on $\mathcal {D}^*_\beta$ with the property that for every $\phi \in \mathcal {D}_+$ the function $\mathbb {R} \rightarrow \mathbb {C}, t \mapsto \hat{\mu }(t \phi )$ has an analytic continuation to some neighbourhood of zero in the complex plane. Then, $( \phi , \theta \phi )_{L^2(\mu )} \ge 0$ for all $\phi \in \mathcal {D}_+$.

Proof

For $\lambda > 0$ let $\psi _1 = \lambda \phi$, $\psi _2 = 0$, $c_1 = \lambda ^{-1}$ and $c_2 = -\lambda ^{-1}$. Since $\mu$ is reflection positive, we obtain

$$\begin{aligned} \begin{aligned} 0&\le \sum _{m,n = 1}^{2} c_m^* \hat{\mu } \left( \psi _m - \theta \psi _n \right) c_n \\&= \frac{1}{\lambda ^2} \int _{\mathcal {D}^*_\beta } \left( \exp \left[ i \lambda T \left( \phi - \theta \phi \right) \right] - \exp \left[ - i \lambda T \left( \phi \right) \right] - \exp \left[ - i \lambda T \left( \theta \phi \right) \right] + 1 \right) \text {d} \mu \left( T \right) . \end{aligned} \end{aligned}$$

(9)

By a classical theorem of Lukacs [8, p. 192], the moment-generating functions of the pushforward measures $\phi _* \mu$, $(\theta \phi )_* \mu$ and $(\phi - \theta \phi )_* \mu$ exist as integrals in some neighbourhood of zero. Consequently, we can take $\lambda \rightarrow 0$ under the integral and obtain

$$\begin{aligned} \lim _{\lambda \rightarrow 0} \sum _{m,n = 1}^{2} c_m^* \hat{\mu } \left( \psi _m - \theta \psi _n \right) c_n = \int _{\mathcal {D}^*_\beta } T \left( \phi \right) T \left( \theta \phi \right) \text {d} \mu \left( T \right) = \langle \phi , \theta \phi \rangle _{L^2(\mu )} \ge 0. \end{aligned}$$

(10)

$\square$

Theorem 3.3

[1, Theorem 6.2.2] Let $\mu$ be a $\theta$-invariant Gaußian measure on $\mathcal {D}^*_\beta$. Then, $\mu$ is reflection positive if and only if $( \phi , \theta \phi )_{L^2(\mu )} \ge 0$ for all $\phi \in \mathcal {D}_+$.

Proof

$\Rightarrow$: This is clear by the preceeding theorem.

$\Leftarrow$: Let $(\cdot , \cdot )$ denote the inner product in $L^2(\mu )$ and let $(\phi _n)_{n \in \mathbb {N}}$ be a sequence in $\mathcal {D}_+$, $(c_n)_{n \in \mathbb {N}}$ a sequence of complex numbers and $N \in \mathbb {N}$. Then, $\theta$-invariance implies

$$\begin{aligned} \sum _{m,n = 1}^{N} c_m^* \hat{\mu } \left( \psi _m - \theta \psi _n \right) c_n = \sum _{m,n = 1}^{N} c_m^* \hat{\mu } \left( \phi _m \right) \exp \left[ ( \phi _m, \theta \phi _n ) \right] \hat{\mu } \left( \phi _n \right) c_n. \end{aligned}$$

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Since $\hat{\mu }$ is real, the statement follows if $( \exp \left[ ( \phi _m, \theta \phi _n ) \right] )_{m,n = 1}^N$ is a positive semidefinite matrix. Since $( \phi _m, \theta \phi _n ) = ( \theta \phi _m, \phi _n )$ by the $\theta$-invariance of $\mu$, $\theta$ extends to a positive semidefinite linear operator on the complexification of $\text {span} \{ \phi _n: n \in \mathbb {N} \}$. Consequently, $(( \phi _m, \theta \phi _n ) )_{m,n = 1}^N$ is positive semidefinite. By decomposing the exponential as a power series, the claim now follows from Lemma 2.4. $\square$

The main theorem of this article depends on the following simple property of a function with respect to $\theta$.

Definition 3.4

A function $F: \mathcal {D}^*_\beta \rightarrow [-\infty ,\infty ]$ is called $\varvec{\theta }$-splitting if there exists a function $G: (\mathcal {D}_+)^*_\beta \rightarrow [-\infty ,\infty ]$ such that

$$\begin{aligned} F = G \circ \pi _+ + G \circ \pi _+ \circ \theta . \end{aligned}$$

(12)

Theorem 3.5

Let $\mu$ be a $\theta$-invariant reflection positive centred Gaußian measure on $\mathcal {D}^*_\beta$. Then, for any $\mu$-measurable $\theta$-splitting function $F: \mathcal {D}^*_\beta \rightarrow [-\infty ,\infty ]$ with $exp \circ F \in L^1(\mu )$, the finite Borel measure

$$\begin{aligned} \omega = \exp \left[ F \right] \cdot \mu \end{aligned}$$

(13)

is reflection positive.

Proof

Define

$$\begin{aligned} j : \mathcal {D}^*_\beta \rightarrow ( \mathcal {D}_+ )^*_\beta \times ( \mathcal {D}_+ )^*_\beta \qquad T \mapsto \left( \pi _+ T , \pi _+ \theta T \right) . \end{aligned}$$

(14)

j is clearly continuous such that the pushforward measure $j_* \mu$ is a Radon measure $\nu$ on $( \mathcal {D}_+ )^*_\beta \times ( \mathcal {D}_+ )^*_\beta$. Now, let

$$\begin{aligned} F_2 : ( \mathcal {D}_+ )^*_\beta \times ( \mathcal {D}_+ )^*_\beta \rightarrow \mathbb {R} \qquad \left( T, K \right) \mapsto G \left( T \right) + G \left( K \right) . \end{aligned}$$

(15)

Then, for every $T \in \mathcal {D}^*_\beta$,

$$\begin{aligned} \left( F_2 \circ j \right) \left( T \right) = G \left( \pi _+ T \right) + G \left( \pi _+ \theta T \right) = F \left( T \right) , \end{aligned}$$

(16)

such that $F_2$ is $\nu$-measurable by Corollary 2.3. Turning to reflection positivity, let $(\phi _n)_{n \in \mathbb {N}}$ be a sequence in $\mathcal {D}_+$ and note that

$$\begin{aligned} \begin{aligned} \hat{\omega } \left( \phi _m - \theta \phi _n \right)&= \int _{\mathcal {D}^*_\beta } \exp \left[ i T \left( \phi _m \right) - i T\left( \theta \phi _n \right) + F \left( T \right) \right] \text {d} \mu \left( T \right) \\&= \int _{\mathcal {D}^*_\beta } \exp \left[ i \; j \left( T \right) \left( \phi _m, -\phi _n \right) + \left( F_2 \circ j \right) \left( T \right) \right] \text {d} \mu \left( T \right) \\&= \int _{(( \mathcal {D}_+ )^*_\beta )^2} \exp \left[ i T \left( \phi _m \right) - i K \left( \phi _n \right) + F_2 \left( T, K \right) \right] \text {d} \nu \left( T, K \right) \\&= \int _{(( \mathcal {D}_+ )^*_\beta )^2} \exp \left[ i T \left( \phi _m \right) - i K \left( \phi _n \right) + G \left( T \right) + G \left( K \right) \right] \text {d} \nu \left( T, K \right) . \end{aligned} \end{aligned}$$

(17)

The above expression suggests to find a disintegration of $\nu$ that separates the T and K variables. To that end, recall that $\mu$ is Gaußian such that for any $\phi , \psi \in \mathcal {D}_+$, we have

$$\begin{aligned} \hat{\nu } \left( \phi , \psi \right) = \int _{\mathcal {D}^*_\beta } \exp \left[ i T \left( \phi \right) + i T \left( \theta \psi \right) \right] \text {d} \mu \left( T \right) = \exp \left[ - \frac{1}{2} \left\| \phi + \theta \psi \right\| _{L^2(\mu )}^2 \right] . \end{aligned}$$

(18)

Furthermore, by Theorem 3.2, Cauchy–Schwartz and the $\theta$-invariance of $\mu$,

$$\begin{aligned} 0 \le \langle \phi , \theta \phi \rangle _{L^2(\mu )} \le \langle \phi , \phi \rangle _{L^2(\mu )}. \end{aligned}$$

(19)

Moreover, since $(\mathcal {D}_+)^*_\beta$ is a reflexive, nuclear, barrelled space, there exist uniquely determined Radon Gaussian measures P and Q on $(\mathcal {D}_+)^*_\beta$ with

$$\begin{aligned} \hat{P} \left( \phi \right)&= \exp \left[ - \frac{1}{2} \langle \phi , \phi \rangle _{L^2(\mu )} + \frac{1}{2} \langle \phi , \theta \phi \rangle _{L^2(\mu )} \right] , \end{aligned}$$

(20)

$$\begin{aligned} \hat{Q} \left( \phi \right)&= \exp \left[ - \frac{1}{2} \langle \phi , \theta \phi \rangle _{L^2(\mu )} \right] \end{aligned}$$

(21)

by Minlos theorem [5, Theorem 7.13.9]. Defining the diagonal map

$$\begin{aligned} \Delta : (\mathcal {D}_+)^*_\beta \rightarrow (\mathcal {D}_+)^*_\beta \times (\mathcal {D}_+)^*_\beta \qquad T \mapsto \left( T, T \right) \end{aligned}$$

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it is clear that

$$\begin{aligned} \hat{\nu } \left( \phi , \psi \right) = \hat{P} \left( \phi \right) \hat{P} \left( \psi \right) \hat{Q} \left( \phi + \psi \right) = \hat{P} \left( \phi \right) \hat{P} \left( \psi \right) \widehat{\Delta _* Q} \left( \phi , \psi \right) \end{aligned}$$

(23)

for all $\phi , \psi \in \mathcal {D}_+$. Equivalently, $\nu = (P\times P) * (\Delta _* Q)$ by Theorem 2.1. Hence, it is straightforward to verify that

$$\begin{aligned}{} & {} \hat{\omega } \left( \phi _m - \theta \phi _n \right) = \int _{(( \mathcal {D}_+ )^*_\beta )^3} \exp \big [ i \left( T + L \right) \left( \phi _m \right) - i \left( K + L \right) \left( \phi _n \right) \nonumber \\{} & {} \qquad \qquad \qquad + G \left( T + L \right) + G \left( K + L \right) \big ] \text {d} \left( P \times P \times Q \right) \left( T, K, L \right) . \end{aligned}$$

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Now, the functions

$$\begin{aligned} H_m \left( L \right) = \int _{( \mathcal {D}_+ )^*_\beta } \exp \left[ - i \left( T + L \right) \left( \phi _m \right) + G \left( T + L \right) \right] \text {d} P \left( T \right) \end{aligned}$$

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for $m \in \mathbb {N}$ are well-defined Q-almost everywhere. Thus, using Fubini, we arrive at

$$\begin{aligned} \sum _{m,n = 1}^N c_m^* \hat{\omega } \left( \phi _m - \theta \phi _n \right) c_n = \int _{( \mathcal {D}_+ )^*_\beta } \left| \sum _{n=1}^N c_n H_n \left( L \right) \right| ^2 \text {d} Q \left( L \right) \ge 0 \end{aligned}$$

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for any $N \in \mathbb {N}$ and any sequence $(c_n)_{n \in \mathbb {N}}$ of complex numbers. $\square$

This theorem is strikingly simple and can be applied very easily. Let us call a locally convex space X together with a continuous, linear map $j: X \rightarrow \mathcal {D}^*_\beta$ a $\varvec{\theta }$-model space, if there is a continuous, linear operator (slight abuse of terminology) $\theta : X \rightarrow X$ such that $\theta \circ j = j \circ \theta$.

Example 3.6

Examples of such $\theta$-model spaces are, e.g. function spaces on $\theta$-symmetric lattice subsets of $\mathbb {R}^{d+1}$, $\mathcal {D}$ or the space of Schwartz functions on $\mathbb {R}^{d+1}$ together with their respective usual injections into $\mathcal {D}^*_\beta$.

Remark 3.7

The above examples cover most of what is used in the literature on Euclidean interacting quantum field theories and are also Souslin spaces.

We may now extend the definition of a $\varvec{\theta }$-splitting function to $\theta$-model spaces.

Definition 3.8

A function $F: X \rightarrow [-\infty ,\infty ]$ on a $\theta$-model space (X, j) is called $\varvec{\theta }$-splitting if there exists a function $G: X \rightarrow [-\infty ,\infty ]$ such that

$$\begin{aligned} F = G \circ \pi ^X_+ + G \circ \pi ^X_+ \circ \theta \end{aligned}$$

(27)

Here, $\pi ^X_+: X \rightarrow X / j^{-1}(\ker \pi _+)$ is the canonical quotient map.

Corollary 3.9

Let (X, j) be a Souslin $\theta$-model space. Furthermore, let $\mu$ be a Gaußian measure on X with the property that $j_* \mu$ is $\theta$-invariant and reflection positive. Then, for any $\mu$-measurable $\theta$-splitting function $F: X \rightarrow [-\infty ,\infty ]$ with $exp \circ F \in L^1(\mu )$, the finite Borel measure

$$\begin{aligned} \omega = j_* \left( \exp \left[ F \right] \cdot \mu \right) \end{aligned}$$

(28)

is reflection positive.

Proof

Let G and $\pi ^X_+$ be given as in Definition 3.8 and define the function $G_2: (\mathcal {D}_+)^*_\beta \rightarrow [-\infty ,\infty ]$ given by

$$\begin{aligned} T \mapsto {\left\{ \begin{array}{ll} G( \pi ^X_+ x ) &{} \text {if } \exists \, x \in X : T = \pi _+ j x \\ 0 &{} \text {else.} \end{array}\right. } \end{aligned}$$

(29)

To see that $G_2$ is well defined, note that if $\pi _+ j x = \pi _+ j y$ for some $x,y \in X$, we have that there is some $T \in \ker \pi _+$ with $j (x - y) = T$, i.e. $x - y \in j^{-1}(\ker \pi _+) = \text {ker} \, \pi ^X_+$. Now, define the function $F_2: \mathcal {D}^*_\beta \rightarrow [-\infty ,\infty ]$ given by

$$\begin{aligned} T \mapsto G_2 \left( \pi _+ T \right) + G_2 \left( \pi _+ \theta T \right) . \end{aligned}$$

(30)

Clearly, $F_2 \circ j = F$ such that $F_2$ is $(j_* \mu )$-measurable by Corollary 2.3. Consequently, $\omega = \exp [F_2] \cdot (j_* \mu )$ and Theorem 3.5 applies. $\square$

We finish this article by a simple example.

Example 3.10

Let $\mathcal {S}$ denote the space of Schwartz functions on $\mathbb {R}^{d+1}$. Define $j: \mathcal {S} \rightarrow \mathcal {D}^*_\beta$ by $j(\phi )(\psi ) = \int _{\mathbb {R}^{d+1}} \psi \phi$ for all $\phi \in \mathcal {S}$ and $\psi \in \mathcal {D}$. Moreover, let $\mu$ be a Gaußian measure on $\mathcal {S}$ with the property that $j_* \mu$ is $\theta$-invariant and reflection positive. Note that this excludes the Gaußian measure on the space $\mathcal {S}^*$ of tempered distributions modelling the Klein-Gordon field. However, regularised versions of that measure will work, see, e.g. [9, Example 6.2]. Furthermore, let $F: \mathcal {S} \rightarrow \mathbb {R}, \phi \mapsto -\lambda \int _{\mathbb {R}^{d+1}} \phi ^4$ for some $\lambda > 0$. Then,

$$\begin{aligned} F(\phi ) = -\lambda \int _{\mathbb {R}_{> 0} \times \mathbb {R}^d} \phi ^4 -\lambda \int _{\mathbb {R}_{> 0} \times \mathbb {R}^d} \left( \theta \phi \right) ^4 \end{aligned}$$

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provides a $\theta$-splitting of F.