A Remark on Reflection Positivity

As the names would suggest, the positivity of the Dirichlet-to-Neumann map (itself the consequence of the positivity of the Dirichlet energy) gives an interesting inequality comparing the resolvants of Laplacians with Dirichlet and Neumann conditions (Corollary A.2), via the Poisson integral formula (Lemma 4.6). Let \(\Omega \), \(\partial \Omega \) be as in Lemma 4.6. We adopt the shorthand notations

$$\begin \quad C_D\overset}(\Delta _+m^2)^,\quad C_N\overset}(\Delta _+m^2)^, \end$$

(A.1)

where, similar to \(C_D\), \(C_N f\) for \(f\in C_c^(\Omega ^)\) solves the Neumann boundary value problem

$$\begin \left\ (\Delta +m^2)(C_N f)=f,& \text \Omega ,\\ \partial _(C_N f)|_=0,& \text \partial \Omega . \end \right. \end$$

(A.2)

There is the following simple, elementary relation:

Lemma A.1

In the situation as above, we have the operator equality on \(C_c^(\Omega ^)\),

$$\begin \,}}_^(\,}}_^)^(\,}}_^)^*=C_N-C_D. \end$$

(A.3)

Proof

Pick \(f\in C_c^(\Omega ^)\), let \(u:=C_D f\) and put \(w:=\,}}_^(\,}}_^)^(-\partial _ u|_)\). Then, by the definition of \(\,}}_^\), w solves the following boundary value problem:

$$\begin \left\ (\Delta +m^2)w=0,& \text \Omega ,\\ \partial _w|_=-\partial _u|_,& \text \partial \Omega . \end \right. \end$$

(A.4)

However,

$$\begin \left\ (\Delta +m^2)u=f,& \text \Omega ,\\ u|_=0,& \text \partial \Omega , \end \right. \quad \text \quad \left\ (\Delta +m^2)(u+w)=f,& \text \Omega ,\\ \partial _(u+w)|_=0,& \text \partial \Omega , \end \right. \nonumber \\ \end$$

(A.5)

namely \(u+w=C_N f\), that is, \(w=C_N f-C_D f\). We obtain the result. \(\square \)

Now the positivity of \(\,}}_^\) (Lemma 4.3) implies \(C_N\geqslant C_D\), namely \(\left\langle f,(C_N-C_D)f \right\rangle _\geqslant 0\) for all \(f\in C_c^(\Omega ^)\). One step further,

Corollary A.2

We have \(C_N\geqslant C_D\) as operators on \(L^2(\Omega )\).  \(\Box \)

It is emphasized in Jaffe and Ritter [30] section 3 that \(C_N\geqslant C_D\) is the crucial relation that leads to the so-called reflection positivity (RP) of the GFF. The geometric setup is as follows. Let now \(\partial \Omega \subset \Omega \) be totally geodesic, \(\Omega ^*\) a copy of \(\Omega \) (reversing the coorientation of \(\partial \Omega \)), \(|\Omega |^2:=\Omega ^*\cup _\Omega \), the isometric double which is a closed Riemannian manifold, and \(\Theta :|\Omega |^2\longrightarrow |\Omega |^2\) an isometric involution fixing \(\partial \Omega \), such that \(\Theta (\Omega )=\Omega ^*\) and \(\Theta (\Omega ^*)=\Omega \).

Remark A.1

Such \(\Omega \) is named in Gibbons [20] (in the 4-dimensional case) as a real tunneling geometry (see [20] section 4). The isometric double \(|\Omega |^2\) and the involution \(\Theta \) exist, the latter being a reflection, dissecting \(|\Omega |^2\) into \(\Omega \) and \(\Omega ^*\), its fixed point set being \(\partial \Omega \) (see also Ritter [42] section 2.1.1).

The action of \(\Theta \) extends in the usual way to \(C^(|\Omega |^2)\) and \(\mathcal '(|\Omega |^2)\) by pulling-back. This setup brings in another resolvant operator which is

$$\begin C\overset}(\Delta _+m^2)^. \end$$

(A.6)

Denote also by \(\Pi _+:L^2(|\Omega |^2)\longrightarrow L^2(\Omega )\) the orthogonal projection. Then we have

Lemma A.3

([30] lemma 3) Let \(|\Omega |^2\), \(\Theta \), \(\Pi _+\) and C be as above. Then

$$\begin \Pi _+ \Theta C=\frac(C_N-C_D) \end$$

(A.7)

on \(C_c^(\Omega ^)\) and \(L^2(\Omega )\). \(\Box \)

In summary,

Corollary A.4

(equivalent formulations of RP) In the situation as above, we have

$$\begin 2\big \langle f,\Theta Cf \big \rangle _= & \big \langle f,(C_N-C_D)f \big \rangle _\nonumber \\= & \big \langle (\,}}_^)^*f,(\,}}_^)^(\,}}_^)^* f \big \rangle _ \end$$

(A.8)

for all \(f\in C_c^(\Omega ^)\), and all of the above quantities are nonnegative.  \(\Box \)

Some Recollections1.1 Wick Ordering

Let \((Q,\mathcal ,\mathbb )\) be a probability space and \(\mathcal \subset L^2(Q,\mathcal ,\mathbb )\) a Gaussian Hilbert space (closed subspace of Gaussian variables). The Wiener Chaos decomposition ([28] Theorem 2.6) says that there is an orthogonal decomposition

$$\begin L^2(Q,\mathcal (\mathcal ),\mathbb )\cong \bigoplus _^\mathcal ^n}, \end$$

(B.1)

where \(\mathcal (\mathcal )\) is the \(\sigma \)-algebra generated by variables in \(\mathcal \), \(\mathcal ^n}=\overline}_n(\mathcal )\cap \overline}_(\mathcal )^\), where \(\mathcal _j(\mathcal )\) denotes the span of polynomials of random variables in \(\mathcal \) of degree \(\leqslant j\); in particular \(\mathcal ^0}\) denotes the constants.

If \(F\in \overline}_n(\mathcal )\), denote by  : F :  the projection of F onto \(\mathcal ^n}\), and is called a Wick ordered polynomial. Define the Hermite polynomials \(h_n(x)\) by

$$\begin \exp \Big ( zx-\fracz^2 \Big )=\sum _^\frach_n(x). \end$$

(B.2)

We have \(h_0(x)=1\), \(h_2(x)=x^2-1\), \(h_4(x)=x^4-6x^2+3\), etc. Computation rules for Wick ordering generally go under the name Wick’s theorem or Feynman rules. Useful formulae include

$$\begin&X^n=\sum _^ \frac\mathbb [X^2]^j X^=\mathbb [X^2]^} h_n\big (X\big /\mathbb [X^2]^}\big ), \end$$

(B.3)

$$\begin&\mathbb [X^n~Y^m]=\delta _n!\mathbb [XY]^n, \end$$

(B.4)

for X, \(Y\in \mathcal \). See [52] Propositions I.2-4, or [28] Theorems 1.28, 3.9, 3.19.

1.2 Sobolev Spaces over Domains

In this paper we make essential use of the usual \(L^2\) Sobolev spaces over Riemannian manifolds. First let (M, g) be a closed Riemannian manifold and \(s\in \mathbb \). Then the Sobolev space \(W^s(M)\) of order s is defined generally as the closure of \(C^(M)\) under a norm \(\left\Vert \cdot \right\Vert _\), where the norm \(\left\Vert \cdot \right\Vert _\) could be defined in various equivalent ways. We refer to Taylor [58] chapter 4 for a general discussion. For us, \(s=\pm 1\), \(\pm \frac\). We rely heavily on the following fact.

Lemma B.1

Let \(\Lambda _\) be an elliptic strictly positive formally self-adjoint pseudodifferential operator on M with order 2s. Then the inner product

$$\begin \left\langle -,- \right\rangle _\overset}\left\langle -,\Lambda _- \right\rangle _ \end$$

(B.5)

induces an equivalent norm for \(W^s(M)\).

In particular, the real power \((\Delta _M+m^2)^s\) of the Helmholtz operator (massive Laplacian) \(\Delta _M+m^2\) provides such a candidate for \(\Lambda _\). Theorem 4.3.1 of [54] shows that \((\Delta _M+m^2)^s\) is a \(\Psi \)DO with the required properties. Convention: whenever we use the space \(W^s(M)\), the inner product (B.5) with \(\Lambda _=(\Delta _M+m^2)^s\) is understood, unless otherwise specified.

Next we discuss important subspaces of \(W^s(M)\). Let \(A\subset M\) be a closed set and \(U\subset M\) an open set. Define

$$\begin W^s_A(M)&\overset}\\,}}u\subset A\text \},\end$$

(B.6)

$$\begin W^s_U(M)&\overset}\text C_c^(U)\text W^s(M), \end$$

(B.7)

$$\begin W^s(U)&\overset}W^s_(M)^\subset W^s(M). \end$$

(B.8)

These are closed subspaces of \(W^s(M)\).

Remark B.1

We point out right away that by definition, then,

$$\begin W^s(U)\cong W^s(M)/W^s_(M), \end$$

(B.9)

the latter equipped with the quotient norm, which is a more familiar characterization of \(W^s(U)\), see Taylor [58] page 339. Our definition as in (B.8) poses the obvious problem that in general \(C_c^(U)\not \subset W^s(U)\), at least for \(s\not \in \mathbb _+\). We emphasize therefore that what is important in this definition is not the space \(W^s(U)\) per se but the following choice for its inner product:

$$\begin \big \langle f,h \big \rangle _\overset}\big \langle P_^ f,P_^ h \big \rangle _, \end$$

(B.10)

for any f, \(h\in W^s(M)\), in particular for f, \(h\in C_c^(U)\), which produces a norm equivalent to the quotient norm, where \(P_^:W^s(M)\longrightarrow W^s(U)\) denotes the orthogonal projection.

We remark that inclusion relations between (B.6–B.8) is a delicate question. See for example [58] page 339 and section 4.7.

The rest of this appendix could be read along with Sect. 5.1. Let \(s=-1\). Although \(C_c^(U)\not \subset W^(U)\),

Lemma B.2

Let \(U\subset M\) be an open set. Then \(P_^(C_c^(U))\) is dense in \(W^(U)\).

Proof

We note \(\Delta _M+m^2\) is local and therefore \((\Delta +m^2)(C_c^(U))\subset C_c^(U)\). It follows from Remark 5.1 and our definition of \(W_U^1(M)\) that \((\Delta +m^2)(C_c^(U))\subset W^_(M)^\) and is dense there, proving the result. \(\square \)

Next we state the duality results for the various spaces. Recall that \(\left\langle -,- \right\rangle _\) denotes both the inner product of \(L^2(M)\) and the distributional pairing between \(\mathcal '(M)\) and \(C^(M)\). Below, we extend it to denote also the pairing between dual Sobolev spaces (see (i) of the lemma below).

Lemma B.3

Let M be a closed Riemannian manifold, \(U\subset M\) an open set, \(A\subset M\) a closed set, and \(s\in \mathbb \).

(i)

\(W^(M)\) is the dual Banach space, denoted \(W^s(M)^*\), of \(W^s(M)\) under \(\left\langle -,- \right\rangle _\);

(ii)

the annihilator of \(W^s_U(M)\) under \(\left\langle -,- \right\rangle _\) is \(W^_(M)\), that is,

$$\begin W^_(M)=\(M)~|~\left\langle u,f \right\rangle _=0\text f\in W^s_U(M)\}; \end$$

(B.11)

the annihilator of \(W^s_A(M)\) is accordingly \(W^_(M)\);

(iii)

\(W^s(U)^*\cong W^_U(M)\), \(W^s_U(M)^*\cong W^(U)\), these spaces being therefore reflexive.

Finally, when \(\Omega \subset M\) is a domain with smooth boundary \(\partial \Omega \), we define, in view of Lemma 5.1, the Dirichlet Green operator \((\Delta _+m^2)^:=(\Delta +m^2)^P_}^:W^(\Omega ^)\longrightarrow W_}^1(M)\). Clearly this agrees with the usual definition. In terms of quadratic forms,

Lemma B.4

([52] theorem VII.1) Let \(\Omega \subset M\) be a domain with smooth boundary \(\partial \Omega \). We have

$$\begin \big \langle f,(\Delta _+m^2)^h \big \rangle _=\big \langle P_}^f, P_}^h \big \rangle _}, \end$$

(B.12)

for f, \(h\in C_c^(\Omega ^)\).

1.3 Symbol Convergence Lemma Proof of Lemma 3.1

By coordinate invariance of the definition of \(\Psi ^m(M)\) it suffices to pick \(x\in M\) and prove the result for a chart around x and \(\chi (x)=1\). Denote the kernel of \(\chi E_\chi \) by \(E_\) then in this chart we could write

$$\begin E_(x,y)=\widetilde_(x,h)=\frac(x)}\widetilde\left( \frac \right) \widetilde(h), \end$$

(B.13)

where \(h=x-y\). Indeed, by definition of our function \(\psi \) and freedom of choosing \(\chi \) we could further assume that for small enough \(\varepsilon \) one has \(\widetilde(h)\equiv 1\) on the support of \(\widetilde(\cdot /\varepsilon )\). Thus under this condition

$$\begin \sigma _\chi }(x,\xi )=\int _^d} \text ^h\cdot \xi }\frac(x)} \widetilde\left( \frac \right) \textrmh =\frac(x)}\underbrace(x,\varepsilon \xi )}_x}.\nonumber \\ \end$$

(B.14)

Note that \(\sigma _(x,\eta )\) is Schwartz in \(\eta \) and \(\sigma _(x,0)=1\). On the other hand clearly \(\sigma _\chi }(x,\xi )\equiv 1\). Thus for some \(U'\subset U\) depending only on the chart and \(\chi \), one has

$$\begin & \sup _\sup _\frac-\mathbbm )\chi }(x,\xi )|}} \nonumber \\ & \qquad \leqslant \left\ \displaystyle C\sup _\sup _ \left\langle \xi \right\rangle ^ |\sigma _\chi }(x,\varepsilon \xi )-1| \leqslant C_ \sqrt,\\ \displaystyle C\sup _\sup _ (\cdots )\leqslant C_\varepsilon ^\sup _|\sigma _(x,\eta )|, \end \right. \end$$

(B.15)

with \(R=\varepsilon ^\). Next we deal with derivatives. Note that by (B.14) all the x-derivatives fall on \(1/F_(x)\) and all \(\xi \)-derivatives fall on \(\sigma _(x,\varepsilon \xi )\). Indeed, one has \(|\partial _x^(1/F_(x))|\leqslant C_ \varepsilon ^\) (see Dyatlov and Zworski [14] page 28), and so when there are only x-derivatives we obtain the same bounds as (B.15) only with new constants depending on \(\alpha \). When there is at least one \(\xi \)-derivative,

$$\begin |\partial _^\left( \sigma _(x,\varepsilon \xi ) \right) |=|\varepsilon ^(\partial _^ \sigma _)(x,\varepsilon \xi )|\leqslant C_\varepsilon ^\left\langle \varepsilon \xi \right\rangle ^,\quad |\beta |\geqslant 1. \end$$

(B.16)

Hence, on account of (B.14) again,

$$\begin \sup _\sup _\frac\partial _^\sigma _-\mathbbm )\chi }(x,\xi )|}}\leqslant C\varepsilon ^d C_\varepsilon ^C_\varepsilon ^=C_\varepsilon ^. \end$$

(B.17)

Consequently, all the \(\mathcal ^_\) seminorms of \(\sigma _-\mathbbm )\chi }\) goes to zero as \(\varepsilon \rightarrow 0\). We obtain the result. \(\square \)