Adiabatic Ground States in Non-smooth Spacetimes

Abstract

Ground states are a well-known class of Hadamard states in smooth spacetimes. In this paper, we show that the ground state of the Klein–Gordon field in a non-smooth ultrastatic spacetime is an adiabatic state. The order of the state depends linearly on the regularity of the metric. We obtain the result by combining a propagation of singularities result for non-smooth pseudodifferential operators, properties of the causal propagator, and eigenvalue asymptotics for elliptic operators of low regularity.

1 Introduction

In the smooth setting, the analysis of the singular structure of physical quantum states goes back to Fulling, Narcowich and Wald [10]. Later on, Kay and Wald further developed the notion of Hadamard states in globally hyperbolic spacetimes possessing a one-parameter group of isometries with a bifurcate Killing horizon [19]. Then in 1996, Radzikowski introduced a characterisation of the singularity structure of these states in terms of the wavefront set [23]. This result was key to using microlocal tools for the construction of quantum states as done by Junker [16], Junker and one of the authors [15], and Gérard and Wrochna [11].

The analysis of quantum states in non-smooth spacetimes has two main motivations. First, there are several models of physical phenomena that require spacetime metrics with finite regularity. These include models of gravitational collapse [1], astrophysical objects [22] and general relativistic fluids [3]. Second, the well-posedness of Einstein’s equations, viewed as a system of hyperbolic PDE requires spaces with finite regularity [20].

In this paper, we focus on scalar fields $\phi$ that satisfy the Klein–Gordon equation

$$\begin{aligned} (\square _g+m^2)\phi :=g^{\mu \nu }\nabla _{\mu }\nabla _{\nu }\phi +m^{2}\phi =0 \end{aligned}$$

(1.1)

on a manifold $M=\mathbb {R}\times {\Sigma }$ where $\Sigma$ is a compact Cauchy hypersurface, $g^{\mu \nu }$ is the inverse metric tensor of an ultrastatic metric, $\nabla _{\mu }$ is the covariant derivative and $m^{2}$ is a positive real number.

In a non-smooth spacetime the quantisation requires in a first instance that the classical system be well-posed. Several results in this direction have been obtained for different degrees of regularity in the time and space variables [5]. Moreover, even when one has classical well-posedness, the quantisation procedure is a significant further challenge. However, some progress has been made for certain degrees of spacetime regularity. For example: Dereziński and Siemssen showed the existence of classical and nonclassical propagators under weak regularity assumptions [6, 7]. Hörmann, Spreitzer, Vickers and one of the authors gave the construction of quantisation functors that satisfy the Haag-Kastler axioms in the $C^{1,1}$ case [14]. In this paper we prove that the ground state of the quantum linear scalar field is an adiabatic state and that the adiabatic order is a linear function with respect to the metric regularity (Theorem 4.17).

Outline of the paper. In Sect. 2, we show the algebraic quantisation of fields satisfying Eq.(1.1) in spacetimes of finite regularity. We give details about the construction of the algebra of observables and precise definitions of the states considered. In Sect. 3, we state the main definitions and theorems regarding non-smooth pseudodifferential operators. In Sect. 4, we focus on ultra-static spacetimes and show that the ground state is an adiabatic state.

2 Quantum Field Theory in Non-smooth Spacetimes

The quantisation of the linear scalar field is a procedure to change the mathematical structure of the theory. On the one hand in the classical theory, the states are represented by vectors in a symplectic space, $(V,\Xi )$, and the classical observables are defined as smooth functionals on $(V,\Xi )$. On the other hand, in the framework of algebraic quantisation, the quantum observables of the theory are represented as the elements of a unique up to $*$-isomorphism $C^{*}$-algebra which satisfies the canonical commutation relations (CCR) and the quantum states, $\omega$, are given by certain positive linear functionals on the $C^{*}$-algebra [2, 30]. Below we give details of the quantisation procedure.

2.1 Observables

For a classical system with equations of motion given by Eq. (1.1) in a globally hyperbolic spacetime (M, g) of regularity $C^{1,1}$, it was shown that the space $(V,\Xi )$ is given by $V=H^1_{\text {comp}}(M)/{\text {ker }G}$ and $\Xi ([f],[g])=([f],G[g])_{L_{\mathbb {R}}^{2}(M)}$ where $H_{\text {comp}}^1(M)$ denotes compactly supported functions in the Sobolev space $H^1(M)$ and ${\text {ker } G}$ is the kernel of the causal propagator [14]. In fact, this symplectic space is symplectically isomorphic to the classical phase space $(\Gamma , \sigma )$ given by the space $\Gamma :=H_{\text {comp}}^2(\Sigma ) \oplus H_{\text {comp}}^1(\Sigma )$ of real-valued initial data with compact support and the symplectic bilinear form

$$\begin{aligned} \sigma (F_1,F_2)=\int _{\Sigma }[q_1p_2 - q_2p_1] dv \end{aligned}$$

with $F_i:=(q_i, p_i) \in \Gamma , i = 1, 2$ and dv the induced volume form on $\Sigma$.

Moreover, to the symplectic space $(V,\Xi )$ one can associate a $C^{*}$-algebra $\mathcal {A}=\mathcal {A}[V, \Xi ]$ that satisfies the CCR, known as the Weyl algebra. It is generated by the elements W([f]), $[f] \in V$, that satisfy

$$\begin{aligned} W([f])^{*}= & {} W([f])^{-1}= W([-f])\\ W([f_1])W([f_2])= & {} e^{-\frac{i}{2}\Xi ([f_1],[f_2])}W([f_1 + f_2]) \end{aligned}$$

for all $[f], [f_1], [f_2] \in V$ (see e.g. [2, 14]).

As $(V,\Xi )$ and $(\Gamma , \sigma )$ are isomorphic as symplectic spaces, one can construct a $C^{*}$-algebra, ${\mathcal {B}}={\mathcal {B}}[\Gamma , \sigma ]$, using the map $\beta :{\mathcal {A}}\rightarrow {\mathcal {B}}$ given by $\beta \left( W([f])\right) := W((\rho ^t_0 Gf, \rho ^t_1 Gf))$ where $\rho ^t_0\phi :=\phi |_{\Sigma _t},\;\rho ^t_1\phi :=\frac{\partial \phi }{\partial t}|_{\Sigma _t}$. The algebra ${\mathcal {B}}$ is $*$-isomorphic to the Weyl algebra $\mathcal {A}$ described above. Each of these algebras represents the quantum observables of the theory.

Moreover, one can localise this construction to suitable subsets of M following the approach of local quantum physics. In fact, one can do these local constructions in a functorial way and the functors satisfy the Haag-Kastler axioms (see [14, Theorem 6.12]).

2.2 States

The quantum states as defined above need to be further restricted in order to be physically relevant. A candidate for physical quantum states, $\omega$, is quasifree states that satisfy the microlocal spectrum condition.

To be precise, given a real scalar product $\mu :\Gamma \times \Gamma \rightarrow \mathbb {R}$ satisfying

$$\begin{aligned} |\sigma (F_1, F_2)|^{2}\le \mu (F_1, F_1)\mu (F_2, F_2) \end{aligned}$$

(2.1)

for all $F_1, F_2 \in \Gamma$, there exists a quasifree state $\omega _{\mu }$ acting on the algebra $\mathcal {B}$ associated with $\mu$ given by $\omega _{\mu }(W(F))=e^{-\frac{1}{2}\mu (F,F)}$. Moreover, one can determine the (“symplectically smeared”) two-point function of $\omega _{\mu }$ by

$$\begin{aligned} \lambda (F_1, F_2) = \mu (F_1, F_2) + \frac{i}{2}\sigma (F_1, F_2) \end{aligned}$$

(2.2)

for $F_1, F_2 \in \Gamma$. The Wightman two-point function $\omega ^{(2)}_{\mu }$ associated with the state $\omega _{\mu }$, is given by

$$\begin{aligned} \omega ^{(2)}_{\mu }(f_1,f_2)=\lambda \left( \left( {\begin{array}{c}\rho _0 Gf_1\\ \rho _1 Gf_1\end{array}}\right) ,\left( {\begin{array}{c}\rho _0 Gf_2\\ \rho _1 Gf_2\end{array}}\right) \right) \end{aligned}$$

(2.3)

for $f_1,f_2\in H^1_\text{ comp }(M)$. By restricting the two point function $\omega ^{(2)}_{\mu }$ to ${{\mathcal {D}}(M)}\otimes {{\mathcal {D}}(M)}$ one obtains a bidistribution in $M\times M$.

To define the microlocal spectrum condition, we use the inverse of the spacetime metric $g^{-1}:=(g^{\mu \nu })_{\mu ,\nu =0}^n$ in order to introduce the sets

$$\begin{aligned}{} & {} C=\big \{(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta }){\in } T^{*}(M{\times } M)\backslash 0; g^{\mu \nu }(\tilde{x})\tilde{\xi }_{\mu }{\tilde{\xi }}_{\nu }{=}g^{\mu \nu }(\tilde{y})\tilde{\eta }_{\mu }\tilde{\eta }_{\nu }{=}0, (\tilde{x},\tilde{\xi })\sim (\tilde{y},\tilde{\eta })\big \} \nonumber \\{} & {} \quad C^{+}=\left\{ (\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })\in C; \tilde{\xi }^{0}\ge 0,\tilde{\eta }^{0}\ge 0\right\} , \end{aligned}$$

(2.4)

where $(\tilde{x},\tilde{\xi })\sim (\tilde{y},\tilde{\eta })$ means that $\tilde{\xi },\tilde{\eta }$ are cotangent to the null geodesic $\gamma$ at $\tilde{x}$ resp. $\tilde{y}$ and parallel transports of each other along $\gamma$.

Using the above sets one can define the microlocal spectrum condition which goes back to Radzikowski [23]:

Definition 2.1

A quasifree state $\omega _{H}$ on the algebra of observables satisfies the microlocal spectrum condition if its two point function $\omega ^{(2)}_{H}$ is a distribution in ${\mathcal {D}}'(M\times M)$ and satisfies the following wavefront set condition

$$\begin{aligned} WF'(\omega ^{(2)}_{H})=C^{+}, \end{aligned}$$

where $WF'(\omega ^{(2)}_{H}):= \{(\tilde{x}, \tilde{\eta }; \tilde{y}, -\tilde{\eta }) \in T^{*}(M\times M); (\tilde{x}, \tilde{\eta }; \tilde{y}, \tilde{\eta }) \in WF(\omega _{2\,H})\}.$

These states are called Hadamard states and include ground states in smooth spacetimes [9,10,11, 16, 24].

A larger class of states called adiabatic states of order N characterised in terms of their Sobolev-wavefront set has been obtained by Junker and one of the authors [15]. These states are the natural generalisation of Hadamard states suitable for spacetimes with limited regularity.

Definition 2.2

A quasifree state $\omega _{N}$ on the algebra of observables is called an adiabatic state of order $N\in \mathbb {R}$ if its two-point function $\omega ^{(2)}_{N}$ is a bidistribution that satisfies the following $H^s$-wavefront set condition for all $s\le N +\frac{3}{2}$

$$\begin{aligned} WF'^{s}(\omega ^{(2)}_{N})\subset C^{+}, \end{aligned}$$

where $WF^{s}$ is a refinement of the notion of the wavefront set in terms of Sobolev spaces. To be precise, a distribution u is microlocally in $H^{s}$ at $(x_0,\xi _0)\in T^*M\backslash 0$ if there exist a conic neighbourhood ${\Gamma }$ of $\xi _0$ and a smooth function $\varphi \in {{\mathcal {D}}(M)}$ with $\varphi (x_0)\ne 0$ such that

$$\begin{aligned} \int _{{\Gamma }}\langle \xi \rangle ^{2s}|{\mathcal {F}}(\varphi u)(\xi )|^{2}d^{n}\xi <\infty . \end{aligned}$$

Otherwise we say that $(x_0,\xi _0)$ lies in the s-wave front set $WF^s(u)$.

If u is microlocally in $H^{s}$ in an open conic subset ${\Gamma }\subset T^*M\backslash 0$ we write $u\in H^s_{mcl}({\Gamma })$.

3 Pseudodifferential Operators with Non-smooth Symbols

3.1 Symbol Classes

Let $\{\psi _j; j=0,1,\ldots \}$ be a Littlewood-Paley partition of unity on ${\mathbb {R}}^n$, i.e. a partition of unity $1=\sum _{j=0}^\infty \psi _j$, where $\psi _0\equiv 1$ for $|\xi |\le 1$ and $\psi _0\equiv 0$ for $|\xi |\ge 2$ and $\psi _j(\xi ) = \psi _0(2^{-j}\xi )-\psi _0(2^{1-j}\xi )$. The support of $\psi _j$, $j\ge 1$, then lies in an annulus around the origin of interior radius $2^{j-1}$ and exterior radius $2^{1+j}$.

Definition 3.1

• (a) For $\tau \in (0,\infty )$, the Hölder space $C^\tau (\mathbb {R}^n)$ is the set of all functions f with

  $$\begin{aligned} \Vert f\Vert _{C^\tau }:=\displaystyle \sum _{|\alpha |\le [\tau ]}\Vert \partial ^\alpha _{x}f\Vert _{L^\infty (\mathbb {R}^n)}+\displaystyle \sum _{|\alpha |= [\tau ]}\sup _{x\ne y}\frac{|\partial ^\alpha _{x}f(x)-\partial ^\alpha _{x}f(y)|}{|x-y|^{\tau -[\tau ]}}<\infty . \end{aligned}$$

  (3.1)

• (b) For $\tau \in \mathbb {R}$ the Zygmund space $C^\tau _{*}(\mathbb {R}^n)$ consists of all functions f with

  $$\begin{aligned} \Vert f\Vert _{C^\tau _*}=\sup _j 2^{j\tau }\Vert \psi _j (D) f\Vert _{L^\infty }<\infty . \end{aligned}$$

  (3.2)

Here $\psi _j(D)$ is the Fourier multiplier with symbol $\psi _j$, i.e. $\psi _j(D)u = {\mathcal {F}}^{-1}\psi _j{\mathcal {F}} u$, where $({\mathcal {F}}{{u}})(\xi ) = (2\pi )^{-n/2} \int e^{-ix\xi } u(x)\, d^nx$ is the Fourier transform.

We have the following relations $C^\tau =C^\tau _{*}$ if $\tau \notin \mathbb {Z}$ and $C^\tau \subset C^\tau _{*}$ if $\tau \in \mathbb {N}$.

We next introduce symbol classes of finite Hölder or Zygmund regularity, following Taylor [27]. We use the notation $\langle \xi \rangle :=(1+|\xi |^2)^{\frac{1}{2}}$, $\xi \in \mathbb R^n$.

Definition 3.2

Let $p(x,\xi ):\mathbb {R}^{2n}\rightarrow \mathbb {R}$ be a continuous function, smooth with respect to $\xi$.

• (a) Let $0\le \delta <1$. A symbol $p(x,\xi )$ belongs to $C^\tau _* S^{m}_{1,\delta }$ if for all $\alpha$

  $$\begin{aligned} \Vert D^{\alpha }_{\xi }p(x,\xi )\Vert _{C_*^{\tau }}\le C_\alpha \langle \xi \rangle ^{m-|\alpha |+\tau \delta } \text { and } |D^{\alpha }_{\xi }p(x,\xi )|\le C_\alpha \langle \xi \rangle ^{m-|\alpha |}. \end{aligned}$$

• (b) We obtain the symbol class $C^\tau S^{m}_{1,\delta }$ for $\tau >0$ by requiring that for all $\alpha$

  $$\begin{aligned} \Vert D^{\alpha }_{\xi }p(x,\xi )\Vert _{C^{s}}\le C_\alpha \langle \xi \rangle ^{m-|\alpha |+s\delta }, \quad 0\le s\le \tau , \text { and } |D^{\alpha }_{\xi }p(x,\xi )|\le C_\alpha \langle \xi \rangle ^{m-|\alpha |}. \end{aligned}$$

• (c) A symbol $p(x,\xi )$ is in $C^{\tau }S_{cl}^{m}$ provided $p(x,\xi )\in C^{\tau }S^{m}_{1,0}$ and $p(x,\xi )$ has a classical expansion

  $$\begin{aligned} p(x,\xi )\sim \sum _{j\ge 0}p_{m-j}(x,\xi ) \end{aligned}$$

  in terms $p_{m-j}$ homogeneous of degree $m-j$ in $\xi$ for $|\xi |\ge 1$, in the sense that the difference between $p(x,\xi )$ and the sum over $0\le j< N$ belongs to $C^{\tau }S^{m-N}_{1,0}$.

3.2 Characteristic Set and Pseudodifferential Operators

Let $p\in C^\tau S^{m}_{1,\delta }$, $\tau >0$, with $\delta <1$. Suppose that there is a conic neighbourhood $\Gamma$ of $(x_0,\xi _0)$ and constants $c,C>0$ such that $|p(x,\xi )|\ge c|\xi |^m$ for $(x,\xi ) \in \Gamma$, $|\xi |\ge C$. Then $(x_0,\xi _0)$ is called non-characteristic. If p is classical and has the homogeneous principal symbol $p_m$, the condition is equivalent to $p_m(x_0,\xi _0)\ne 0$. The complement of the set of non-characteristic points is the set of characteristic points denoted by ${{{\,\textrm{Char}\,}}}(p)$.

The pseudodifferential operator $p(x,D_x)$ with the symbol $p(x,\xi )\in C^\tau S^m_{1,\delta }$ is given by

$$\begin{aligned} p(x,D_x)u=(2\pi )^{-n/2} \int _{\mathbb {R}^n}e^{ix\cdot \xi }p(x,\xi )({\mathcal {F}}{u})(\xi )d^n\xi , \quad u\in {\mathcal {S}}({\mathbb {R}}^n). \end{aligned}$$

(3.3)

It extends to continuous maps

$$\begin{aligned} p(x,D_x): H^{s+m}(\mathbb {R}^n)\rightarrow H^{s}(\mathbb {R}^n) , \quad -\tau (1-\delta )<s<\tau . \end{aligned}$$

(3.4)

3.3 Symbol Smoothing

Given $p(x,\xi )\in C^{\tau }S_{1,\delta }^{m}$ and $\gamma \in (\delta ,1)$ let

$$\begin{aligned} p^{\#}(x,\xi )=\displaystyle \sum _{j=0}^{\infty }J_{\epsilon _{j}}p(x,\xi )\psi _j(\xi ). \end{aligned}$$

(3.5)

Here $J_\epsilon$ is the smoothing operator given by $(J_\epsilon f)(x)=(\phi (\epsilon D)f)(x)$ with $\phi \in C^{\infty }_0(\mathbb {R}^n)$, $\phi (\xi )=1$ for $|\xi |\le 1$, and we take $\epsilon _j=2^{-j\gamma }$.

Letting $p^{b}(x,\xi )=p(x,\xi )-p^{\#}(x,\xi )$ we obtain the decomposition

$$\begin{aligned} p(x,\xi )=p^{\#}(x,\xi )+p^{b}(x,\xi ), \end{aligned}$$

(3.6)

where $p^{\#}(x,\xi )\in S^{m}_{1,\gamma }$ and $p^{b}(x,\xi )\in C^{\tau }S^{m-\tau (\gamma -\delta )}_{1,\gamma }$.

If $p\in C^\tau S^m_{1,0}$, then we additionally have $p^b\in C^{\tau -t}S^{m-t\gamma }_{1,0}$ with $\tau -t>0$ by [27, Proposition 1.3.B]. Furthermore, we have better estimates, see [27, Proposition 1.3.D]:

$$\begin{aligned} D_x^\beta p^{\#}(x,\xi )\in {\left\{ \begin{array}{ll}S^{m}_{1,\gamma },\quad &{}|\beta |\le \tau \text { and}\\ S^{m+\gamma (|\beta |-\tau )}_{1,\gamma }, \quad &{} |\beta |>\tau \end{array}\right. }. \end{aligned}$$

(3.7)

4 Ground States in Ultrastatic Spacetimes

Let $M=\mathbb {R}\times \Sigma$, where $\Sigma$ is a 3-dimensional compact manifold and the Lorentzian metric $g:=(g_{\mu \nu })_{\mu ,\nu =0}^3$ is of the form

$$\begin{aligned} ds^2=dt^2-h_{ij}(x)dx^idx^j, \end{aligned}$$

(4.1)

where $h_{ij}(x)$ are the components of a time independent Riemannian metric of Hölder regularity $C^{\tau }$ (when $\tau \in \mathbb {N}$ we will consider the Zygmund spaces $C_*^\tau$, introduced in Definition 3.1). As usual, the tensor $(g^{\mu \nu })$ in (1.1) is the inverse to $(g_{\mu \nu })$.

Moreover, the vector field $\partial _t$ induces a one-parameter group of isometries $\tau _t:M\rightarrow M,t\in {\mathbb {R}}$, such that $\tau _t(\Sigma _{t_o})=\Sigma _{t_o+t}$.

This group induces a one-parameter group of automorphisms in the $C^*$-algebras as follows. Define ${\mathcal {T}}(t):\Gamma _{t_o}\rightarrow \Gamma _{t_o+t}$ by

$$\begin{aligned} {\mathcal {T}}(t)\tilde{F}_{t_o}:=\tilde{F}_{t_o+t}, \end{aligned}$$

where $\Gamma _s$ is the initial data at $\Sigma _s$, $\tilde{F}_{s}:=(\rho ^s_0\phi ,\rho ^s_1\phi )$ and $\phi \in \text { ker } (\square _g+m^2)$.

Since the symplectic form $\sigma$ is invariant under the action of ${\mathcal {T}}(t)$ and since ${\mathcal {T}}(t){\mathcal {T}}(s)={\mathcal {T}}(t+s)\; t,s\in {\mathbb {R}},\;{\mathcal {T}}$ is a one-parameter group of symplectic transformations (also called Bogoliubov transformations), it gives rise to a group of automorphisms $\tilde{\alpha } (t), t\in {\mathbb {R}}$, (Bogoliubov automorphisms) on the algebra ${\mathcal {B}}$ via

$$\begin{aligned} \tilde{\alpha }(t) W(F)=W({\mathcal {T}}(t) F). \end{aligned}$$

In this case, there exists a preferred class of states on ${\mathcal {A}}$, namely those invariant under $\alpha (t)=\tilde{\alpha }(t)\circ \beta$ with $\beta$ as defined in Section 2.1. A quasifree state $\omega _\mu$ will be invariant under this symmetry if and only if

$$\begin{aligned}\mu ({\mathcal {T}}(t)F_1,{\mathcal {T}}(t)F_2)=\mu (F_1,F_2)\; \forall t\in {\mathbb {R}} \;\forall F_1,F_2\in \Gamma . \end{aligned}$$

The specification of $\mu$ is equivalent to the specification of a one-particle structure as established by the following theorem of Kay and Wald [19, Proposition 3.1]:

Theorem 4.1

Let $\omega _\mu$ be a quasifree state on ${\mathcal {B}}[\Gamma ,\sigma ]$. Then there exists a one-particle Hilbert space structure, i.e. a Hilbert space ${\mathcal {H}}$ and a real-linear map $k:\Gamma \rightarrow {\mathcal {H}}$ such that

• (i) $k\Gamma +ik\Gamma$ is dense in ${\mathcal {H}}$,

• (ii) $\mu (F_1,F_2)=\textrm{Re}\langle kF_1,kF_2\rangle _{\mathcal {H}}\;\forall F_1,F_2\in \Gamma$,

• (iii) $\sigma (F_1,F_2)=2\textrm{Im}\langle kF_1,kF_2\rangle _{\mathcal {H}}\; \forall F_1,F_2\in \Gamma$. The pair $(k,{\mathcal {H}})$ is uniquely determined up to unitary equivalence. Moreover, $\omega _\mu$ is pure if and only if $k(\Gamma )$ is dense.

Remark 4.2

Notice that the specification of a Hilbert space $\mathcal {H}$ together with a real-linear map $k:\Gamma \rightarrow {\mathcal {H}}$ such that $k\Gamma +ik\Gamma$ is dense in ${\mathcal {H}}$ and $2\textrm{Im}\langle kF_1,kF_2\rangle _{\mathcal {H}}=\sigma (F_1,F_2)$ gives rise via Eq.(2.2) to a real scalar product $\mu$ satisfying Eq.(2.1).

Moreover, the automorphism group $\tilde{\alpha }(t)$ can be unitarily implemented in the one-particle Hilbert space structure $(k,{\mathcal {H}})$ of an invariant state $\omega _\mu$, i.e. there exists a unitary group $U(t), t\in {\mathbb {R}},$ on ${\mathcal {H}}$ satisfying

$$\begin{aligned} U(t)k= & {} k{\mathcal {T}}(t)\nonumber \\ U(t)U(s)= & {} U(t+s). \end{aligned}$$

(4.2)

If U(t) is strongly continuous it takes the form $U(t)=e^{-iHt}$ for some self-adjoint operator H on ${\mathcal {H}}$.

We define now the notion of ground states following Kay [18]:

Definition 4.3

Let the phase space $(\Gamma ,\sigma ,{\mathcal {T}}(t))$ be given. A quasifree ground state is a quasifree state over ${\mathcal {B}} [\Gamma ,\sigma ]$ with one-particle Hilbert space structure $(k,{\mathcal {H}})$ and a strongly continuous unitary group $U(t)=e^{-iHt}$ (satisfying (4.2)) such that H is a positive operator (the “one-particle Hamiltonian”).

In the ultrastatic case we define the ground state, $\omega _G$ by the one-particle Hilbert space structure $(k_{G},{\mathcal {H}}_{G})$

$$\begin{aligned} k_G:\Gamma\rightarrow & {} {\mathcal {H}}_G:=L^2_\textbf{C}(\Sigma _{t_0})\nonumber \\ F=(q,p)\mapsto & {} \frac{1}{\sqrt{2}}\left( A^{1/4}q-iA^{-1/4}p\right) , \end{aligned}$$

(4.3)

where $A:=-\Delta _h+m^2$ and $t_0\in \mathbb {R}$ (invariance under time translations makes any choice of $t\in \mathbb {R}$ equivalent to any other) and the strongly continuous unitary group is given by $U(t):=e^{iA^{\frac{1}{2}}t}$.

The Wightman two-point function of $\omega _G$ is:

$$\begin{aligned} \omega ^{(2)}_{G}(h_1,h_2) = \lambda _G\left( {\rho ^t_0Gh_1\atopwithdelims ()\rho _1^tGh_1 },{\rho _0^tGh_2\atopwithdelims ()\rho _1^tGh_2}\right) , \end{aligned}$$

(4.4)

for $h_1,h_2\in {\mathcal {D}}({\mathcal {M}})$.

Moreover using Eq. 2.3, Eq. (4.3) and Theorem 4.1 the “symplectically smeared two-point function” $\lambda _G$ is given on the initial data $F_i={q_i\atopwithdelims ()p_i}\in \Gamma$ by Eq.(2.2),

$$\begin{aligned} \lambda _G(F_1,F_2)= & {} \left\langle k_G F_1,k_GF_2\right\rangle _{L^2_{\mathbb {C}}(\Sigma )} \nonumber \\= & {} \frac{1}{2}\left\langle A^{1/4}q_1-iA^{-1/4}p_1,A^{1/4}q_2-iA^{-1/4}p_2\right\rangle _{L^2_{\mathbb {C}}(\Sigma )} \nonumber \\= & {} \frac{1}{2}\left\langle (A^{1/2}q_1-ip_1), A^{-1/2}\left( A^{1/2}q_2-ip_2\right) \right\rangle _{L^2_{\mathbb {C}}(\Sigma )}, \end{aligned}$$

(4.5)

since A is selfadjoint. Combining (4.4) and (4.5) we obtain

$$\begin{aligned} \omega ^{(2)}_{G}(h_1,h_2)=\frac{1}{2}\left\langle \left( A^{1/2}\rho _0^t-i\rho _1^t\right) Gh_1,A^{-1/2} \left( A^{1/2}\rho ^t_o-i\rho ^t_1\right) Gh_2\right\rangle _{L^2_{\mathbb {C}}(\Sigma _t)}. \nonumber \\ \end{aligned}$$

(4.6)

The two-point function, $\omega ^{(2)}_{G}$, of the ground state, $\omega _G$, is the Schwartz kernel of the operator $\displaystyle {{e^{i A^\frac{1}{2}(t-s)}}{A^{-\frac{1}{2}}}}$.

Explicitly, for $u,v\in {{\mathcal {D}}}(M)$ we have

$$\begin{aligned}&\omega ^{(2)}_{G}(u, v)=\int _{M}\left( {e^{i A^\frac{1}{2}(t-s)}}A^{-\frac{1}{2}}u\right) (s,y)v(s,y)dsdy, \end{aligned}$$

which gives the singular integral kernel representation

$$\begin{aligned} \omega ^{(2)}_{G}(t,x;s,y)=\sum _j \lambda _j^{-1} e^{i\lambda _j(t-s)}\phi _j(x)\phi _j(y). \end{aligned}$$

(4.7)

where $\{\phi _j, j=1,2,\ldots \}$ is an orthonormal basis of eigenfunctions of $L^2(\Sigma )$ associated with the eigenvalues $\lambda _j^2$ of the operator $m^2-\Delta _h$.

The ground state in a smooth ultrastatic space-time is a Hadamard state [10, 11, 16, 24]. In the following section we will show that, in the non-smooth case, the ground state is an adiabatic state.

4.1 Microlocal Analysis for Bisolutions of the Klein–Gordon Operator

We write local coordinates on ${\mathbb {R}}\times \Sigma$ in the form

$$\begin{aligned} \tilde{x}= (t,x),\; \tilde{y}=(s,y) \end{aligned}$$

(4.8)

and the associated covariables as

$$\begin{aligned} \tilde{\xi }= (\xi _0,\xi ),\; \tilde{\eta }= (\eta _{0},\eta ). \end{aligned}$$

(4.9)

Using the notation above and in (4.1), we have:

Remark 4.4

The Klein–Gordon operator on M is given by

$$\begin{aligned} P\phi =(\Box _g+m^2)\phi = \partial _{tt}\phi -\Delta _h\phi +m^2\phi . \end{aligned}$$

(4.10)

It has the symbol $P(\tilde{x},\tilde{\xi })=(-\xi _0^2+ h^{ij}\xi _i\xi _j)+ i\frac{1}{\sqrt{h}}\partial _{x^i}(h^{ij}\sqrt{h})\xi _j+m^2$. For a metric of regularity $C^{\tau }$, the symbol $P(\tilde{x},\tilde{\xi })$ belongs to $C^{\tau -1}S^{2}_{cl}$ and

$$\begin{aligned} {{{\,\textrm{Char}\,}}}(P)=\{(t,x,\xi _0,\xi )\in T^*M \backslash \ \{0\}; -\xi _0^2+ h^{ij}\xi _i\xi _j=0\}. \end{aligned}$$

In the sequel we shall apply the Klein–Gordon operator to functions and distributions on $M\times M$, say with variables ((t, x), (s, y)) in the notation (4.8). In order to make clear whether P acts on the first set of variables (t, x) or on the second set (s, y) we will write $P_{(t,x)}$ and $P_{(s,y)}$, respectively. Using the coordinates in Eqs.(4.8) and (4.9), the associated symbols $P_{(t,x)}(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })$ and $P_{(s,y)}(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })$ formally depend on the full set of (co-)variables $(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })$, however, only the (co-)variables associated with either (t, x) or (s, y) show up:

$$\begin{aligned} P_{(t,x)}(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })&=\underbrace{({-\xi ^{0}}^{2}+h^{ij}(x)\xi _i\xi _j)}_{p_2(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })}+\underbrace{i\frac{1}{\sqrt{h}}\partial _{x^i}(h^{ij}\sqrt{h}(x))\xi _j}_{p_1(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })}+\underbrace{m^2}_{p_0(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })}. \\ P_{(s,y)}(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })&=(-{\eta _{0}}^{2}+h^{ij}(x)\eta _i\eta _j)+{i\frac{1}{\sqrt{h}}\partial _{y^i}(h^{ij}\sqrt{h}(y))\eta _j}+{m^2}. \end{aligned}$$

In particular,

$$\begin{aligned} {{\,\textrm{Char}\,}}(P_{(t,x)})= & {} \left( {{\,\textrm{Char}\,}}(P)\times T^{*}M\right) \cup \{(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })\in T^{*}(M\times M)\backslash \{0\}, \tilde{\xi }=0\}\nonumber \\ {{\,\textrm{Char}\,}}(P_{(s,y)})= & {} \left( T^{*}M\times {{\,\textrm{Char}\,}}(P)\right) \cup \{(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })\in T^{*}(M\times M)\backslash \{0\}, \tilde{\eta }=0\}.\nonumber \\ \end{aligned}$$

(4.11)

Now we will state a microelliptic estimate tailored for bisolutions of the Klein–Gordon operator

Theorem 4.5

Let the metric g be of class $C^\tau$, $\tau >1$, $0\le \sigma <\tau -1$ and $v\in H_{loc}^{2+\sigma -\tau +\epsilon }(M\times M)$ for some ${\epsilon }>0$ with $P_{(t,x)}(\tilde{x},\tilde{y},D)v= P_{(s,y)}(\tilde{x},\tilde{y},D)v=0$. Then

$$\begin{aligned} WF^{\sigma +2}(v)\subset {{\,\textrm{Char}\,}}(P_{(t,x)})\cap {{\,\textrm{Char}\,}}(P_{(s,y)}). \end{aligned}$$

The proof can be found in [26, Theorem 3.4].

Remark 4.6

To obtain Theorem 4.5 we smooth each of the non-smooth symbols (the principal symbol and the sub-leading term) separately to obtain the remainder $p_2^b+p_1^b$ for $p^b_2\in C^\tau S^{2-\tau \delta }_{1,\delta }$ and $p^b_1\in C^{\tau -1} S^{1-(\tau -1)\delta }_{1,\delta }$. Applying the symbol smoothing directly to $P_{(t, x)}\in C^{\tau -1} S^2_{1,0}$ would leave us with $P_{(t, x)}^b\in C^{\tau -1}S^{2-{(\tau -1)}\delta }_{1,\delta }.$

Furthermore, the main results on the microlocal propagation of singularities in the non-smooth setting that we will apply can be found in [27, Proposition 6.1.D] or [28, Proposition 11.4]. In particular, the theorem below holds for spacetime metrics belonging to the space $C_*^2$ [28, p.215].

Theorem 4.7

Let $u\in {\mathcal {D}}'(M\times M)$ solve $P_{(t,x)}u=f$. Let $\gamma$ be an integral curve of the Hamiltonian vector field $H_{p_{2}}$ with $p_2$ the principal symbol of $P_{(t,x)}$. If for some $s\in \mathbb {R}$, $f\in H_{mcl}^s({\Gamma })$ and $P_{(t,x)}^b u\in H_{mcl}^s({ \Gamma )}$ where $\gamma \subset {\Gamma }$ with ${\Gamma }$ a conical neighbourhood and $u\in H_{mcl}^{s+1}(\gamma (0))$ then $u\in H_{mcl}^{s+1}({\gamma })$.

Remark 4.8

If $u\in H_{loc}^{2+s-\tau \delta }(M\times M)$, then $P_{(t,x)}^b u\in H_{loc}^s(M\times M)$ for $-(1-\delta )(\tau -1)\le s\le \tau -1$, see Remark 4.6.

4.2 The Microlocal Spectrum Condition

Now we will show that the Wightman two-point function of the ground state described above satisfies Definition 2.2. We will assume throughout this section that the metric is of regularity $C^\tau$ with $\tau >2$.

Let $\{\phi _j\otimes \phi _k; j,k=1,2,\ldots \}$ be an orthonormal basis of $L^2(\Sigma )\otimes L^2(\Sigma )$ associated with the eigenfunctions $\{\phi _j\}$ and the eigenvalues $\{\lambda _j^2\}$ of the operator $m^2-\Delta _h$. Then, for $u\in L^2(M\times M)$ we have the representation

$$\begin{aligned} {u(t,s, x,y) =\sum _{j,k}u_{jk}(t,s)\phi _j(x)\phi _k(y)} \quad \text {with } u_{jk}=\langle u,\phi _j\otimes \phi _k\rangle \in L^2(\mathbb {R}^2).\nonumber \\ \end{aligned}$$

(4.12)

Moreover, we have the following generalisation for $u\in H^{2\theta }(M\times M)$ shown in [26, Proposition 4.1, Corollary 4.4]

Theorem 4.9

For $-1\le \theta \le 1$

$$\begin{aligned} H^{2\theta }(\mathbb {R}^2\times \Sigma ^2)&=\left\{ u\in {\mathcal {S}}'(\mathbb {R}^2\times \Sigma ^2); \sum _{j,k}\int _{\mathbb {R}^2}(|\xi _0|^2+|\eta _0|^2+\lambda _j^2+\lambda _k^2)^{2\theta }\right. \\&\qquad \left. \times |\mathcal Fu_{jk}(\xi _0, \eta _0)|^2d\xi _0d\eta _0<\infty \ \right\} , \end{aligned}$$

with $u_{jk} = \langle u,\phi _j\otimes \phi _k\rangle \in {\mathcal {S}}'(\mathbb {R}^2)$.

The previous theorem allows us to establish the local Sobolev regularity of the two-point function.

Theorem 4.10

$\omega ^{(2)}_{G}\in H_{loc}^{-\frac{1}{2}-\epsilon }(M\times M)$ for every $\epsilon >0$

Proof

Let $\psi \in {\mathcal {D}}(M\times M)$. Without loss of generality assume that $\psi = \psi (s,t)$. We will show that $\psi \omega ^{(2)}_{G}\in H^{-\frac{1}{2}-\epsilon }(M\times M)$.

According to Eq. (4.7) and Theorem 4.9

$$\begin{aligned}&\Vert \psi \omega ^{(2)}_{G}\Vert ^2_{H^{-\frac{1}{2}-\epsilon }(M\times M)} \end{aligned}$$

(4.13)

$$\begin{aligned}&=\sum _{j=k}\int _{\mathbb {R}^2} (|\xi _0|^2+|\eta _0|^2+\lambda _j^2+\lambda _k^2)^{-\frac{1}{2}-\epsilon } \nonumber \\ {}&\quad \Big |{{\mathcal {F}}_{(t,s)\rightarrow (\xi _0,\eta _0)}} \Big (\frac{\psi (t,s)}{\lambda _j}e^{i{\lambda _j(t-s)}}\Big )(\xi _0, \eta _0)\Big |^2d\xi _0d\eta _0. \end{aligned}$$

(4.14)

We have by direct computation that

$$\begin{aligned}&\Big |{{\mathcal {F}}_{(t,s)\rightarrow (\xi _0,\eta _0)}} \Big (\frac{\psi (t,s)}{\lambda _j}e^{i{\lambda _j(t-s)}}\Big )(\xi _0, \eta _0)\Big |^2\nonumber \\&=\frac{1}{\lambda _j^2}\left| {{\mathcal {F}}} (\psi )(\xi _0-\lambda _j, \eta _0+\lambda _j)\right| ^2.\nonumber \\ \end{aligned}$$

(4.15)

Taking into account that $\Vert \psi \Vert _{L^2(\mathbb {R}^2)}=\Vert {\mathcal {F}}(\psi )\Vert _{L^2(\mathbb {R}^2)}<\infty$ we have (with constants possibly changing from line to line)

$$\begin{aligned}&\Vert \psi \omega ^{(2)}_{G}\Vert _{H^{-\frac{1}{2}-\epsilon }(M\times M)}\\ {}&=\sum _{j=k}\int _{\mathbb {R}^2} (|\xi _0|^2+|\eta _0|^2+\lambda _j^2+\lambda _k^2)^{-\frac{1}{2}-\epsilon }\\ {}&\quad \Big |{{\mathcal {F}}_{(t,s)\rightarrow (\xi _0,\eta _0)}} \Big (\frac{\psi (t,s)}{\lambda _j}e^{i{\lambda _j(t-s)}}\Big )(\xi _0, \eta _0)\Big |^2 d\xi _0d\eta _0\\ {}&\le \displaystyle {\sum ^\infty _{j=1}}\int _{\mathbb {R}^2}(\lambda _j^2+\lambda _j^2)^{-\frac{1}{2}-\epsilon } \frac{1}{\lambda _j^2}\left| {{\mathcal {F}}}(\psi )(\xi _0-\lambda _j, \eta _0+\lambda _j)\right| ^2d\xi _0d\eta _0\\ {}&\le C\displaystyle {\sum ^\infty _{j=1}}\frac{(\lambda _j^2+\lambda _j^2)^{-\frac{1}{2}-\epsilon }}{\lambda _j^2}\\ {}&\le C \displaystyle {\sum ^\infty _{j=1}}\frac{1}{\lambda _j^{3+2\epsilon }}. \end{aligned}$$

From Weyl’s law for non-smooth metrics [31, Theorem 1.1] we obtain the estimate $j^{\frac{2}{3}}\le C \lambda _j^2$ for a suitable constant C which gives

$$\begin{aligned} \Vert \psi \omega ^{(2)}_ {G}\Vert ^2_{H^{-\frac{1}{2}-\epsilon }(M\times M)}&\le \displaystyle {\sum ^\infty _{j=1}}\frac{C}{\lambda _j^{3+2\epsilon }}\le \displaystyle {\sum ^\infty _{j=1}}\frac{C'}{j^{1+\frac{2\epsilon }{3}}}<\infty \end{aligned}$$

for a suitable constant $C'$. $\square$

It will be useful to consider the following bidistribution:

Corollary 4.11

Let $\omega _A\in {\mathcal {D}}'(M\times M)$ be the bidistribution given by

$$\begin{aligned} \omega _A(u\otimes v):= & {} -\int _{M\times M}\sum _j \lambda _j^{-2} e^{i\lambda _j(t-s)}\phi _j(x)\phi _j(y)\sqrt{h(y)}\sqrt{h(x)}\\ {}{} & {} \quad \times u(t,x) v(s,y)dsdydtdx \end{aligned}$$

Then,

$$\begin{aligned} \omega _A\in H_{loc}^{\frac{1}{2}-\epsilon }(M\times M) \text { for every } \epsilon >0. \end{aligned}$$

(4.16)

Proof

Direct computation shows that for $\psi$ as in the previous proof and $s\ge 0$ we have

$$\begin{aligned}&\Vert \psi \omega _A\Vert _{H^{s}(M\times M)}\\&=\displaystyle {\sum _{j=1}^{\infty }}\int _{\mathbb {R}^2}(|\xi _0|^2+|\eta _0|^2+\lambda _j^2+\lambda _j^2)^{s} \\ {}&\quad \times \Big |{{\mathcal {F}}_{(t,s)\rightarrow (\xi _0,\eta _0)}} \Big (\frac{\psi (t,s)}{\lambda _j^2}e^{-i{\lambda _j(t-s)}}\Big )(\xi _0, \eta _0)\Big |^2d\xi _0d\eta _0\\&=\displaystyle {\sum _{j=1}^{\infty }}\int _{\mathbb {R}^2}(|\xi _0|^2+|\eta _0|^2+\lambda _j^2+\lambda _j^2)^{s} \Big |{{\mathcal {F}}} \Big (\frac{\psi }{\lambda _j^2}\Big )(\xi _0-\lambda _j, \eta _0+\lambda _j)\Big |^2d\xi _0d\eta _0\\&=\displaystyle {\sum _{j=1}^{\infty }}\int _{\mathbb {R}^2}(|\xi _0-\lambda _j|^2+|\eta _0+\lambda _j|^2+\lambda _j^2+\lambda _j^2)^{s} \Big |{{\mathcal {F}}} \Big (\frac{\psi }{\lambda _j^2}\Big )(\xi _0, \eta _0)\Big |^2d\xi _0d\eta _0\\&\le \displaystyle {\sum _{j=1}^{\infty }}\int _{\mathbb {R}^2}(2|\xi _0|^2+2|\eta _0|^2+3\lambda _j^2+3\lambda _j^2)^{s} \Big |{{\mathcal {F}}}\Big (\frac{\psi }{\lambda _j^2}\Big )(\xi _0, \eta _0)\Big |^2d\xi _0d\eta _0\\&\le C \displaystyle {\sum _{j=1}^{\infty }}\frac{(1+\lambda _j^2)^{s}}{\lambda _j^{4}}\int _{\mathbb {R}^2}(1+|\xi _0|^2+|\eta _0|^2)^s \Big |{{\mathcal {F}}}\Big ({\psi }\Big )(\xi _0, \eta _0)\Big |^2d\xi _0d\eta _0\\&\le C\displaystyle {\sum _{j=1}^{\infty }}\frac{(1+\lambda _j^2)^{s}}{\lambda _j^{4}}|\psi |^{2}{H^s(\mathbb {R}^2)}\\&{\le \displaystyle C{\sum _{j=1}^{j_0}}\frac{(1+\lambda _j^2)^{s}}{\lambda _j^{4}}+C\displaystyle {\sum _{j=j_0}^{\infty }}\frac{(\lambda _j^2)^{s}}{\lambda _j^{4}}}, \end{aligned}$$

where we have chosen $j_0$ large enough such that $\lambda _{j_0}>1$.

According to Weyl’s law for non-smooth metrics [31, Theorem 1.1] we have the estimate $j^{\frac{2}{3}}\le C \lambda _j^2$ for a suitable constant C. This gives for $s=\frac{1}{2}-\epsilon$

$$\begin{aligned} \Vert \psi \omega _A\Vert ^2_{H^{\frac{1}{2}-\epsilon }(M\times M)}&\le C+\sum _j\frac{C}{\lambda _j^{3+2\epsilon }}\le C+ \sum _j\frac{C}{j^{1+\frac{2\epsilon }{3}}}<\infty \end{aligned}$$

(4.17)

for a suitable constant C. $\square$

Remark 4.12

Notice that $i\partial _{t}\omega _A=\omega ^{(2)}_{G}$.

Lemma 4.13

For any $\tilde{\epsilon }>0$

$$\begin{aligned} WF^{-\frac{1}{2}-\tilde{\epsilon }+\tau }(\omega ^{(2)}_{G})\subset {{\,\textrm{Char}\,}}(P)\times {{\,\textrm{Char}\,}}(P). \end{aligned}$$

(4.18)

Proof

Since $\omega ^{(2)}_{G}$ satisfies $(\partial _t+\partial _s)\omega ^{(2)}_{G}=0$ we conclude that for all $l\in \mathbb {R}$

$$\begin{aligned} WF^l(\omega ^{(2)}_{G})\subset & {} WF(\omega ^{(2)}_{G})\subset {{\,\textrm{Char}\,}}(\partial _t+\partial _s)\nonumber \\ {}{} & {} \quad =\{(\tilde{x},\xi _0,\xi , \tilde{y},\eta _0,\eta )\in T^*(M\times M)\backslash \{0\}; \xi _0+\eta _0=0\},\nonumber \\ \end{aligned}$$

(4.19)

where the second inclusion follows from the standard theory of pseudodifferential operators.

Now we have $P_{(t,x)}(\tilde{x},\tilde{y},D)\omega _A=P_{(s,y)}(\tilde{x},\tilde{y},D)\omega _A=0$. Choose $\epsilon <\tilde{\epsilon }/2$. Since $\omega _A\in H_{loc} ^{\frac{1}{2}-\frac{\epsilon }{2}}(M\times M) = H_{loc} ^{(\frac{1}{2}- \epsilon )+\frac{\epsilon }{2}}(M\times M)$, an application of Theorem 4.5 with $\sigma =-\frac{3}{2}+\tau -{\epsilon }< \tau -1$ shows that

$$\begin{aligned} WF^{\frac{1}{2}+\tau -\tilde{\epsilon }}(\omega _A)\subset&{{\,\textrm{Char}\,}}(P_{(t, x)})\cap {{\,\textrm{Char}\,}}(P_{(s, y)}); \end{aligned}$$

here we assume without loss of generality that $\epsilon$ is so small that $-\frac{3}{2}+\tau -{\epsilon }\ge 0$. Equation (4.11) implies that

$$\begin{aligned} WF^{\frac{1}{2}-\tilde{\epsilon }+\tau }(\omega _A)\subset&({{\,\textrm{Char}\,}}(P)\times {{\,\textrm{Char}\,}}(P))\\&\cup ( \{(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })\in T^{*}(M\times M)\backslash \{0\}; \tilde{\xi }=0,(\tilde{y},\tilde{\eta })\in {{\,\textrm{Char}\,}}(P)\}\\&\cup \{(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })\in T^{*}(M\times M)\backslash \{0\}; (\tilde{x},\tilde{\xi })\in {{\,\textrm{Char}\,}}(P), \tilde{\eta }=0\}. \end{aligned}$$

If $\tilde{\eta }=0$, then $\eta _0=0$, and $\xi _0=0$ by Eq. (4.19). Since ${{\,\textrm{Char}\,}}P=\{(\tilde{x},\tilde{\xi });(\xi _0)^2=h^{ij}(x)\xi _i\xi _j\}$ we then have $\tilde{\xi }=0$. Together with the corresponding argument for the case $\tilde{\xi }=0$ this shows that

$$\begin{aligned} WF^{\frac{1}{2}-\tilde{\epsilon }+\tau }(\omega _A)\subset {{\,\textrm{Char}\,}}(P)\times {{\,\textrm{Char}\,}}(P); \end{aligned}$$

(4.20)

otherwise $0\in T^*(M\times M)$ would be in $WF^{\frac{1}{2}-\tilde{\epsilon }+\tau }(\omega _A)$.

Since $WF^{-\frac{1}{2}-\tilde{\epsilon }+\tau }(i\partial _{t}\omega _A)\subset WF^{\frac{1}{2}-\tilde{\epsilon }+\tau }(\omega _A)$ by [15, Proposition B.3], we have

$$\begin{aligned} WF^{-\frac{1}{2}-\tilde{\epsilon }+\tau }(\omega ^{(2)}_{G})= & {} WF^{-\frac{1}{2}-\tilde{\epsilon }+\tau }(i\partial _{t}\omega _{A})\subset WF^{\frac{1}{2}-\tilde{\epsilon }+\tau }(\omega _A)\\ {}\subset & {} ({{\,\textrm{Char}\,}}(P)\times {{\,\textrm{Char}\,}}(P)). \end{aligned}$$

$\square$

Theorem 4.14

For all $s\in \mathbb {R}$, $WF^{s}(\omega ^{(2)}_{G})\subset \{(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })\in T^*(M\times M); \tilde{\xi }^0> 0\}$.

Proof

We define $F:\mathbb {R}+i\,{]0,\delta [}\subset \mathbb {C}\rightarrow {\mathcal {D}}'(\Sigma \times M)$ for $\delta >0$ by

(4.21)

Notice that $\partial _zF:\mathbb {R}+i]0,\delta [\subset \mathbb {C}\rightarrow \mathcal D'(\Sigma \times M)$ is given by

(4.22)

and therefore F is a holomorphic function with values in ${\mathcal {D}}'(\Sigma \times M)$, see [17, Theorem 10.11]. Moreover, for $\varphi (t)\in {\mathcal {D}}(\mathbb {R})$ we have

$$\begin{aligned}&|\langle \langle F(t+i\epsilon ),\psi _1(s)\psi _2(x)\psi _3(y)\rangle ,\varphi (t)\rangle | \end{aligned}$$

(4.23)

$$\begin{aligned}&\quad =\left| \int _{\mathbb {R}}\int _{M}\sum _{j}\lambda _j^{-1}e^{i(t+i\epsilon )\lambda _j}e^{-is\lambda _j}\left( \int _{\Sigma }\psi _2(x)\phi _j(x)dx\right) \phi _{j}(y)\psi _1(s)\psi _3(y)dyds\varphi (t)dt\right| \end{aligned}$$

(4.24)

$$\begin{aligned}&\quad =\left| \int _\mathbb {R}\int _\mathbb {R}\sum _{j}\underbrace{\lambda _j^{-1}e^{i(t+i\epsilon )\lambda _j}e^{-is\lambda _j}(\psi _2,\phi _j)_{L^2(\Sigma )}(\psi _3,\phi _j)_{L^2(\Sigma )}\psi _1(s)\varphi (t)}_{h_\epsilon (s,t,j)}dsdt\right| . \end{aligned}$$

(4.25)

Now let $g(s,t,j):=|(\psi _2,\phi _j)_{L^2(\Sigma )}(\psi _3,\phi _j)_{L^2(\Sigma )}|\cdot |\psi _1(s)\varphi (t)|$, then

$$\begin{aligned} |h_\epsilon (s,t,j)|\le g(s,t,j). \end{aligned}$$

(4.26)

Moreover,

$$\begin{aligned} \sum _{j}(\psi _2,\phi _j)_{L^2(\Sigma )}(\psi _3,\phi _j)_{L^2(\Sigma )}=\int _\Sigma \psi _2(w)\psi _3(w)\sqrt{h(w)}dw. \end{aligned}$$

(4.27)

This implies the sequence is unconditionally convergent and therefore absolutely convergent.

Hence, $g(s,t,j)\in L^1(dt\times ds\times \mu )$, where $\mu$ is the counting measure on $\mathbb {N}$.

Using dominated convergence in Eq. (4.30) and Parseval’s Identity in Eq. (4.31) we obtain for $\epsilon _n\rightarrow 0^+$

$$\begin{aligned}&\lim _{\epsilon \rightarrow 0^+ }\langle \langle F(t+i\epsilon ),\psi _1(s)\psi _2(x)\psi _3(y)\rangle ,\varphi (t)\rangle \end{aligned}$$

(4.28)

$$\begin{aligned}&\quad =\lim _{n\rightarrow \infty }\int _{\mathbb {R}}\int _{M}\sum _{j}\lambda _j^{-1}e^{i(t+i\epsilon _n)\lambda _j}e^{-is\lambda _j}\left( \int _{\Sigma }\psi _2(x)\phi _j(x)dx\right) \nonumber \\&\qquad \times \phi _{j}(y)\psi _1(s)\psi _3(y)dyds\varphi (t)dt \end{aligned}$$

(4.29)

$$\begin{aligned}&\quad =\int _{\mathbb {R}}\int _{\mathbb {R}}\sum _{j}\lim _{n\rightarrow \infty }\lambda _j^{-1}e^{i(t+i\epsilon _n)\lambda _j}e^{-is\lambda _j}\left( \int _{\Sigma }\psi _3(y)\phi _{j}(y)dy\right) \left( \int _{\Sigma }\psi _2(x)\phi _j(x)dx\right) \nonumber \\ {}&\qquad \times \psi _1(s)ds\varphi (t)dt \end{aligned}$$

(4.30)

$$\begin{aligned}&\quad =\int _{\mathbb {R}}\int _{M}\sum _{j}\lambda _j^{-1}e^{it\lambda _j}e^{-is\lambda _j}\left( \int _{\Sigma }\psi _2(x)\phi _j(x)dx\right) \phi _{j}(y)\psi _1(s)\psi _3(y)dyds\varphi (t)dt \end{aligned}$$

(4.31)

$$\begin{aligned}&\quad =\omega ^{(2)}_{G}(\varphi \psi _1\psi _2\psi _3). \end{aligned}$$

(4.32)

Therefore $\lim _{\epsilon \rightarrow 0^+ }\langle F(t+i\epsilon ),\cdot \rangle =\omega ^{(2)}_{G}\in {\mathcal D}'(M\times M)$. Applying [12, Proposition 7.5] we obtain

$$\begin{aligned} WF^{s}(\omega ^{(2)}_{G})\subset WF(\omega ^{(2)}_{G})\subset \{(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })\in T^*(M\times M); \xi _0> 0\} \end{aligned}$$

(4.33)

which gives $\sum \nolimits _{\mu =0}^3 g^{0\mu }\xi _\mu =\xi ^0>0$. $\square$

Lemma 4.15

Let $(\tilde{x},\tilde{y})\in M\times M$ be such that $\tilde{x}$ and $\tilde{y}$ are not causally related, i.e. $\tilde{x}\notin J(\tilde{y})$. Then $(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })\notin WF^{-\frac{1}{2}-\epsilon +\tau } (\omega ^{(2)}_{G})$ for every $\epsilon >0$.

Proof

From Eq. (4.19), Lemma 4.13 and Theorem 4.14 we conclude that

$$\begin{aligned} WF^{-\frac{1}{2}-\epsilon +\tau } (\omega ^{(2)}_{G})\subset N_+\times N_{-}, \end{aligned}$$

(4.34)

where $N_{\pm }:=\{(t,x,\xi _0,\xi )\in {{{\,\textrm{Char}\,}}}(P); \pm \xi _0>0\}$

Now consider the restriction $\omega ^{(2)}_{G}|_{\mathcal {Q}}:=\omega ^{(2)}_{G}:{{\mathcal {D}}(M\times M)}|_{\mathcal {Q}}\rightarrow \mathbb {C}$, where the set $\mathcal {Q}$ is defined as the set of pairs of causally separated points $(\tilde{x},\tilde{y})\in M\times M$.

Notice that $\omega ^{(2)}_{G}= \omega ^{+}+i K_G$, where $\omega ^+$ is the Schwartz kernel of $A^{-\frac{1}{2}}\cos (A^{\frac{1}{2}}(t-s))$ and $K_G$ is the causal propagator, which is the Schwartz kernel of $A^{-\frac{1}{2}}\sin (A^{\frac{1}{2}}(t-s))$. Since $K_G|_{\mathcal {Q}}=0$ by [26, Lemma 5.1] we have $\omega ^{(2)}_{G}|_{\mathcal {Q}}=\omega ^{+}|_{\mathcal {Q}}$.

Also, the “flip” map $\rho (\tilde{x},\tilde{y})=(\tilde{y},\tilde{x})$ is a diffeomorphism of $\mathcal {Q}$ and we have $\rho ^*\omega ^{+}=\omega ^{+}$. Moreover, using the covariance of the Sobolev wavefront set under diffeomorphisms (see Appendix 5.1), we have

$$\begin{aligned} WF^{-\frac{1}{2}-\epsilon +\tau }(\omega ^{+}|_{\mathcal {Q}})=WF^{-\frac{1}{2}-\epsilon +\tau }(\rho ^*\omega ^{+}|_{\mathcal {Q}})=\rho ^*WF^{-\frac{1}{2}-\epsilon +\tau }(\omega ^{+}|_{\mathcal {Q}}). \end{aligned}$$

(4.35)

Moreover, $\rho ^*( N_+\times N_{-})= N_-\times N_{+}$ which implies

$$\begin{aligned} WF^{-\frac{1}{2}-\epsilon +\tau }(\omega ^{(2)}_{G}|_{\mathcal {Q}})\subset ( N_+\times N_{-})\cap ( N_-\times N_{+})=\emptyset . \end{aligned}$$

(4.36)

$\square$

Lemma 4.16

If $(\tilde{x},\tilde{\xi },\tilde{x},\tilde{\eta })\in WF^{-\frac{3}{2}-\tilde{\epsilon }+\tau }(\omega ^{(2)}_{G})$ for some $\tilde{\epsilon }>0$, then $\tilde{\eta }=-\tilde{\xi }$.

Proof

Note that $WF^{-\frac{3}{2}-\tilde{\epsilon }+\tau }(\omega ^{(2)}_{G})\subset WF^{-\frac{3}{2}-{\epsilon }+\tau }(\omega ^{(2)}_{G})$ for $0<\epsilon <\tilde{\epsilon }$, so that we may possibly decrease $\tilde{\epsilon }$. Suppose $\tilde{\eta }$ and $\tilde{\xi }$ are linearly independent, i.e. $\tilde{\eta }\ne \lambda \tilde{\xi }$ for $\lambda \in \mathbb {R}$. By Lemma 4.13, $(\tilde{x}, \tilde{\xi },\tilde{x}, \tilde{\eta })\in {{\,\textrm{Char}\,}}(P)\times {{\,\textrm{Char}\,}}(P)$.

Now we choose a Cauchy hypersurface $\Sigma _{ t_0}=\{t_0\}\times \Sigma$ such that the null geodesic with initial data $(\tilde{x},\tilde{\xi })$ and the null geodesic with initial data $(\tilde{x},\tilde{\eta })$ intersect it. These points of intersections are unique by global hyperbolicity (see [4, 21, 25] for low regularity definitions). Moreover, using the condition $\tilde{\eta }\ne \lambda \tilde{\xi }$, we can choose $\Sigma _ {t_0}$ such that these points are distinct. We denote these points by $(t_0,x_0), (t_0, y_0)$. Clearly, these points are not causally related.

Notice that $\omega _A$ satisfies $P_{(t,x)}\omega _A=0$. Given any $\epsilon >0$ we can achieve $P^b_{(t,x)}\omega _A\in H^{-\frac{3}{2}-\epsilon +\tau }$ by fixing $\delta$ in Remark 4.6 close to 1. Taking $\epsilon$ small, this allows us to choose $s=-\frac{3}{2}-\epsilon +\tau$ with $0<s<\tau -1$ in Theorem 4.7. This propagation of singularities result applied to the distribution $\omega _A$ and the operator $P_{(t,x)}$ guarantees that if $(\tilde{x},\tilde{\xi },\tilde{x},\tilde{\eta })\in WF^{-\frac{3}{2}-\epsilon +\tau }(\omega ^{(2)}_{G})\subset WF^{-\frac{1}{2}-\epsilon +\tau }(\omega _A)$ then the full null bicharacteristic is contained in the wavefront set i.e. $(\gamma (\tilde{x},\tilde{\xi }),(\tilde{x},\tilde{\eta }))\in {WF^{-\frac{1}{2}-\epsilon +\tau }(\omega _A)}$, where $\gamma (\tilde{x},\tilde{\xi })$ is the null bicharacteristic with initial data $(\tilde{x},\tilde{\xi })$. Similarly, using the operator $P_{(s,y)}$, we obtain $(\gamma (\tilde{x},\tilde{\xi }),\gamma (\tilde{x},\tilde{\eta }))\in {WF^{-\frac{1}{2}-\epsilon +\tau }(\omega _A)}$, where $\gamma (\tilde{x},\tilde{\eta })$ is the null bicharacteristic with initial data $(\tilde{x},\tilde{\eta })$.

Now we show that $(\gamma (\tilde{x},\tilde{\xi }),\gamma (\tilde{x},\tilde{\eta }))\in {WF^{-\frac{1}{2}-\epsilon +\tau }(\omega ^{(2)}_{G})}$.

By Theorem 6.1.1’ from [8] we have

$$\begin{aligned} WF^{-\frac{1}{2}-\epsilon +\tau }(\omega _A)\backslash WF^{-\frac{1}{2}-\epsilon +\tau }(\omega ^{(2)}_{G})= & {} WF^{-\frac{1}{2}-\epsilon +\tau }(\omega _A)\backslash WF^{-\frac{1}{2}-\epsilon +\tau }(i\partial _{t}\omega _A)\nonumber \\ {}{} & {} \subset {{\,\textrm{Char}\,}}(i\partial _{t}). \end{aligned}$$

(4.37)

However, using Eq.(4.20) and that $(\partial _t+\partial _s)\omega _A=0$, we have the inclusion

$$\begin{aligned} {WF^{-\frac{1}{2}-\epsilon +\tau }(\omega _A)\subset WF^{\frac{1}{2}-\epsilon +\tau }(\omega _A)} \subset ({{\,\textrm{Char}\,}}(P)\times {{\,\textrm{Char}\,}}(P))\cap {{\,\textrm{Char}\,}}(\partial _t+\partial _s). \end{aligned}$$

(4.38)

Since $({{\,\textrm{Char}\,}}(P)\times {{\,\textrm{Char}\,}}(P))\cap {{\,\textrm{Char}\,}}(\partial _t+\partial _s)\cap {{\,\textrm{Char}\,}}(i\partial _{t})=\emptyset$, taking the intersection between Eq.(4.37) and Eq.(4.38), we obtain that the left hand side of Eq.(4.37) must be empty. Therefore,

$$\begin{aligned} {WF^{-\frac{1}{2}-\epsilon +\tau }(\omega _A)\subset WF^{-\frac{1}{2}-\epsilon +\tau }(\omega ^{(2)}_{G}).} \end{aligned}$$

(4.39)

Hence, $(\gamma (\tilde{x},\tilde{\xi }),\gamma (\tilde{x},\tilde{\eta }))\in {WF^{-\frac{1}{2}-\epsilon +\tau }}(\omega ^{(2)}_{G})$. In particular $(t_0,x_0,\tilde{\xi }, t_0, y_0, \tilde{\eta })\in {WF^{-\frac{1}{2}-\epsilon +\tau }}(\omega ^{(2)}_{G}))$. However, this is a contradiction to Lemma 4.15. Therefore, $\tilde{\eta }=\lambda \tilde{\xi }$ for some $\lambda \in \mathbb {R}$. Using Eq.(4.19) we have $\xi _0=-\eta _0$ which gives $\lambda =-1$, i.e. $\tilde{\eta }=-\tilde{\xi }$. $\square$

We have used the distribution $\omega _{A}$, because a direct application of Theorem 4.5 for $\omega ^{(2)}_{G}$ is not possible, since for $\delta$ close to 1, $\sigma$ cannot take the value $-\frac{1}{2}$.

Now we state the main result

Theorem 4.17

$WF'^{-\frac{3}{2}-{\epsilon }+\tau }(\omega ^{(2)}_{G})\subset C^+$ for every ${\epsilon }>0$ and $C^+$ as in Eq.(2.4).

Proof

Let $(\tilde{x},\tilde{\xi },\tilde{y},-\tilde{\eta })\in WF^{-\frac{3}{2}-{\epsilon }+\tau }(\omega ^{(2)}_{G}) {\subset WF^{-\frac{1}{2}-\epsilon +\tau }(\omega _{A})}$, where the inclusion follows from [15, Proposition B.3] since $\omega ^{(2)}_{G}=i\partial _t\omega _{A}$. The propagation of singularities result (Theorem 4.7) implies that $(\gamma (\tilde{x},\tilde{\xi }),\gamma (\tilde{y},-\tilde{\eta }))\in { WF^{-\frac{1}{2}-\epsilon +\tau }}(\omega _{A})$, where $\gamma (\tilde{x},\tilde{\xi })$ is the null bicharacteristic with initial data $(\tilde{x},\tilde{\xi })$ and $\gamma (\tilde{y},-\tilde{\eta })$ is the null bicharacteristic with initial data $(\tilde{y},-\tilde{\eta })$. Hence, by Eq.(4.39), we have $(\gamma (\tilde{x},\tilde{\xi }),\gamma (\tilde{y},-\tilde{\eta }))\in {W F^{-\frac{1}{2}-\epsilon +\tau }}(\omega ^{(2)}_{G})$.

Now we choose a Cauchy surface $\Sigma _{t_1}=\{t_1\}\times \Sigma$ and suppose that $(t_1,x_1,\tilde{\xi }_1,t_1,x_2,\tilde{\xi }_2)\in (\gamma (\tilde{x},\tilde{\xi }),\gamma (\tilde{y},-\tilde{\eta }))\cap (\Sigma _{t_1}^2)$. By Lemmas 4.15 and 4.16, $(t_1, x_1,\tilde{\xi }_1),(t_1, x_2,\tilde{\xi }_2)\in {{\,\textrm{Char}\,}}(P)$, $x_1=x_2$, and $\tilde{\xi }_2 = -\tilde{\xi }_1$.

Next we define a curve $\tilde{\gamma }:(-\infty ,\infty )\rightarrow M$ as follows. First, we shift the parametrisation $\lambda$ in the definition of the null bicharacteristics so that

$$\begin{aligned} \gamma (\tilde{x},\tilde{\xi })(t_1)= (t_1,x_1,\tilde{\xi }_1) \quad \gamma (\tilde{y},-\tilde{\eta })(t_1)= (t_1,x_1,-\tilde{\xi }_1). \end{aligned}$$

Then, we denote by $\Pi :T^*M\rightarrow M$ the canonical projection and define two curves in M by

$$\begin{aligned} \gamma _1(\lambda ):=\Pi (\gamma (\tilde{x},\tilde{\xi })(\lambda ))\quad \gamma _2(\lambda ):=\Pi (\gamma (\tilde{y},-\tilde{\eta })(\lambda )). \end{aligned}$$

Notice that we have ${\gamma _1}(t_1)=(t_1,x_1), \dot{\gamma _1}(t_1)=g^{-1}(\tilde{\xi }_1,\cdot )$ and ${\gamma _2}(t_1)=(t_1,x_1),\dot{\gamma _2}({t_1})=g^{-1}(-\tilde{\xi }_1,\cdot )$. Moreover, we can assume that $\tilde{x}= \gamma _1(a)$ and $\tilde{y}=\gamma _2(b)$ for suitable $a, b\in {\mathbb {R}}$ with $a<t_1<b$.

Finally, let

$$\begin{aligned} \tilde{\gamma }(\lambda )={\left\{ \begin{array}{ll} \gamma _1(\lambda )&{} \lambda \in (-\infty , t_1]\\ -\gamma _2(\lambda )&{} \lambda \in (t_1,\infty ) \end{array}\right. } \end{aligned}$$

(4.40)

where $-\gamma _2$ denotes the curve with opposite orientation.

Then $\tilde{\gamma }(a) = \tilde{x}$, $\tilde{\gamma }(b) = \tilde{y}$; moreover $g(\cdot , {\dot{\tilde{\gamma }}})|_{T_{\tilde{x}}M}=\tilde{\xi }$, $g(\cdot , {\dot{\tilde{\gamma }}})|_{T_{\tilde{y}}M}=\tilde{\eta }$ and therefore, $\tilde{\gamma }$ is a null geodesic between $\tilde{x}$ and $\tilde{y}$ with cotangent vectors $\tilde{\xi }$ at $\tilde{x}$ and $\tilde{\eta }$ at $\tilde{y}$, i.e. $(\tilde{x},\tilde{\xi },\tilde{y},-\tilde{\eta })\in C':=\{(\underline{\tilde{x}}, \underline{\tilde{\xi }},\underline{\tilde{y}}, -\underline{\tilde{\eta }});(\underline{\tilde{x}}, \underline{\xi } ; \underline{\tilde{y}}, \underline{\tilde{\eta }}) \in C\}$. This shows

$$\begin{aligned} {WF^{-\frac{3}{2}-\epsilon +\tau }}(\omega ^{(2)}_{G})\subset C' \end{aligned}$$

(4.41)

Using the definition of $WF^{l '}(u){:=} \{(\tilde{x}, \tilde{\eta }; \tilde{y}, -\tilde{\eta }) {\in } T^{*}(M\times M); (\tilde{x}, \tilde{\xi }; \tilde{y}, \tilde{\eta }) \in WF^l(u)\}$ and Theorem 4.14 gives the result. $\square$

Remark 4.18

For a $C^{1,1}$ metric the same arguments as used in [26, Theorem 7.1] apply and therefore in that scenario we have for every $\epsilon >0$

$$\begin{aligned} WF^{\frac{1}{2}-\epsilon }(\omega ^{(2)}_{G})\subset C'^+. \end{aligned}$$

Appendix

1.1 Covariance of the Sobolev Wavefront Set under Diffeomorphisms

The following is a variant of Theorem [13, Theorem 8.2.4] adapted to the $H^s$-wave front set.

Lemma 5.1

Let $\varphi :M\rightarrow M$ be a $C^\infty$ diffeomorphism and $u\in {\mathcal {D}}'(M)$. Then

$$\begin{aligned} WF^s(\varphi ^* u)=\varphi ^*WF^s(u),\quad s\in \mathbb {R}. \end{aligned}$$

(5.1)

Proof

Let $(x,\xi )\notin \varphi ^*WF^s(u)$ which by definition implies $(\varphi (x),^t\partial \varphi (x)^{-1}\xi )\notin WF^s(u)$. By [8, p. 202], we can write $u=u_1+u_2$ where $u_1\in H^s_{loc}$ and $(\varphi (x),^t\partial \varphi (x)^{-1}\xi )\notin WF(u_2)$. By the covariance of the Sobolev spaces in compact sets [29, Chapter 4, Section 2] we have $\varphi ^*u_1\in H^s_{loc}$ and by the covariance under diffeomorphism of the wavefront set $(x,\xi )\notin WF(\varphi ^*u_2)=\varphi ^*WF(u_2)$. Putting this together gives $(x,\xi )\notin WF^s(\varphi ^*u)$, i.e. $WF^s(\varphi ^*u)\subset \varphi ^*WF^s(u)$.

Applying this relation to $\varphi ^{-1}$ we conversely see that

$\varphi ^*WF^s(u) = \varphi ^*WF^s(\varphi ^{-1*}\varphi ^*u)\subset \varphi ^*\varphi ^{-1*}WF^s(\varphi ^*u) = WF^s(\varphi ^*u).$

This completes the argument. $\square$