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.
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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
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
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
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
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
for \(F_1, F_2 \in \Gamma \). The Wightman two-point function \(\omega ^{(2)}_{\mu }\) associated with the state \(\omega _{\mu }\), is given by
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
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
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 \(s\le N +\frac{3}{2}\)
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
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
It extends to continuous maps
3.3 Symbol Smoothing
Given \(p(x,\xi )\in C^{\tau }S_{1,\delta }^{m}\) and \(\gamma \in (\delta ,1)\) let
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
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]:
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
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
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
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
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
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 \((k_{G},{\mathcal {H}}_{G})\)
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:
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),
since A is selfadjoint. Combining (4.4) and (4.5) we obtain
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
which gives the singular integral kernel representation
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
and the associated covariables as
Using the notation above and in (4.1), we have:
Remark 4.4
The Klein–Gordon operator on M is given by
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
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:
In particular,
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
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
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 \(-1\le \theta \le 1\)
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
We have by direct computation that
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)
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
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
Then,
Proof
Direct computation shows that for \(\psi \) as in the previous proof and \(s\ge 0\) we have
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 \(s=\frac{1}{2}-\epsilon \)
for a suitable constant C. \(\square \)
Remark 4.12
Notice that \(i\partial _{t}\omega _A=\omega ^{(2)}_{G}\).
Lemma 4.13
For any \(\tilde{\epsilon }>0\)
Proof
Since \(\omega ^{(2)}_{G}\) satisfies \((\partial _t+\partial _s)\omega ^{(2)}_{G}=0\) we conclude that for all \(l\in \mathbb {R}\)
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
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
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
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
\(\square \)
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
![](http://media.springernature.com/lw498/springer-static/image/art%3A10.1007%2Fs00023-023-01280-7/MediaObjects/23_2023_1280_Equ33_HTML.png)
Notice that \(\partial _zF:\mathbb {R}+i]0,\delta [\subset \mathbb {C}\rightarrow \mathcal D'(\Sigma \times M)\) is given by
![](http://media.springernature.com/lw496/springer-static/image/art%3A10.1007%2Fs00023-023-01280-7/MediaObjects/23_2023_1280_Equ34_HTML.png)
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
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
Moreover,
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 \(\epsilon _n\rightarrow 0^+\)
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
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
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
Moreover, \(\rho ^*( N_+\times N_{-})= N_-\times N_{+}\) which implies
\(\square \)
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
However, using Eq.(4.20) and that \((\partial _t+\partial _s)\omega _A=0\), we have the inclusion
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,
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
Then, we denote by \(\Pi :T^*M\rightarrow M\) the canonical projection and define two curves in M by
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
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
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 \(\epsilon >0\)
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We are grateful to Chris Fewster, Bernard Kay and James Vickers for helpful discussions. We also thank the anonymous reviewers for their comments.
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Appendix
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
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. \(\square \)
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Sanchez Sanchez, Y., Schrohe, E. Adiabatic Ground States in Non-smooth Spacetimes. Ann. Henri Poincaré 24, 2929–2948 (2023). https://doi.org/10.1007/s00023-023-01280-7
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DOI: https://doi.org/10.1007/s00023-023-01280-7