Adiabatic Ground States in Non-Smooth Spacetimes

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 microlocal estimates for the causal propagator, propagation of singularities results for non-smooth pseudodifferential operators, and eigenvalue asymptotics for elliptic operators of low regularity.


Introduction
The analysis of quantum fields in spacetimes where the metric is not smooth 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 [21] 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 [19].
In this paper we focus on the on scalar fields φ that satisfy the Klein-Gordon equation on a manifold M = R × Σ where Σ is a compact Cauchy hypersurface, g µν is the inverse metric tensor of a ultrastatic metric, ∇ µ is the covariant derivative and m 2 is a positive real number.
In the smooth setting, Fulling, Narcowich and Wald showed that the ground state in an ultrastatic spacetime is a Hadamard state [10]. Later, Kay and Wald showed the (non)existence of Hadamard states in stationary spacetimes with a bifurcate Killing horizon [18]. Then, Radzikowski introduced a microlocal characterisation in terms of the wavefront set [22]. This result allowed for further constructions of these states, for example by Junker [15] and Gérard and Wrochna [11].
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 [13]. 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 4.16).
Outline of the paper: In Section 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 Section 3, we state the main definitions and theorems regarding non-smooth pseudodifferential operators. In Section 4, we focus on ultra-static spacetimes and show that the ground state is an adiabatic state.

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, Ξ), and the classical observables are defined as smooth functionals on (V, Ξ). 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, ω, are given by certain positive linear functionals on the C * -algebra [29,2]. Below we give details of the quantisation procedure.

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, Ξ) is given by where H 1 comp (M ) denotes compactly supported function in the Sobolev space H 1 (M ) and ker G is the kernel of the causal propagator [13]. In fact, this symplectic space is symplectically isomorphic to the classical phase space (Γ, σ) given by the space Γ := H 2 comp (Σ) ⊕ H 1 comp (Σ) of real-valued initial data with compact support and the symplectic bilinear form with F i := (q i , p i ) ∈ Γ, i = 1, 2 and dv the induced volume form on Σ.
Moreover, to the symplectic space (V, Ξ) one can associate a C * -algebra A that satisfies the CCR, known as the Weyl algebra. It is generated by the elements [2,13]).
As (V, Ξ) and (Γ, σ) are isomorphic as symplectic spaces, one can construct a C * -algebra, B, using the map α : The algebra B is * -isomorphic to the Weyl algebra 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 [13,Theorem 6.12]).

States
The quantum states as defined above need to be further restricted in order to be physically relevant. A candidate for physical quantum states, ω, are quasifree states that satisfy the microlocal spectrum condition.
To be precise, given a real scalar product µ : Γ × Γ → R satisfying for all F 1 , F 2 ∈ Γ, there exist a quasifree state ω µ acting on the algebra B associated with µ given by ω µ (W (F )) = e − 1 2 µ(F,F ) . Moreover, one can determine the ("symplectically smeared") two-point function of ω µ by for F 1 , F 2 ∈ Γ. The Wightman two-point function ω (2) µ associated to the state ω µ , is given by for f 1 , f 2 ∈ H 1 comp (M ). By restricting the two point function ω To define the microlocal spectrum condition, it is useful to introduce the sets where (x,ξ) ∼ (ỹ,η) means thatξ,η are cotangent to the null geodesic γ atx resp.ỹ and parallel transports of each other along γ.
Using the above sets one can define the microlocal spectrum condition which goes back to Radzikowski [22]: Definition 2.1. A quasifree state ω H on the algebra of observables satisfies the microlocal spectrum condition if its two point function ω (2) H is a distribution in D ′ (M × M ) and satisfies the following wavefront set condition These states are called Hadamard states and include ground states in smooth spacetimes [10,23,15,11,9].
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 [14]. These states are the natural generalisation of Hadamard states suitable for spacetimes with limited regularity.
Definition 2.2. A quasifree state ω N on the algebra of observables is called an adiabatic state of order N ∈ R if its two-point function ω (2) N is a bidistribution that satisfies the following H s -wavefront set condition for all s ≤ N + 3 where W F 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 , ξ 0 ) ∈ T * M \0 if there exists a conic neighbourhood Γ of ξ 0 and a smooth function ϕ ∈ D(M ) with ϕ(x 0 ) = 0 such that Γ ξ 2s |F(ϕu)(ξ)| 2 d n ξ < ∞.
Otherwise we say that (x 0 , ξ 0 ) lies in the s-wave front set W F s (u).
If u is microlocally in H s in an open conic subset Γ ⊂ T * M \0 we write u ∈ H s mcl (Γ).
3 Pseudodifferential Operators with Non-smooth Symbols

Symbol Classes
Let {ψ j ; j = 0, 1, . . .} be a Littlewood-Paley partition of unity on R n , i.e., a partition of unity The support of ψ j , j ≥ 1, then lies in an annulus around the origin of interior radius 2 j−1 and exterior radius 2 1+j .
Here ψ j (D) is the Fourier multiplier with symbol ψ j , i.e., We have the following relations We next introduce symbol classes of finite Hölder or Zygmund regularity, following Taylor [26]. We use the notation ξ : (b) We obtain the symbol class C τ S m 1,δ for τ > 0 by requiring that in terms p m−j homogeneous of degree m − j in ξ for |ξ| ≥ 1, in the sense that the difference between p(x, ξ) and the sum over

Characteristic Set and Pseudodifferential Operators
Let p ∈ C τ S m ρ,δ , τ > 0, with δ < ρ. Suppose that there is a conic neighborhood Γ of (x 0 , ξ 0 ) and constants c, C > 0 such that |p(x, ξ)| ≥ c|ξ| m for (x, ξ) ∈ Γ, |ξ| ≥ C. Then (x 0 , ξ 0 ) is called non-characteristic. If p has a principal homogeneous symbol p m , the condition is equivalent to p m (x 0 , ξ 0 ) = 0. The complement of the set of non-characteristic points is the set of characteristic points denoted by Char(p).

Remark 3.3. The Klein-Gordon operator on M is given by
For a metric of regularity C τ , the symbol P (x,ξ) belongs to C τ −1 S 2 cl and The pseudodifferential operator p(x, D x ) with the symbol p(x, ξ) ∈ C τ S m 1,δ is given by It extends to continuous maps

Ground States in Ultrastatic Spacetimes
Let M = R × Σ where Σ is a 3-dimensional compact manifold and the Lorentzian metric g is of the form where h ij (x) are the components of a time independent Riemannian metric of Hölder regularity C τ (when τ ∈ N we will consider the Zygmund spaces C τ * , introduced in Definition 3.1).
Moreover, the vector field ∂ t induces a one-parameter group of isometries τ t : M → M, t ∈ R, such that τ t (Σ to ) = Σ to+t . This group induces a one-parameter group of automorphisms in the C * -algebras as follows. Define T (t) : Γ → Γ by . Since the symplectic form σ is invariant under the action of T (t) and since T (t)T (s) = T (t + s) t, s ∈ R, T is a one-parameter group of symplectic transformations (also called Bogoliubov transformations). It gives rise to a group of automorphismsα(t), t ∈ R, (Bogoliubov automorphisms) on the algebra B viã In this case, there exists a preferred class of states on A, namely those invariant under α(t). A quasifree state ω µ will be invariant under this symmetry if and only if The specification of µ is equivalent to the specification of a one-particle structure as established by the following theorem of Kay and Wald [18, Proposition 3.1]: Then there exists a one-particle Hilbert space structure, i.e. a Hilbert space H and a real-linear map k :  Moreover, the automorphism groupα(t) can be unitarily implemented in the one-particle Hilbert space structure (k, H) of an invariant state ω µ , i.e. there exists a unitary group U (t), t ∈ R, on H satisfying If U (t) is strongly continuous it takes the form U (t) = e −iht for some self-adjoint operator h on H.
We define now the notion of ground states following Kay [17]: Let the phase space (Γ, σ, T (t)) be given. A quasifree ground state is a quasifree state over B[Γ, σ] with one-particle Hilbert space structure (k, H) and a strongly continuous unitary group U (t) = e −iht (satisfying (4.1)) such that h is a positive operator (the "one-particle Hamiltonian").
In the ultrastatic case we define the ground state, ω G by the one-particle Hilbert space structure (k G , H G ) where A := −∆ h φ + m 2 and t 0 ∈ R (invariance under time translations makes any choice of t ∈ R equivalent to any other) and the strongly continuous unitary group is given by The Wightman two-point function of ω G is: for h 1 , h 2 ∈ D(M).
Moreover using Eq. (2.3), Eq.(4.2) and Theorem 4.1 the "symplectically smeared two-point function" λ G is given on the initial data F i = q i p i ∈ Γ by Eq. (2.2), . (4.5) The two-point function, ω G , of the ground state, ω G , is the Schwartz kernel of the operator e iA where {φ j , j = 1, 2, . . .} is an orthonormal basis of eigenfunctions of L 2 (Σ) associated to the eigenvalues λ 2 j of the operator m 2 I − ∆ h . The proof that the ground state in an ultrastatic smooth globally hyperbolic space-time is a Hadamard state has been shown by different methods [10,23,11,15]. In the following section we show that the ground states is an adiabatic state in the non-smooth case.

Microlocal analysis for Bisolutions of the Klein-Gordon Operator
We write local coordinates on R × Σ in the form x = (t, x),ỹ = (s, y) (4.7) and the associated covariables asξ = (ξ 0 , ξ),η = (η 0 , η). In the sequel we shall apply the Klein-Gordon operator also to functions and distributions on M × M . Using the coordinates in Eqs.(4.7),(4.8) and (4.9), we distinguish the cases, where P acts on the first set of variables (t, x) or on the second set (s, y), and write P (t,x) and P (s,y) , respectively. The associated symbols P (t,x) (x, ξ) and P (s,y) (x, ξ) formally depend on the full set of (co-)variables (x, ξ), however, only the (co-)variables associated with either (t, x) or (s, y) show up: In particular, Remark 4.5. Applying the symbol smoothing directly to P (t,x) ∈ C τ −1 S 2 1,0 would leave us with . Therefore, we smooth each of the non-smooth symbols ( the principal symbol and the sub-leading term) separately to obtain the remainder p b Furthermore, the main results on the microlocal propagation of singularities in the non-smooth setting that we will apply can be found in [  If for some s ∈ R, f ∈ H s mcl (Γ) and P b (t,x) u ∈ H s mcl (Γ) where γ ⊂ Γ with Γ a conical neighbourhood and u ∈ H s+1 mcl (γ(0)) then u ∈ H s+1 mcl (γ).

The Microlocal Spectrum Condition
Now we will show that the Wightman two-point function of the ground state described above satisfies Defintion 2.2. We will assume throughout this section that the metric is of regularity C τ with τ > 2.
Let {φ j ⊗ φ k ; j, k = 1, 2, . . .} be an orthonormal basis of L 2 (Σ) ⊗ L 2 (Σ) associated to the eigenfunctions {φ j } and the eigenvalues {λ 2 j } of the operator m 2 I − ∆ h . Then, for u ∈ L 2 (M × M ) we have the representation The previous theorem allows us to establish the local Sobolev regularity of the two-point function.
Proof. Let ψ ∈ D(M × M ). We will show ψω We have by direct computation that (4.14) Taking into account that ψ L 2 (R 2 ) = F(ψ) L 2 (R 2 ) < ∞ we have (with constants possibly changing from line to line) From Weyl's law for non-smooth metrics [30, Theorem 1.1] we obtain the estimate l 2 3 ≤ Cλ 2 j for a suitable constant C which gives It will be useful to consider the following bidistribution: Then, Proof. Direct computation shows that for ψ as in the previous proof where we have chosen j 0 large enough such that λ j 0 > 1.
According to Weyl's law for non-smooth metrics [30, Theorem 1.1] we have the estimate l 2 3 ≤ Cλ 2 l for a suitable constant C. This gives for s = 1 2 − ǫ for a suitable constant C.
Using dominated convergence we obtain Lemma 4.14.