Construction of Hadamard states by characteristic Cauchy problem

Construction of Hadamard states by characteristic Cauchy problem

Abstract.

We construct Hadamard states for Klein-Gordon fields in a spacetime equal to the interior of the future lightcone from a base point in a globally hyperbolic spacetime . Under some regularity conditions at future infinity of , we identify a boundary symplectic space of functions on , which allows to construct states for Klein-Gordon quantum fields in from states on the algebra associated to the boundary symplectic space. We formulate the natural microlocal condition on the boundary state on ensuring that the bulk state it induces in satisfies the Hadamard condition. Using pseudodifferential calculus on the cone we construct a large class of Hadamard boundary states on the boundary with pseudodifferential covariances, and characterize the pure states among them. We then show that these pure boundary states induce pure Hadamard states in .

Key words and phrases:
Hadamard states, microlocal spectrum condition, pseudo-differential calculus, characteristic Cauchy problem, curved spacetimes
2010 Mathematics Subject Classification:
81T13, 81T20, 35S05, 35S35

1. Introduction

Hadamard states are widely accepted as physically admissible states for non-interacting quantum fields on a curved spacetime, one of the main reasons being their link with the renormalization of the stress-energy tensor, a basic step in the formulation of semi-classical Einstein equations. Furthermore, they are nowadays considered a necessary ingredient in the perturbative formulation of interacting (non-linear) theories (cf. recent review articles [KM, HW]).

For Klein-Gordon fields, the construction of Hadamard states amounts to finding bi-solutions of the Klein-Gordon equation (called in this context two-point functions and denoted here ) with a specified wave front set (that is, verifying the microlocal spectrum condition) and satisfying additionally a positivity property [Ra].

There exist several methods to construct Hadamard states for Klein-Gordon fields: the first method relies on the Fulling-Narcowich-Wald deformation argument [FNW], which reduces the construction of Hadamard states on an arbitrary spacetime to the case of ultrastatic spacetimes, where vacuum or thermal states are easily shown to be Hadamard states.

The second approach, worked out in [Ju, JS, GW], uses pseudodifferential calculus on a fixed Cauchy surface in and relies on the construction of a parametrix for the Cauchy problem on . To use pseudodifferential calculus, some restrictions on and on the behavior of the metric at spatial infinity are necessary. On the other hand, the method produces a large classes of rather explicit Hadamard states, whose covariances, expressed in terms of Cauchy data are pseudodifferential operators.

Another method, initiated by Moretti [Mo1, Mo2] applies to conformal field equations, like the conformal wave equation, on an asymptotically flat vacuum spacetime . By asymptotic flatness, there exists a metric , conformal to , and a spacetime such that can be causally embedded as an open set in , with the boundary of being null in . States on the boundary symplectic space, containing the traces on of solutions of the wave equation in , naturally induce states inside .

This method has been successfully applied in [Mo1, Mo2] to construct a distinguished Hadamard state for asymptotically flat vacuum spacetimes with past time infinity and then extended to several other geometrical situations in [DMP1, DMP2, BJ]. Further results also include generalization to Maxwell fields [DS] and linearized gravity [BDM].

In the present paper we rework systematically the above strategy in terms of the associated characteristic Cauchy problem in order to construct a large class of Hadamard states (instead of a preferred single one) and to characterize the pure ones. For the sake of clarity, we do not impose geometrical assumptions on that allow to correctly embed it in a larger spacetime .

Instead we go the other way around and work in an a priori arbitrary globally hyperbolic spacetime , fix a base point and consider the interior of the future lightcone

as the spacetime of main interest, i.e. , where (resp. ) is the time-like (resp. causal) shadow of , cf. [Wa, Sec. 8.1].

We make the following assumption on the geometry of .

Hypothesis 1.1.

We assume that there exists such that:

Using Hypothesis 1.1 one can construct coordinates near , such that and

where is a Riemannian metric on .

Such choice of coordinates allows one to identify with . A natural space of smooth functions on is then provided by  — the intersection of Sobolev spaces of all orders, defined using the standard metric on .

We consider the Klein-Gordon operator (with real-valued) and its restriction on , denoted . The bulk-to-boundary correspondence can be expressed in this setup as follows. For an appropriate choice of , the restriction map

is a monomorphism1 between the symplectic space of smooth, space-compact solutions of (endowed with the usual symplectic form induced by the causal propagator) and , equipped with the symplectic form

(1.1)

Thus, a quasi-free state on with two-point functions induces a unique quasi-free state on the usual symplectic space associated to .

Product-type pseudodifferential operators

In [GW] we have constructed Hadamard states whose two-point functions on a Cauchy surface are pseudodifferential operators. In the present case, the obvious difference is that on the cone , the coordinate is distinguished both from the point of view of the microlocal spectrum condition (from now on abbreviated ) and in the expression (1.1) for the symplectic form. This suggests that one should rather consider product-type pseudodifferential operators with symbols satisfying estimates:

in the covariables relative to the decomposition . Actually, to cope with the issue that is defined using an operator whose spectrum is not separated from (analogously to the infrared problem in massless theories), we need to introduce a larger class that includes some operators whose symbol is discontinuous at . Namely, we set

where is the class of pseudodifferential operators of order (in the variables) with values in operators on that infinitely increase Sobolev regularity. Then for instance although it is not in the pseudodifferential class .

Summary of results

Our main results can be summarized as follows. We always assume Hypothesis 1.1. If are topological vector spaces, we write to mean is linear and continuous.

  1. For pairs2 of two-point functions on satisfying , we give in Thm. 5.3 conditions on that guarantee that the corresponding two-point functions on satisfy . This is essentially an adaptation of the results of [Mo2] to our framework.

  2. In Thm. 7.4 we construct a large class of Hadamard states by specifying their two-point functions on the cone.

  3. In Thm. 8.2 we characterize the subclass of Hadamard states constructed in 2), which additionally are pure on the symplectic space on the cone. It turns out that they can be parametrized by a single operator in .

  4. In Thm. 8.4 we prove that if , then the pure states considered in 3) induce pure states in the interior of the cone.

In Subsect. 2.3 we argue that Hypothesis 1.1 covers the case when is an asymptotically flat vacuum spacetime with future time infinity, after a conformal transformation. Thus, our result 4) solves an open question by Moretti [Mo2] for .

Characteristic Cauchy problem

The proof of our main result 4) relies on rather standard results on the characteristic Cauchy problem (also called Goursat problem in the literature) in appropriate Sobolev spaces. We use

Let be a Cauchy surface for () in the future of and . We set

see Fig. 1. is relatively compact in with , and are compact in with smooth boundary . We denote by , the respective restricted Sobolev spaces of order , i.e. the space of distributions in , that vanish on the boundary.  

Fig. 1

If is a pair of Cauchy data, we denote by its extension by to and by the restriction to of the solution of the Cauchy problem

where . By standard energy estimates one obtains that is continuous.

In Subsect. 8.3 we prove the following result.

Theorem 1.1.

The map

is a homeomorphism. Moreover, if then is dense in .

The first part of Thm. 1.1 is equivalent to the existence and uniqueness of solutions in of the characteristic Cauchy problem:

The proof proceeds by reduction to a case already considered by Hörmander in [Hö2], namely when the characteristic surface is the graph of a Lipschitz function defined on a compact domain. Beside [Hö2] there is a considerable literature on the characteristic Cauchy problem for the Klein-Gordon equation, for example [BW, Ca, Do, Ni2], let us also mention related works on the Dirac equation [Ni1, HN, Jo]. The first part of Thm. 1.1 could actually also be deduced from [BW, Thm. 23].

The second part of Thm. 1.1 asserts that there is no loss of information on the level of purity of states when going from the cone to its interior . The precise form of the statement comes from the fact that the one-particle Hilbert space associated to our Hadamard states, i.e. the completion of for the inner product , equals . The validity of such result appears to be very delicate, it would be for instance problematic for with instead of and we do not know whether it holds for . The generalization of Thm. 1.1 to other geometrical situations is thus an interesting open problem, particularly relevant for the quantum field theoretical bulk-to-boundary correspondence.

Plan of the paper

In Sect. 2 we fix the geometric setup and outline the construction of null coordinates near the cone . In Sect. 3 we briefly review the Klein-Gordon field in and the definition of Hadamard states. Sect. 4 is devoted to the so-called bulk-to-boundary correspondence, i.e. to the definition of a convenient symplectic space of functions on , containing the traces on of space-compact solutions in .

In Sect. 5, we formulate the Hadamard condition on , i.e. the natural microlocal condition on the two-point functions of a quasi-free state on which ensures that the induced state in is a Hadamard state.

Sect. 6 is devoted to the pseudodifferential calculus on , more precisely to the ‘product-type’ classes, associated to bi-homogeneous symbols. We also describe more general operator classes which are pseudodifferential only in the variables in .

In Sect. 7 we construct large classes of Hadamard states on the cone, whose covariances belong to the operator classes introduced in Sect. 6. In Sect. 8 we characterize pure Hadamard states, and show that they induce pure states in . Finally in Sect. 9 we discuss the invariance of our classes of Hadamard states under change of null coordinates on . Various technical results are collected in Appendix A.

2. Geometric setup

In this section we describe our geometrical setup and construct null coordinates near the cone .

2.1. Future lightcone

We consider a globally hyperbolic spacetime of dimension . If , resp. denote the future/past time-like resp. causal shadow of in , see e.g. [Wa, Chap. 8] or [BGP, Sec. 1.3] for more details. If the spacetime is clear from the context these sets will simply be denoted by , .

As outlined in the introduction, we fix a base point , and consider

so that is the future lightcone from , with tip removed, and is the interior of . From [Wa, Sect. 8.1] we know that is open, with

We assume Hypothesis 1.1, i.e. that there exists such that:

It follows that is a smooth hypersurface, although is not smooth. Moreover since is a null hypersurface, is tangent to .

2.2. Causal structure

We now collect some useful results on the causal structure of and .

Lemma 2.1.

Let be compact. Then:

(2.1)
(2.2)

Proof.  (2.1) follows from [BGP, Lemma A.5.7]. Moreover if is open with , we have . Since and is open, this implies (2.2).

The following lemma is due to Moretti [Mo1, Thm. 4.1 (a)]. If , the notation or is used in the place of to specify which causal structure one refers to.

Lemma 2.2.

The Lorentzian manifold is globally hyperbolic. Moreover

(2.3)

The next proposition is also due to Moretti [Mo2, Lemma 4.3].

Proposition 2.3.

Let be compact. Then there exists a neighborhood of in such that no null geodesic starting from intersects .

2.3. Asymptotically flat spacetimes

In what follows we explain the relation between Hypothesis 1.1 and the geometrical assumptions met in the literature on Hadamard states [Mo1, Mo2, DS, BDM].

Let us consider two globally hyperbolic spacetimes and , where is an embedded submanifold of . One introduces the following set of assumptions.

Hypothesis 2.1.

Suppose the spacetime is such that:

  • there exists with on and ,

  • there exists such that is closed and

  • solves the vacuum Einstein equations at least in a neighborhood of

  • and on , , ,

  • if , then there exists , with on and

    (a) on ,

    (b) the vector field is complete on ,

Above, the symbols refer to the metric .

One says that is an asymptotically flat vacuum spacetime with past time infinity if there exists a spacetime such that is an embedded submanifold of and Hypothesis 2.1 is satisfied3.

Lemma 2.4.

Suppose is an asymptotically flat vacuum spacetime with past time infinity and let satisfy Hypothesis 2.1. Then Hypothesis 1.1 is satisfied for and .

Note that actually only conditions (1), (2), (4) and (5b) in Hypothesis 2.1 are needed in Lemma 2.4.

In the present paper we construct Hadamard states for the Klein-Gordon operator in for any smooth, real valued . In the special case of the conformal wave operator (with the scalar curvature) this yields however also Hadamard states on since the two metrics are conformally related, cf. Appendix A.2.

2.4. Null coordinates near

For later use it is convenient to introduce null coordinates near . The construction seems to be well-known, we sketch it for the reader’s convenience. Note however the estimates in Lemma 2.5, which will be useful later on.

We first choose normal coordinates at such that on a neighborhood of , .

Set

(2.4)

so that on a neighborhood of one has . Abusing notation slightly, we denote by coordinates on , and use the same letter for their pullback to local coordinates on near . We set

(2.5)

where will be chosen small enough. Note that is diffeomorphic to .

Lemma 2.5.
  • There exists a unique solution of:

  • There exists unique solutions , of:

  • Moreover there exists and such that

Proof.  The proof is given in Appendix A.4.

It remains to extend to smooth functions on a neighborhood of .

We argue as in [Wa, Sect. 11.1]: for , the submanifold is spacelike, of codimension in . At a given point of the orthogonal to its tangent space is two dimensional, timelike, and hence contain two null lines. One of them is generated by , the other is transverse to . We extend to a neighborhood of by imposing that are constant along the above family of null geodesics, transverse to .

Lemma 2.6.

The functions constructed above are a system of local coordinates near with and

(2.6)

where is a smooth, dependent Riemannian metric on .

Proof.  The proof will be given in Appendix A.3.

2.5. Estimates on traces

In this subsection we derive estimates, in the coordinates on constructed above, for the restriction to of a smooth, space compact function in . These estimates will be applied later to traces on of solutions of the Klein-Gordon equation in .

We recall that denotes the space of smooth space compact functions i.e. the space of such that for some compact .

We will slightly abuse notation by writing for the function expressed in some coordinate system near . We will similarly write for example for .

By Lemma 2.1 we see that is compact in if . This means that it suffices to control the derivatives in of near , i.e. of near . Clearly the only task is to control what happens near , i.e. when . We first derive estimates in the coordinates introduced in (2.4), in a neighborhood of . If we denote by the function expressed in normal coordinates at , which is defined on a neighborhood of . We set then

so that

We denote by the space of functions which are bounded with all derivatives.

Lemma 2.7.
  • if then belongs to .

  • Let . Then for .

Proof.  Considering the map and denoting still by some coordinates on we have:

From this we obtain (1). To prove (2) we need to express on . An easy computation using the estimates in Lemma 2.5 shows that on we have:

where and invertible. Plugging this into (A.9), we obtain

where . This implies (2).

We will also need later the following lemma. We denote by the standard Riemannian metric on and set:

(2.7)
Lemma 2.8.

Let

Then for all one has:

Proof.  We note that , for . From this and Lemma 2.7 it follows that if , then . It remains to estimate the derivatives of w.r.t. and . By a standard computation we obtain for :

for , and invertible. From this point on the lemma is a routine computation.

3. Klein-Gordon fields inside the future lightcone

3.1. Klein-Gordon equation in

We fix a smooth real function and consider the Klein-Gordon operator on :

We denote by the retarded/advanced Green’s functions for , by the Pauli-Jordan commutator function, and by the space of smooth, complex valued, space-compact solutions of

Recall that we have set in Subsect. 2.1:

and by Lemma 2.2 we know that is globally hyperbolic.

We denote by the restriction of to , by the Pauli-Jordan function for , and by the space of smooth, complex valued, space-compact solutions of

By the global hyperbolicity of we know that . From (2.3) and the uniqueness of we obtain that , hence

It follows that any uniquely extends to , in fact

(3.1)

As usual we equip with the symplectic form

(3.2)

where is a Cauchy hypersurface for (see Subsect. A.1 for notation). It is well known that

is a symplectomorphism.

3.2. Hadamard states in

We first briefly recall some standard facts, and refer for example to [GW, Sect. 2] for details and notation.

Covariances of a quasi-free state

If is a complex symplectic space, the complex covariances of a (gauge invariant) quasi-free state on (the polynomial CCR -algebra of ) are defined by:

From the CCR we obtain that , and the necessary and sufficient condition for to be the complex covariances of a (gauge invariant) quasi-free state is that .

If , the complex covariances of a state are induced from two-point functions, still denoted by such that

where we identify operators on with sesquilinear forms using the scalar product

Hadamard condition

We now recall the Hadamard condition for quasi-free states. We denote by the cotangent bundle of and the zero section. The principal symbol of is , the set

is called the characteristic manifold of .

The Hamilton vector field of will be denoted by , whose integral curves inside are called bicharacteristics.

We will use the notation for points in and write if and are in and