Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes

Quantum field theory on curved spacetimes is a semiclassical theory, which investigates the coupling of a quantum field with classical gravitation. It already predicts remarkable effects such as particle creation by the curved spacetime itself, a phenomenon most prominently represented by Hawking’s evaporation of black holes [Haw1975] and the Unruh effect [Ful1973, Dav1975, Unr1976]. Due to their rather small, generally trivial, isometry group, curved spacetimes lack an invariant concept of energy, so a distinct vacuum and consequently the notion of particles turn out to be non-sensible in general curved spacetimes [Dav1984, Wal1994]. This situation is best addressed by the algebraic approach to quantum field theory [HK1964, Dim1980], where first of all observables are introduced as elements of rather abstract algebras $\mathcal{A}(O)\subset\mathcal{A}(M)$ associated to spacetime regions $O\subset M$ in a local and covariant manner, and states will join the game only later as certain functionals on these algebras.
The categorical framework of locally covariant quantum field theory [BFV2003] has established as a suitable generalization of the principles of AQFT to curved spacetimes. Instead of fixing a spacetime and its symmetries from the beginning, a whole category of spacetimes is considered with arrows given by certain isometric embeddings. The actual QFT is then represented by a covariant functor to the category of $C^{*}$-algebras and injective $C^{*}$-homomorphisms fulfilling adapted Haag-Kastler-axioms. For the category of spacetimes, we will adopt the setting of [BG2011]:

Because of their good causal and analytic properties (no causal loops, foliation by Cauchy hypersurfaces, well-posed Cauchy problem), globally hyperbolic Lorentzian manifolds have proven to be a reasonable model for curved spacetimes. As further data, we consider real vector bundles, which excludes for instance charged fields, and the large class of Green hyperbolic operators implying the existence of an advanced and retarded Green operator $G_{A},G_{R}$ (see [Bär2015] for a thorough discussion of these operators). For $G:=G_{A}-G_{R}$, Theorem 3.5 of [BG2011] provides the exact sequence

and hence, it leads to a covariant functor into the category of symplectic vector spaces with objects $(V,\sigma)$ essentially given by the solution space of the field equation

$\mathscr{D}(M,E)$ represents test sections in $E$, more precisely smooth sections with compact support, and $C^{\infty}_{sc}(M,E)$ those with only spacelike compact support, meaning that it is contained in the causal future and past of some compact subset of $M$. The $L^{2}$-product $(\cdot,\cdot)_{M}$ of test sections is induced by the non-degenerate inner product on $E$.
For a bosonic quantum field theory, we take the $\mathrm{CCR}$-representation of $(V,\sigma)$, i.e. a pair $(w,A)$ consisting of a $C^{*}$-algebra $A$ and a map $w$ from $V$ into the unitary elements of $A$ such that $A$ is generated as a $C^{*}$-algebra by $\{w(x)\}_{x\in V}$ and the Weyl relations hold:

This construction goes back to [Man1968] (see also section 4.2 of [BGP2007] and 5.2.2.2 of [BR2002]) and it is unique in an appropriate sense. Therefore, altogether, $(M,E,P)\mapsto\mathrm{CCR}\big{(}\mathscr{D}(M,E)/\ker G\big{)}$ provides the desired functor and Haag-Kastler’s axioms are satisfied (Theorem 3.10 of [BG2011])

We introduce states in the theory as normed and positive functionals $\tau$ on $\mathrm{CCR}(V)$, where $\tau(a)$ can be thought of as the expectation value of the observable $a$ in the state $\tau$. The induced GNS-representation $(\pi_{\tau},\mathscr{H}_{\tau},\Omega_{\tau})$ provides the familiar framework of a state space $\mathscr{H}_{\tau}$ with observables as bounded operators $\pi_{\tau}(a)$ and a cyclic vector $\Omega_{\tau}$ (see section 2.3 of [BR2002] for details). Hence, the selection of a distinct vacuum is shifted to that of an algebraic state $\tau$. Particularly adapted to free quantum fields are the so-called quasifree states generated by

for some scalar product $\eta$ on $V$, by which $(\pi_{\tau},\mathscr{H}_{\tau},\Omega_{\tau})$ is determined up to unitary equivalence. For these states, the unitary operators $\big{\{}\pi_{\tau}\big{(}w(tx)\big{)}\big{\}}_{t\in\mathbb{R}}$ constitute a strongly continuous one-parameter group and thus, field operators $\Phi_{\tau}(x)$ are given by the self-adjoint generators due to Stone’s theorem. Furthermore, there is a dense domain $D_{\tau}\subset\mathscr{H}_{\tau}$ such that $\mathrm{ran}\,\Phi_{\tau}(x)\subset D_{\tau}\subset\mathrm{dom}\,\Phi_{\tau}(x)$ for all $x$, so polynomials of field operators are well-defined on $D_{\tau}$. Hence, the Weyl relations (1.3) imply the familiar canonical commutator relations $\big{[}\Phi_{\tau}(x),\Phi_{\tau}(y)\big{]}=i\sigma(x,y)\,\text{id}_{\mathscr{H}_{\tau}}$ and for all $n\in\mathbb{N}$, the $n$-point-function of the state $\tau_{n}(x_{1},\ldots,x_{n}):=\big{\langle}\Phi_{\tau}(x_{1})\ldots\Phi_{\tau}(x_{n})\Omega_{\tau},\Omega_{\tau}\big{\rangle}_{\mathscr{H}_{\tau}}$ represents a well-defined distribution (see section 4.2 of [BG2011] for precise definitions and proofs). In particular, the two-point-function is of the form

so it reproduces $\eta$ and hence $\tau$. Indeed, we have $\tau_{n}=0$ for odd $n$, and for $n$ even, it is given by some polynomial in the elements of $\{\tau_{2}(x_{i},x_{j})\}_{i,j=1,\ldots,n}$. Thinking of $\tau_{n}$ as the propagation of the state of the field, the focus on quasifree states corresponds to the perception that this propagation is essentially given by independent one-particle-propagations, which legitimates them as the natural objects to look at when dealing with free quantum fields (see chapter 17 of [DG2013] for an overview of quasifree states). Going back from $(V,\sigma)$ to $(M,E,P)$, the scalar product $\eta$ corresponds to a bidistribution $S\colon\mathscr{D}(M,E)\times\mathscr{D}(M,E)\rightarrow\mathbb{R}$ with

With regard to (1.2) and (1.4), a quasifree state is therefore determined by $S$ and $G$.
Despite all physically motivated restrictions on the field so far, there is still a huge variety of possible states, so we need to look for constraints also on this level. A reasonable demand would be renormalizability of $\tau_{2}$, most prominently represented by the expectation value of the energy momentum tensor, since products of distributions are in general ill-defined. In flat quantum field theory, this would be carried out by subtracting the vacuum expectation value setting us back to the problem of non-existence of a distinct vacuum. On the other hand, this procedure merely requires regularity of differences of expectation values, and indeed, J. Hadamard’s theory of second order hyperbolic equations [Had1923] led to a family of bisolutions with fixed singular part in the sense that the difference of any two such bisolutions is smooth (see [Wal1994] and, more recently, [Hac2016] for details, as well as [DF2008] for the concrete renormalization). Accordingly, a state is called a Hadamard state if its two-point-function has the Hadamard singularity structure, which, by now, has been shown to be invariant under Cauchy evolution [FSW1978]. Futhermore, any globally hyperbolic spacetime admits a large class of pure Hadamard states [FNW1981, SV2001].
However, the first mathematically precise definition of the Hadamard singularity structure has been specified only in [KW1991], in which the authors also show that for a wide class of spacetimes the Hadamard property singles out an invariant quasifree state. Moreover, in any spatially compact spacetime ("closed universes"), all Hadamard states, more specifically their GNS representations, comprise one unitary equivalence class, which, for general spacetimes, suggests a certain "local equivalence" of all possible notions of a vacuum state [Ver1994]. In addition, Hadamard states yield finite fluctuations for all Wick polynomials [BF2000], which makes them relevant also for the perturbative construction of interacting fields (see also [HW2015, Rej2016, Düt2019] and references therein). Consequently, Hadamard states are by now considered a reasonable counterpart of Minkowski finite energy states and the Hadamard condition an appropriate generalization of the energy condition for Minkowski quantum field theory. Note that the replacement of a distinct vacuum state by a whole class of states somehow reflects the essence of general relativity: Just like there is no preferred coordinate system, the concept of vacuum and particles as absolute quantities has to be re-evaluated and eventually downgraded to one choice among many.
It was Radzikowski who showed that for the massive scalar field the global Hadamard condition is equivalent to a certain requirement on the wave front set of the two-point-function [Rad1992, Rad1996], namely

where

Note that, unlike the criterion given in [KW1991], this is a local condition, which Sahlmann and Verch generalized to sections in general vector bundles [SV2001]. It includes Hadamard states of the Dirac field in the sense of [Köh1995, Kra2000, Hol2001], which have been used, for instance, for a mathematical rigorous description of the chiral anomaly [BS2016]. In addition, [SVW2002] proposed an even more elegant characterization of the Hadamard property in terms of Hilbert space valued distributions $\varphi\mapsto\Phi_{\tau}[\varphi]\Omega_{\tau}\in\mathscr{H}_{\tau}$, involving the GNS-representation induced by $\tau$. Also for non-quasifree states, one can formulate (1.6) as a constraint on the whole $n$-point-function, which is compatible with the special case of quasifree states [San2010]. Moreover, in analytic spacetimes, this generalized Hadamard condition can be sharpened to a condition on the analytic wave front set, thereby implying the Reeh-Schlieder-property [SVW2002]. Likewise, for non-globally hyperbolic spacetimes, there is a formulation of the Hadamard condition via restriction to globally hyperbolic subregions. Hadamard states have therefore been studied in connection with the Casimir effect and on anti-de Sitter spacetime (see [DNP2014, DFM2018] and references therein). By using the weaker concept of Sobolev wave front sets, a definition of adiabatic states on globally hyperbolic spacetimes similar to (1.6) is given in [JS2002], thus implying that Hadamard states are adiabatic.
However, most importantly for the purpose of this work, the Hadamard condition in the form (1.6) allows us to employ the techniques of microlocal analysis provided by Duistermaat and Hörmander [DH1972]. Soon after Radzikowski’s work, Junker derived pure Hadamard states for the massive scalar field on spatially compact globally hyperbolic spacetimes, using a factorization of the Klein-Gordon operator by pseudo-differential operators [Jun1996, Jun2002]. Gérard, Wrochna et al. generalized this construction to a large class of spacetimes [GOW2017] and even gauge fields [GW2015]. Furthermore, they proved the existence of (not necessarily pure) Hadamard states [GW2014] in a much more concrete manner than [FNW1981]. See [Gér2019] for a recent review of these techniques.
On the other hand, there have been further proposals for physically reasonable states like the Sorkin-Johnston-states [AAS2012], which in general lack the Hadamard property [FV2012]. Nevertheless, a modification of their construction produces Hadamard states [BF2014]. For a contemporary synopsis concerning preferred vacuum states on general spacetimes, the nature of the Hadamard property and this construction in particular, see also [Few2018]. Apart from these rather general prescriptions, Hadamard states have been constructed explicitly for a large variety of spacetimes with special (asymptotic) symmetries, and furthermore, well-established states have been tested for the Hadamard property (see the introduction sections of [GW2014, GOW2017] and the references therein as well as section 8.4 of [FV2015] and 2.4 of [Hac2016] for an overview).

In his seminal work [Rad1996], Radzikowski already realized that a bidistribution $\widetilde{H}$ satisfies the Hadamard condition if and only if it is of the form

with $\widetilde{G}_{aF},\widetilde{G}_{F},\widetilde{G}_{A},\widetilde{G}_{R}$ the distinguished parametrices in the sense of Duistermaat-Hörmander (Theorem 6.5.3 in [DH1972]). With $\Delta^{\prime}:=\big{\{}(p,\xi;p,-\xi)\big{\}}$ the primed diagonal, they are characterized by

However, he remarked that it is not clear how to prove that one may choose the smooth part such that (1.5) is exactly satisfied [Rad1992], an issue closely related to the question, whether $\widetilde{G}_{aF},\widetilde{G}_{F}$ can be chosen as actual Green operators, which has been already addressed in section 6.6 of [DH1972]. Clearly, microlocal analysis proved to be a powerful tool for the investigation of singularities and indispensable for the results listed in the former paragraph. Nevertheless, for these remaining questions, the non-singular part of $\widetilde{H}$ is primarily concerned, so different techniques are required.
This work is dedicated to provide such techniques and resolves the question for wave operators on globally hyperbolic spacetimes. It therefore gives a further and more constructive existence proof of, not necessarily pure, Hadamard states than [FNW1981]. By avoiding any kind of deformation argument, it covers situations involving, for example, analytic spacetimes or constraint equations as in general relativity, where such an argument is usually not applicable.
The starting point in chapter 3 is the local construction of $\widetilde{G}_{aF},\widetilde{G}_{F}$ in terms of Hadamard series very much inspired by [BGP2007]. In chapter 4, these local objects are globalized and finally patched together by introducing a well-posed Cauchy problem for bisolutions and techniques from Čech cohomology theory. In the final chapter 5, we check existence of a "positive and symmetric choice", that is the smooth part of any Hadamard bisolution can be chosen such that it obtains the properties of a two-point-function. The necessary preparation is given in chapter 2 as well as a proof for the symmetry of the Hadamard coefficients in the vector valued case alternative to [Kam2019] in the Appendix.

A crucial feature of globally hyperbolic Lorentzian manifolds $M$ is a well-posed Cauchy problem for wave operators on smooth sections, i.e. for any Cauchy hypersurface $\Sigma$ with normal field $\nu$, which is timelike, and Cauchy data $f\in C^{\infty}(M,E),u_{0},u_{1}\in C^{\infty}(\Sigma,E)$, the Cauchy problem

has a unique solution $u\in C^{\infty}(M,E)$, which is supported in $J\big{(}\text{supp}\,u_{0}\cup\text{supp}\,u_{1}\cup\text{supp}\,f\big{)}$ and depends continuously on the data (see section 3.2 of [BGP2007] and chapter 3 of [BF2009]). For $F$ a vector bundle over some further globally hyperbolic Lorentzian manifold $N$, recall (2.1) for the definition of the vector bundle $E\boxtimes F$ over $M\times N$.

It seems that this procedure directly generalizes to operators of order $k$, for which derivatives up to order $k^{2}$ have to be provided as data. Furthermore, symmetry of the data is inherited by the solution:

Additionally, we investigate the propagation of a family of singular solutions from a neighborhood of $\Sigma$ to all of $M$ by applying the well-posed Cauchy problem for singular sections treated in [BTW2015]. For that, we just have to ensure the existence of the restriction to $\Sigma$ by checking Hörmander’s criterion.

The Hadamard condition in the form (1.6) is a local condition and, moreover, the singularity structure of a bisolution is related to the corresponding differential operator essentially via its principal symbol. On these grounds, we start with the prototype setting $P=\Box$ on $M=\mathbb{R}^{d}_{\mathrm{Mink}},\leavevmode\nobreak\ d\geq 3$, since, from the viewpoint of the singularity structure of the solutions, this already incorporates the characteristic properties of the solutions for the general setting of wave operators on curved spacetimes. In order to obtain a decomposition like (1.8), we study fundamental solutions for $\Box$. It is not hard to check that these objects are invariant under the special orthochronous Lorentz group $\mathcal{L}^{\uparrow}_{+}$ and their singular support is cointained in the light cone $C$. This makes them directly comparable outside $C$ meaning that we need to understand $\mathcal{L}^{\uparrow}_{+}$-invariant distributions supported on $C$.
For $\mathring{\mathbb{R}}^{d}:=\mathbb{R}^{d}\backslash\{0\}$, we consider the submersions $\gamma_{\pm}:=\gamma\big{|}_{J_{\pm}^{c}}$, where $J_{\pm}^{c}:=\mathring{\mathbb{R}}^{d}\backslash J_{\pm}$. Then the pullback $\gamma^{*}_{\pm}$ maps distributions on $\mathbb{R}$ to $\mathcal{L}^{\uparrow}_{+}$-invariant distributions on $J_{\pm}^{c}$ and moreover establishes a close connection to the well-known classification of distributions on $\mathbb{R}$ supported in $\{0\}$:

For that, it is important to note that the proof of Theorem 2.5 particularly demonstrates surjectivity of $(\gamma_{\pm})_{*}$.

There are two immediate consequences: First, every $\mathcal{L}^{\uparrow}_{+}$-invariant distribution $T$ supported in $C$ has to be of the form

where only finitely many coefficients $a_{k}^{\pm},b_{k}$ are non-zero. Recall that $\gamma_{\pm}^{*}\delta_{0}^{(k)}[\varphi]=\big{(}(\gamma_{\pm})_{*}\varphi\big{)}^{(k)}(0)$ and the push-forward along a submersion is given by integration along the fibers. This leads to the second consequence, which is that every $\mathcal{L}^{\uparrow}_{+}$-invariant measure supported on $C_{\pm}$ is of the form $a\,\text{d}\Omega^{0}_{\pm}+b\delta_{0}$ (section IX.8 of [RS1975]), where

The most known examples of $\mathcal{L}^{\uparrow}_{+}$-invariant supported on $C_{\pm}$ are certain Riesz distributions (not all of them) and in fact, there are no other: For all $\alpha\in\mathbb{C}$ with $\text{Re}\left(\alpha\right)>d$, they are defined as continuous functions via

One directly checks holomorphicity in $\alpha$ and calculates $\Box R^{\alpha+2}_{\pm}=R^{\alpha}_{\pm}$, which provides an extension as distributions to all of $\mathbb{C}$ via $R^{\alpha}_{\pm}:=\Box^{k}R^{\alpha+2k}_{\pm},\leavevmode\nobreak\ k>\frac{d}{2}-\text{Re}\left(\alpha\right)$. Moreover, as shown in section 13.2 of [DK2010], we have $R^{0}_{\pm}=\delta_{0}$ and