In a recent paper, we proved that a large class of spacetimes, not necessarily homogeneous or isotropous and relevant at a cosmological level, possesses a preferred codimension one submanifold, i.e., the past cosmological horizon, on which it is possible to encode the information of a scalar field theory living in the bulk. Such bulk-to-boundary reconstruction procedure entails the identification of a preferred quasifree algebraic state for the bulk theory, enjoying remarkable properties concerning invariance under isometries (if any) of the bulk and energy positivity, and reducing to well-known vacua in standard situations. In this paper, specialising to open FRW models, we extend previously obtained results and we prove that the preferred state is of Hadamard form, hence the backreaction on the metric is finite and the state can be used as a starting point for renormalisation procedures. That state could play a distinguished role in the discussion of the evolution of scalar fluctuations of the metric, an analysis often performed in the development of any model describing the dynamic of an early Universe which undergoes an inflationary phase of rapid expansion in the past.
We review a few rigorous and partly unpublished results on the regularisation of the stress-energy in quantum field theory on curved spacetimes: 1) the symmetry of the Hadamard/Seeley-DeWitt coefficients in smooth Riemannian and Lorentzian spacetimes 2) the equivalence of the local $ζ$-function and the Hadamard-point-splitting procedure in smooth static spacetimes 3) the equivalence of the DeWitt-Schwinger- and the Hadamard-point-splitting procedure in smooth Riemannian and Lorentzian spacetimes.
We develop a quantization scheme for the vector potential on globally hyperbolic spacetimes which realizes it as a locally covariant conformal quantum field theory. This result allows us to employ on a large class of backgrounds, which are asymptotically flat at null infinity, a bulk-to-boundary correspondence procedure in order to identify for the underlying field algebra a distinguished ground state which is of Hadamard form.
A spatial variant of the Functional Renormalization Group (FRG) is introduced on (Lorentzian signature) globally hyperbolic spacetimes. Through its perturbative expansion it is argued that such a FRG must inevitably be state dependent and that it should be based on a Hadamard state. A concrete implementation is presented for scalar quantum fields on flat Friedmann-Lemaître spacetimes. The universal ultraviolet behavior of Hadamard states allows the flow to be matched to the one-loop renormalized flow (where strict removal of the ultraviolet cutoff requires a tower of potentials, one for each p…
Some years ago, Radzikowski has found a characterization of Hadamard states for scalar quantum fields on a four-dimensional globally hyperbolic spacetime in terms of a specific form of the wavefront set of their two-point functions (termed `wavefront set spectrum condition'), thereby initiating a major progress in the understanding of Hadamard states and the further development of quantum field theory in curved spacetime. In the present work, we extend this important result on the equivalence of the wavefront set spectrum condition with the Hadamard condition from scalar fields to vector field…
We construct a state in the Schwarzschild-de Sitter spacetime which is invariant under the action of its group of symmetries. Our state is not defined in the whole Kruskal extension of this spacetime, but rather in a subset of the maximally extended conformal diagram. The construction is based on a careful use of the bulk-to-boundary technique. We will show that our state is Hadamard and that it is not a KMS state, differently from the case of states constructed in spacetimes containing only one event horizon.
In this Thesis we will explicitly construct Hadamard states with different features. In the two first cases, the states will be constructed in static and in cosmological spacetimes. In one of the cases, the states will be constructed in such a way as to minimize the expectation value of the energy-momentum tensor of a free massive scalar field, multiplied by a suitable smooth function of compact support, in a cosmological spacetime. The final form of the state clearly reduces to the vacuum in a static spacetime. Besides, we will show that these are Hadamard states. They do not contradict the a…
We present a modification of the recently proposed Sorkin-Johnston states for scalar free quantum fields on a class of globally hyperbolic spacetimes possessing compact Cauchy hypersurfaces. The modification relies on a smooth cutoff of the commutator function and leads always to Hadamard states, in contrast to the original Sorkin-Johnston states. The modified Sorkin-Johnston states are, however, due to the smoothing no longer uniquely associated to the spacetime.
We give a rigorous definition of the Unruh state in the setting of massless Dirac fields on slowly rotating Kerr spacetimes. In the black hole exterior region, we show that it is asymptotically thermal at Hawking temperature on the past event horizon. Furthermore, we demonstrate that in the union of the exterior and interior regions, the Unruh state is pure and Hadamard. The main ingredients are the Häfner-Nicolas scattering theory, new microlocal estimates for characteristic Cauchy problems and criteria on the level of square-integrable solutions.
We give an introduction to the techniques from microlocal analysis that have successfully been applied in the investigation of Hadamard states of free quantum field theories on curved spacetimes. The calculation of the wave front set of the two point function of the free Klein-Gordon field in a Hadamard state is reviewed, and the polarization set of a Hadamard two point function of the free Dirac field on a curved spacetime is calculated.
We consider a real, massive scalar field on ${\rm PAdS}_{d+1}$, the Poincaré domain of the $(d+1)$-dimensional anti-de Sitter (AdS) spacetime. We first determine all admissible boundary conditions that can be applied on the conformal boundary, noting that there exist instances where "bound states" solutions are present. Then, we address the problem of constructing the two-point function for the ground state satisfying those boundary conditions, finding ultimately an explicit closed form. In addition, we investigate the singularities of the resulting two-point functions, showing that they are c…
We consider the Klein-Gordon equation on asymptotically anti-de Sitter spacetimes subject to Neumann or Robin (or Dirichlet) boundary conditions, and prove propagation of singularities along generalized broken bicharacteristics. The result is formulated in terms of conormal regularity relative to a twisted Sobolev space. We use this to show the uniqueness, modulo regularising terms, of parametrices with prescribed b-wavefront set. Furthermore, in the context of quantum fields, we show a similar result for two-point functions satisfying a holographic Hadamard condition on the b-wavefront set.
We develop the Hadamard renormalization of the stress-energy tensor for a massive scalar field theory defined on a general spacetime of arbitrary dimension. Our formalism could be helpful in treating some aspects of the quantum physics of extra spatial dimensions. More precisely, for spacetime dimension up to six, we explicitly describe the Hadamard renormalization procedure and for spacetime dimension from seven to eleven, we provide the framework permitting the interested reader to perform this procedure explicitly in a given spacetime. We complete our study (i) by considering the ambiguitie…
We give a non-perturbative construction of the fermionic projector in Minkowski space coupled to a time-dependent external potential which is smooth and decays faster than quadratically for large times. The weak and strong mass oscillation properties are proven. We show that the integral kernel of the fermionic projector is of Hadamard form, provided that the time integral of the spatial sup-norm of the potential satisfies a suitable bound. This gives rise to an algebraic quantum field theory of Dirac fields in an external potential with a distinguished pure quasi-free Hadamard state.
Within this chapter (published as [49]) we introduce the overall idea of the algebraic formalism of QFT on a fixed globally hyperbolic spacetime in the framework of unital $*$-algebras. We point out some general features of CCR algebras, such as simplicity and the construction of symmetry-induced homomorphisms. For simplicity, we deal only with a real scalar quantum field. We discuss some known general results in curved spacetime like the existence of quasifree states enjoying symmetries induced from the background, pointing out the relevant original references. We introduce, in particular, th…
We conjecture that (when the notion of Hadamard state is suitably adapted) there is no isometry-invariant Hadamard state for the massive or massless covariant Klein-Gordon equation defined on the region of the Kruskal spacetime to the left of a surface of constant Schwarzschild radius in the right Schwarzschild wedge when Dirichlet boundary conditions are put on that surface. We also prove that, with a suitable definition for 'boost-invariant Hadamard state' (which we call 'strongly boost-invariant globally-Hadamard') which takes into account both the existence of the timelike boundary and the…
This paper continues the analysis of the quantum states determined by the universal asymptotic structure of four-dimensional asymptotically flat vacuum spacetimes at null infinity M. It is now focused on the quantum state lambda_M, of a massles conformally coupled scalar field phi propagating in M. lambda_M is ``holographically'' induced in the bulk by the universal BMS-invariant state lambda at infinity scri of M. It is done by means of the correspondence between observables in the bulk and those on the boundary at null infinity discussed in previous papers. The induction is possible when som…
In the celebrated Unruh effect, we learn that a uniformly accelerating detector in a Minkowski vacuum spacetime registers a constant temperature. Building on prior work, we present a technique based on derivative couplings of the two-point Wightman function and the Hadamard renormalization procedure to define an instantaneous temperature for a massive scalar field, non-minimally coupled to gravity. We find the temperature contains local contributions from the acceleration of the detector, the curvature of spacetime, and the renormalized stress-energy tensor of the field. Our result, which can …
The discovery of the radiation properties of black holes prompted the search for a natural candidate quantum ground state for a massless scalar field theory on Schwarzschild spacetime, here considered in the Eddington-Finkelstein representation. Among the several available proposals in the literature, an important physical role is played by the so-called Unruh state which is supposed to be appropriate to capture the physics of a black hole formed by spherically symmetric collapsing matter. Within this respect, we shall consider a massless Klein-Gordon field and we shall rigorously and globally…
In a recent paper (Phys. Rev. D 94, 125016 (2016)), the authors argued that the singularities of the two-point functions on the Poincaré domain of the $n$-dimensional anti-de Sitter spacetime ($\text{PAdS}_n$) have the Hadamard form, regardless of which (Robin) boundary condition is chosen at the conformal boundary. However, the argument used to prove this statement was based on an incorrect expression for the two-point function $G^{+}(x,x')$, which was obtained by demanding $\text{AdS}$ invariance for the vacuum state. In this comment I show that their argument works only for Dirichlet and Ne…
In this work we define a new state on the Weyl algebra of the free massive scalar Klein-Gordon field on a Robertson-Walker spacetime and prove that it is a Hadamard state. The state is supposed to approximate a thermal equilibrium state on a Robertson-Walker spacetime and we call it an adiabatic KMS state. This opens the possibility to do quantum statistical mechanics on Robertson-Walker spacetimes in the algebraic framework and the analysis of the free Bose gas on Robertson-Walker spacetimes. The state reduces to an adiabatic vacuum state if the temperature is zero and it reduces to the usual…
The study of physical effects of quatum fields in black hole spacetimes, which is related to questions such as the validity of the strong cosmic censorship conjecture, requires a Hadamard state describing the physical situation. Here, we consider the theory of a free scalar field on a Kerr-de Sitter spacetime, focussing on spacetimes with sufficiently small angular momentum of the black hole and sufficiently small cosmological constant. We demonstrate that an extension of the Unruh state, which describes the expected late-time behaviour in spherically symmetric gravitational collapse, can be r…
This book provides a rather self-contained survey of the construction of Hadamard states for scalar field theories in a large class of notable spacetimes, possessing a (conformal) light-like boundary. The first two sections focus on explaining a few introductory aspects of this topic and on providing the relevant geometric background material. The notions of asymptotically flat spacetimes and of expanding universes with a cosmological horizon are analysed in detail, devoting special attention to the characterization of asymptotic symmetries. In the central part of the book, the quantization of…
We characterise the homogeneous and isotropic gauge invariant and quasifree states for free Dirac quantum fields on Robertson-Walker spacetimes in any even dimension. Using this characterisation, we construct adiabatic vacuum states of order $n$ corresponding to some Cauchy surface. We then show that any two such states (of sufficiently high order) are locally quasi-equivalent. We propose a microlocal version of the Hadamard condition for spinor fields on arbitrary spacetimes, which is shown to entail the usual short distance behaviour of the twopoint function. The polarisation set of these tw…
We derive for a pair of operators on a symplectic space which are adjoints of each other with respect to the symplectic form (that is, they are sympletically adjoint) that, if they are bounded for some scalar product on the symplectic space dominating the symplectic form, then they are bounded with respect to a one-parametric family of scalar products canonically associated with the initially given one, among them being its ``purification''. As a typical example we consider a scalar field on a globally hyperbolic spacetime governed by the Klein-Gordon equation; the classical system is describe…
Afshordi, Aslanbeigi and Sorkin have recently proposed a construction of a distinguished "S-J state" for scalar field theory in (bounded regions of) general curved spacetimes. We establish rigorously that the proposal is well-defined on globally hyperbolic spacetimes or spacetime regions that can be embedded as relatively compact subsets of other globally hyperbolic spacetimes, and also show that, whenever the proposal is well-defined, it yields a pure quasifree state. However, by explicitly considering portions of ultrastatic spacetimes, we show that the S-J state is not in general a Hadamard…
We show that a free Dirac quantum field on a globally hyperbolic spacetime has the following structural properties: (a) any two quasifree Hadamard states on the algebra of free Dirac fields are locally quasiequivalent; (b) the split-property holds in the representation of any quasifree Hadamard state; (c) if the underlying spacetime is static, then the nuclearity condition is satisfied, that is, the free energy associated with a finitely extended subsystem (``box'') has a linear dependence on the volume of the box and goes like $\propto T^{s+1}$ for large temperatures $T$, where $s+1$ is the n…
We analyze the implications of the microlocal spectrum/Hadamard condition for states in a (linear) quantum field theory on a globally hyperbolic spacetime $M$ in the context of a (distributional) initial value formulation. More specifically, we work in a $3+1$-split $M\cong\mathbb{R}\timesΣ$ and give a bound, independent of the spacetime metric, on the wave front sets of the initial data for a quasi-free Hadamard state in the quantum field theory defined by a normally hyperbolic differential operator $P$ acting in a vector bundle $E\stackrelπ{\rightarrow}M$. This aims at a possible way to appl…
An analytic method is given for deriving the part of the retarded Green's function v(x,x') that contributes to the tail term in the radiation reaction force felt by a particle coupled to a massless minimally coupled scalar field. The method gives an expansion of v(x,x') for small separations of the points x, x' valid for an arbitrary static spherically symmetric spacetime. It is obtained by using a WKB approximation for the Euclidean Green's function for the massless minimally coupled scalar field and is equivalent to the DeWitt-Schwinger expansion for v(x,x'). The first few terms in this expa…
Worldline quantum inequalities provide lower bounds on weighted averages of the renormalised energy density of a quantum field along the worldline of an observer. In the context of real, linear scalar field theory on an arbitrary globally hyperbolic spacetime, we establish a worldline quantum inequality on the normal ordered energy density, valid for arbitrary smooth timelike trajectories of the observer, arbitrary smooth compactly supported weight functions and arbitrary Hadamard quantum states. Normal ordering is performed relative to an arbitrary choice of Hadamard reference state. The ineq…
We discuss Stueckelberg massive electromagnetism on an arbitrary four-dimensional curved spacetime (gauge invariance of the classical theory and covariant quantization, wave equations for the massive spin-1 field $A_μ$, for the auxiliary Stueckelberg scalar field $Φ$ and for the ghost fields $C$ and $C^\ast$, Ward identities, Hadamard representation of the various Feynman propagators and covariant Taylor series expansions of the corresponding coefficients). This permits us to construct, for a Hadamard quantum state, the expectation value of the renormalized stress-energy tensor associated with…
Hadamard states are generally considered as the physical states for linear quantized fields on curved spacetimes, for several good reasons. Here, we provide a new motivation for the Hadamard condition: for "ultrastatic slab spacetimes" with compact Cauchy surface, we show that the Wick squares of all time derivatives of the quantized Klein-Gordon field have finite fluctuations only if the Wick-ordering is defined with respect to a Hadamard state. This provides a converse to an important result of Brunetti and Fredenhagen. The recently proposed "S-J (Sorkin-Johnston) states" are shown, generica…
Free field theories on a four dimensional, globally hyperbolic spacetime, whose dynamics is ruled by a Green hyperbolic partial differential operator, can be quantized following the algebraic approach. It consists of a two-step procedure: In the first part one identifies the observables of the underlying physical system collecting them in a *-algebra which encodes their relational and structural properties. In the second step one must identify a quantum state, that is a positive, normalized linear functional on the *-algebra out of which one recovers the interpretation proper of quantum mechan…
We define a distinguished "ground state" or "vacuum" for a free scalar quantum field in a globally hyperbolic region of an arbitrarily curved spacetime. Our prescription is motivated by the recent construction of a quantum field theory on a background causal set using only knowledge of the retarded Green's function. We generalize that construction to continuum spacetimes and find that it yields a distinguished vacuum or ground state for a non-interacting, massive or massless scalar field. This state is defined for all compact regions and for many noncompact ones. In a static spacetime we find …
We discuss the quantization of linearized gravity on globally hyperbolic, asymptotically flat, vacuum spacetimes and the construction of distinguished states which are both of Hadamard form and invariant under the action of all bulk isometries. The procedure, we follow, consists of looking for a realization of the observables of the theory as a sub-algebra of an auxiliary, non-dynamical algebra constructed on future null infinity $\Im^+$. The applicability of this scheme is tantamount to proving that a solution of the equations of motion for linearized gravity can be extended smoothly to $\Im^…
First, the present work is concerned with generalising constructions and results in quantum field theory on curved spacetimes from the well-known case of the Klein-Gordon field to Dirac fields. To this end, the enlarged algebra of observables of the Dirac field is constructed in the algebraic framework. This algebra contains normal-ordered Wick polynomials in particular, and an extended analysis of one of its elements, the stress-energy tensor, is performed. Based on detailed calculations of the Hadamard coefficients of the Dirac field, it is found that a construction of a stress-energy tensor…
Quantum weak energy inequalities (QWEI) provide state-independent lower bounds on averages of the renormalised energy density of a quantum field. We derive QWEIs for the electromagnetic and massive spin-one fields in globally hyperbolic spacetimes whose Cauchy surfaces are compact and have trivial first homology group. These inequalities provide lower bounds on weighted averages of the renormalized energy density as ``measured'' along an arbitrary timelike trajectory, and are valid for arbitrary Hadamard states of the spin-one fields. The QWEI bound takes a particularly simple form for averagi…
Building on an insight due to Avramidi, we provide a system of transport equations for determining key fundamental bi-tensors, including derivatives of the world-function, σ(x,x'), the square root of the Van Vleck determinant, Δ^{1/2}(x,x'), and the tail-term, V(x,x'), appearing in the Hadamard form of the Green function. These bi-tensors are central to a broad range of problems from radiation reaction to quantum field theory in curved spacetime and quantum gravity. Their transport equations may be used either in a semi-recursive approach to determining their covariant Taylor series expansions…
Hollands and Wald's technique based on *-algebras of Wick products of field operators is strightforwardly generalized to define the stress-energy tensor operator in curved globally hyperbolic spacetimes. In particular, the locality and covariance requirement is generalized to Wick products of differentiated quantum fields. The proposed stress-energy tensor operator is conserved and reduces to the classical form if field operators are replaced by classical fields satisfying the equation of motion. The definition is based on the existence of convenient counterterms given by certain local Wick pr…
By considering Hadamard vacuum states, we first construct the two-point functions associated with Stueckelberg massive electromagnetism in de Sitter and anti-de Sitter spacetimes. Then, from the general formalism developed in [A. Belokogne and A. Folacci, Phys. Rev. D \textbf{93}, 044063 (2016)], we obtain an exact analytical expression for the vacuum expectation value of the renormalized stress-energy tensor of the massive vector field propagating in these maximally symmetric spacetimes.
We prove that, under the standard conformal scaling, a massless field in de Sitter spacetime admits an O(4)-invariant Fock quantization such that time evolution is unitarily implemented. This result disproves previous claims in the literature. We discuss the relationship between this quantization with unitary dynamics and the family of O(4)-invariant Hadamard states given by Allen and Folacci, as well as with the Bunch-Davies vacuum.
Whenever real particle production occurs in quantum field theory, the imaginary part of the Hadamard Elementary function $G^{(1)}$ is non-vanishing. A method is presented whereby the imaginary part of $G^{(1)}$ may be calculated for a charged scalar field in a static spherically symmetric spacetime with arbitrary curvature coupling and a classical electromagnetic field $A^μ$. The calculations are performed in Euclidean space where the Hadamard Elementary function and the Euclidean Green function are related by $(1/2)G^{(1)}=G_{E}$. This method uses a $4^{th}$ order WKB approximation for the Eu…
In the framework of the algebraic formulation, we discuss and analyse some new features of the local structure of a real scalar quantum field theory in a strongly causal spacetime. In particular we use the properties of the exponential map to set up a local version of a bulk-to-boundary correspondence. The bulk is a suitable subset of a geodesic neighbourhood of any but fixed point p of the underlying background, while the boundary is a part of the future light cone having p as its own tip. In this regime, we provide a novel notion for the extended *-algebra of Wick polynomials on the said con…
We obtain the electric field and scalar field for a static point charge in closed form in the 5D Schwarzschild-Tangherlini black hole spacetime. We then compute the static self-force in each of these cases by assuming that the appropriate singular field is a 4D Hadamard Green's function on the constant time Riemannian slice. It is well known that the Hadamard Green's function involves an arbitrary regular biscalar $W_{0}(x,x')$, whose coincidence limit $w(x)$ appears in the expression for the self-force. We develop an axiomatic approach to reduce this arbitrary function to a single arbitrary d…
Quantum field theory on curved spacetimes lacks an obvious distinguished vacuum state. We review a recent no-go theorem that establishes the impossibility of finding a preferred state in each globally hyperbolic spacetime, subject to certain natural conditions. The result applies in particular to the free scalar field, but the proof is model-independent and therefore of wider applicability. In addition, we critically examine the recently proposed "SJ states", that are determined by the spacetime geometry alone, but which fail to be Hadamard in general. We describe a modified construction that …
We discuss the algebraic quantization of a real, massive scalar field in the Poincaré patch of the (d+1)-dimensional anti-de Sitter spacetime, with arbitrary boundary conditions. By using the functional formalism, we show that it is always possible to associate to such system an algebra of observables enjoying the standard properties of causality, time-slice axiom and F-locality. In addition, we characterize the wavefront set of the ground state associated to the system under investigation. As a consequence, we are able to generalize the definition of Hadamard states and construct a global alg…
We provide general guidelines for generalizing dynamical reduction models to curved spacetimes and propose a class of generally covariant relativistic versions of the GRW model. We anticipate that the collapse operators of our class of models may play a rôle in a yet-to-be-formulated theory of semiclassical gravity with collapses. We show explicitly that the collapse operators map a dense domain of states that are initially Hadamard to final Hadamard states -- a property that we expect will be needed for the construction of such a semiclassical theory. Finally, we provide a simple example in w…
For an analogon of the free Wess-Zumino model on Ricci flat spacetimes, the relation between a conserved `supercurrent' and the point-separated improved energy momentum tensor is investigated and a similar relation as on Minkowski space is established. The expectation value of the latter in any globally Hadamard product state is found to be a priori finite in the coincidence limit if the theory is massive. On arbitrary globally hyperbolic spacetimes the `supercurrent' is shown to be a well defined operator valued distribution on the GNS Hilbertspace of any globally Hadamard product state. View…
