arXiv subjects

Christopher J. Fewster

Publications and source records attributed to Christopher J. Fewster.

At least 73 records · Page 4Linked to original sources

arXiv2012

On a Recent Construction of "Vacuum-like" Quantum Field States in Curved Spacetime

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 state. In the specific case where the Cauchy surface is a round 3-sphere, we prove that the representation induced by the S-J state is generally not unitarily equivalent to that of a Hadamard state, and indeed that the representations induced by S-J states on nested regions of the ultrastatic spacetime also fail to be unitarily equivalent in general. The implications of these results are discussed.

math-ph↗

arXiv2012

Lectures on quantum energy inequalities

Quantum field theory violates all the classical energy conditions of general relativity. Nonetheless, it turns out that quantum field theories satisfy remnants of the classical energy conditions, known as Quantum Energy Inequalities (QEIs), that have been developed by various authors since the original pioneering work of Ford in 1978. These notes provide an introduction to QEIs and also to some of the techniques of quantum field theory in curved spacetime (particularly, the use of microlocal analysis together with the algebraic formulation of QFT) that enable rigorous and general QEIs to be derived. Specific examples are computed for the free scalar field and their consequences are discussed. QEIs are also derived for the class of unitary, positive energy conformal field theories in two spacetime dimensions. In that setting it is also possible to determine the probability distribution for individual measurements of certain smearings of the stress-energy tensor in the vacuum state.

gr-qc↗

arXiv2012

Dynamical locality and covariance: What makes a physical theory the same in all spacetimes?

The question of what it means for a theory to describe the same physics on all spacetimes (SPASs) is discussed. As there may be many answers to this question, we isolate a necessary condition, the SPASs property, that should be satisfied by any reasonable notion of SPASs. This requires that if two theories conform to a common notion of SPASs, with one a subtheory of the other, and are isomorphic in some particular spacetime, then they should be isomorphic in all globally hyperbolic spacetimes (of given dimension). The SPASs property is formulated in a functorial setting broad enough to describe general physical theories describing processes in spacetime, subject to very minimal assumptions. By explicit constructions, the full class of locally covariant theories is shown not to satisfy the SPASs property, establishing that there is no notion of SPASs encompassing all such theories. It is also shown that all locally covariant theories obeying the time-slice property possess two local substructures, one kinematical (obtained directly from the functorial structure) and the other dynamical (obtained from a natural form of dynamics, termed relative Cauchy evolution). The covariance properties of relative Cauchy evolution and the kinematic and dynamical substructures are analyzed in detail. Calling local covariant theories dynamically local if their kinematical and dynamical local substructures coincide, it is shown that the class of dynamically local theories fulfills the SPASs property. As an application in quantum field theory, we give a model independent proof of the impossibility of making a covariant choice of preferred state in all spacetimes, for theories obeying dynamical locality together with typical assumptions.

math-ph↗

arXiv2012

Dynamical locality of the free scalar field

Dynamical locality is a condition on a locally covariant physical theory, asserting that kinematic and dynamical notions of local physics agree. This condition was introduced in [arXiv:1106.4785], where it was shown to be closely related to the question of what it means for a theory to describe the same physics on different spacetimes. In this paper, we consider in detail the example of the free minimally coupled Klein--Gordon field, both as a classical and quantum theory (using both the Weyl algebra and a smeared field approach). It is shown that the massive theory obeys dynamical locality, both classically and in quantum field theory, in all spacetime dimensions $n\ge 2$ and allowing for spacetimes with finitely many connected components. In contrast, the massless theory is shown to violate dynamical locality in any spacetime dimension, in both classical and quantum theory, owing to a rigid gauge symmetry. Taking this into account (equivalently, working with the massless current) dynamical locality is restored in all dimensions $n\ge 2$ on connected spacetimes, and in all dimensions $n\ge 3$ if disconnected spacetimes are permitted. The results on the quantized theories are obtained using general results giving conditions under which dynamically local classical symplectic theories have dynamically local quantizations.

math-ph↗

arXiv2012

Probability distributions for quantum stress tensors in four dimensions

We treat the probability distributions for quadratic quantum fields, averaged with a Lorentzian test function, in four-dimensional Minkowski vacuum. These distributions share some properties with previous results in two-dimensional spacetime. Specifically, there is a lower bound at a finite negative value, but no upper bound. Thus arbitrarily large positive energy density fluctuations are possible. We are not able to give closed form expressions for the probability distribution, but rather use calculations of a finite number of moments to estimate the lower bounds, the asymptotic forms for large positive argument, and possible fits to the intermediate region. The first 65 moments are used for these purposes. All of our results are subject to the caveat that these distributions are not uniquely determined by the moments. However, we also give bounds on the cumulative distribution function that are valid for any distribution fitting these moments.We apply the asymptotic form of the electromagnetic energy density distribution to estimate the nucleation rates of black holes and of Boltzmann brains.

quant-ph↗

arXiv2011

On the notion of "the same physics in all spacetimes"

Brunetti, Fredenhagen and Verch (BFV) have shown how the notion of local covariance for quantum field theories can be formulated in terms of category theory: a theory being described as a functor from a category of spacetimes to a category of $(C)^*$-algebras. We discuss whether this condition is sufficient to guarantee that a theory represents `the same physics' in all spacetimes, giving examples to show that it does not. A new criterion, dynamical locality, is formulated, which requires that descriptions of local physics based on kinematical and dynamical considerations should coincide. Various applications are given, including a proof that dynamical locality for quantum fields is incompatible with the possibility of covariantly choosing a preferred state in each spacetime. As part of this discussion we state a precise condition that should hold on any class of theories each representing the same physics in all spacetimes. This condition holds for the dynamically local theories but is violated by the full class of locally covariant theories in the BFV sense. The majority of results described form part of forthcoming papers with Rainer Verch.

math-ph↗

arXiv2011

Singularity theorems from weakened energy conditions

We establish analogues of the Hawking and Penrose singularity theorems based on (a) averaged energy conditions with exponential damping; (b) conditions on local stress-energy averages inspired by the Quantum Energy Inequalities satisfied by a number of quantum field theories. As particular applications, we establish singularity theorems for the Einstein equations coupled to a classical scalar field, which violates the strong energy condition, and the nonminimally coupled scalar field, which also violates the null energy condition.

gr-qc↗

arXiv2010

Probability distributions of smeared quantum stress tensors

We obtain in closed form the probability distribution for individual measurements of the stress-energy tensor of two-dimensional conformal field theory in the vacuum state, smeared in time against a Gaussian test function. The result is a shifted Gamma distribution with the shift given by the previously known optimal quantum inequality bound. For small values of the central charge it is overwhelmingly likely that individual measurements of the sampled energy density in the vacuum give negative results. For the case of a single massless scalar field, the probability of finding a negative value is 84%. We also report on computations for four-dimensional massless scalar fields showing that the probability distribution of the smeared square field is also a shifted Gamma distribution, but that the distribution of the energy density is not.

quant-ph↗

arXiv2008

Quantum Inequalities from Operator Product Expansions

Quantum inequalities are lower bounds for local averages of quantum observables that have positive classical counterparts, such as the energy density or the Wick square. We establish such inequalities in general (possibly interacting) quantum field theories on Minkowski space, using nonperturbative techniques. Our main tool is a rigorous version of the operator product expansion.

math-ph↗

arXiv2008

Phase space quantization and Loop Quantum Cosmology: A Wigner function for the Bohr-compactified real line

We give a definition for the Wigner function for quantum mechanics on the Bohr compactification of the real line and prove a number of simple consequences of this definition. We then discuss how this formalism can be applied to loop quantum cosmology. As an example, we use the Wigner function to give a new quantization of an important building block of the Hamiltonian constraint.

math-ph↗

arXiv2007

Absolute quantum energy inequalities in curved spacetime

Quantum Energy Inequalities (QEIs) are results which limit the extent to which the smeared renormalised energy density of the quantum field can be negative, when averaged along a timelike curve or over a more general timelike submanifold in spacetime. On globally hyperbolic spacetimes the minimally-coupled massive quantum Klein--Gordon field is known to obey a `difference' QEI that depends on a reference state chosen arbitrarily from the class of Hadamard states. In many spacetimes of interest this bound cannot be evaluated explicitly. In this paper we obtain the first `absolute' QEI for the minimally-coupled massive quantum Klein--Gordon field on four dimensional globally hyperbolic spacetimes; that is, a bound which depends only on the local geometry. The argument is an adaptation of that used to prove the difference QEI and utilises the Sobolev wave-front set to give a complete characterisation of the singularities of the Hadamard series. Moreover, the bound is explicit and can be formulated covariantly under additional (general) conditions. We also generalise our results to incorporate adiabatic states.

gr-qc↗

arXiv2007

Averaged null energy condition in spacetimes with boundaries

The Averaged Null Energy Condition (ANEC) requires that the average along a complete null geodesic of the projection of the stress-energy tensor onto the geodesic tangent vector can never be negative. It is sufficient to rule out many exotic phenomena in general relativity. Subject to certain conditions, we show that the ANEC can never be violated by a quantized minimally coupled free scalar field along a complete null geodesic surrounded by a tubular neighborhood in which the geometry is flat and whose intrinsic causal structure coincides with that induced from the full spacetime. In particular, the ANEC holds in flat space with boundaries, as in the Casimir effect, for geodesics which stay a finite distance away from the boundary

gr-qc↗

arXiv2007

Quantum energy inequalities and local covariance II: Categorical formulation

We formulate Quantum Energy Inequalities (QEIs) in the framework of locally covariant quantum field theory developed by Brunetti, Fredenhagen and Verch, which is based on notions taken from category theory. This leads to a new viewpoint on the QEIs, and also to the identification of a new structural property of locally covariant quantum field theory, which we call Local Physical Equivalence. Covariant formulations of the numerical range and spectrum of locally covariant fields are given and investigated, and a new algebra of fields is identified, in which fields are treated independently of their realisation on particular spacetimes and manifestly covariant versions of the functional calculus may be formulated.

math-ph↗

arXiv2007

Mass Dependence of Quantum Energy Inequality Bounds

In a recent paper [J. Math. Phys. 47 082303 (2006)], Quantum Energy Inequalities were used to place simple geometrical bounds on the energy densities of quantum fields in Minkowskian spacetime regions. Here, we refine this analysis for massive fields, obtaining more stringent bounds which decay exponentially in the mass. At the technical level this involves the determination of the asymptotic behaviour of the lowest eigenvalue of a family of polyharmonic differential equations, a result which may be of independent interest. We compare our resulting bounds with the known energy density of the ground state on a cylinder spacetime. In addition, we generalise some of our technical results to general $L^p$-spaces and draw comparisons with a similar result in the literature.

math-ph↗

arXiv2006

Averaged Energy Inequalities for the Non-Minimally Coupled Classical Scalar Field

The stress energy tensor for the classical non-minimally coupled scalar field is known not to satisfy the point-wise energy conditions of general relativity. In this paper we show, however, that local averages of the classical stress energy tensor satisfy certain inequalities. We give bounds for averages along causal geodesics and show, e.g., that in Ricci-flat background spacetimes, ANEC and AWEC are satisfied. Furthermore we use our result to show that in the classical situation we have an analogue to the phenomenon of quantum interest. These results lay the foundations for analogous energy inequalities for the quantised non-minimally coupled fields, which will be discussed elsewhere.

gr-qc↗

arXiv2006

Quantum Energy Inequalities and local covariance I: Globally hyperbolic spacetimes

We begin a systematic study of Quantum Energy Inequalities (QEIs) in relation to local covariance. We define notions of locally covariant QEIs of both 'absolute' and 'difference' types and show that existing QEIs satisfy these conditions. Local covariance permits us to place constraints on the renormalised stress-energy tensor in one spacetime using QEIs derived in another, in subregions where the two spacetimes are isometric. This is of particular utility where one of the two spacetimes exhibits a high degree of symmetry and the QEIs are available in simple closed form. Various general applications are presented, including a priori constraints (depending only on geometric quantities) on the ground-state energy density in a static spacetime containing locally Minkowskian regions. In addition, we present a number of concrete calculations in both two and four dimensions which demonstrate the consistency of our bounds with various known ground- and thermal state energy densities. Examples considered include the Rindler and Misner spacetimes, and spacetimes with toroidal spatial sections. In this paper we confine the discussion to globally hyperbolic spacetimes; subsequent papers will also discuss spacetimes with boundary and other related issues.

math-ph↗

arXiv2005

p-Nuclearity in a New Perspective

In this paper we try to settle some confused points concerning the use of the notion of p-nuclearity in the mathematical and physical literature, pointing out that the nuclearity index in the physicists' sense vanishes for any p > 1. Our discussion of these issues suggests a new perspective, in terms of epsilon-entropy and operator spaces, which might permit connections to be drawn between phase space criteria and quantum energy inequalities.

math-ph↗

arXiv2005

Quantum Energy Inequalities in two-dimensional conformal field theory

Quantum energy inequalities (QEIs) are state-independent lower bounds on weighted averages of the stress-energy tensor, and have been established for several free quantum field models. We present rigorous QEI bounds for a class of interacting quantum fields, namely the unitary, positive energy conformal field theories (with stress-energy tensor) on two-dimensional Minkowski space. The QEI bound depends on the weight used to average the stress-energy tensor and the central charge(s) of the theory, but not on the quantum state. We give bounds for various situations: averaging along timelike, null and spacelike curves, as well as over a spacetime volume. In addition, we consider boundary conformal field theories and more general 'moving mirror' models. Our results hold for all theories obeying a minimal set of axioms which -- as we show -- are satisfied by all models built from unitary highest-weight representations of the Virasoro algebra. In particular, this includes all (unitary, positive energy) minimal models and rational conformal field theories. Our discussion of this issue collects together (and, in places, corrects) various results from the literature which do not appear to have been assembled in this form elsewhere.

math-ph↗