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 inequality obtained generalises a previous result derived for static trajectories in a static spacetime. The underlying argument is straightforward and is made rigorous using the techniques of microlocal analysis. In particular, an important role is played by the characterisation of Hadamard states in terms of the microlocal spectral condition. We also give a compact form of our result for stationary trajectories in a stationary spacetime. *

We generalise results of Ford and Roman which place lower bounds -known as quantum inequalities -on the renormalised energy density of a quantum field averaged against a choice of sampling function. Ford and Roman derived their results for a specific non-compactly supported sampling function; here we use a different argument to obtain quantum inequalities for a class of smooth, even and non-negative sampling functions which are either compactly supported or decay rapidly at infinity. Our results hold in d-dimensional Minkowski space (d ≥ 2) for the free real scalar field of mass m ≥ 0. We discuss various features of our bounds in 2 and 4 dimensions. In particular, for massless field theory in 2-dimensional Minkowski space, we show that our quantum inequality is weaker than Flanagan's optimal bound by a factor of 3 2 .

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 twodimensional 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.

The process of quantum measurement is considered in the algebraic framework of quantum field theory on curved spacetimes. Measurements are carried out on one quantum field theory, the “system”, using another, the “probe”. The measurement process involves a dynamical coupling of “system” and “probe” within a bounded spacetime region. The resulting “coupled theory” determines a scattering map on the uncoupled combination of the “system” and “probe” by reference to natural “in” and “out” spacetime regions. No specific interaction is assumed and all constructions are local and covariant. Given any initial state of the probe in the “in” region, the scattering map determines a completely positive map from “probe” observables in the “out” region to “induced system observables”, thus providing a measurement scheme for the latter. It is shown that the induced system observables may be localized in the causal hull of the interaction coupling region and are typically less sharp than the probe observable, but more sharp than the actual measurement on the coupled theory. Post-selected states conditioned on measurement outcomes are obtained using Davies–Lewis instruments that depend on the initial probe state. Composite measurements involving causally ordered coupling regions are also considered. Provided that the scattering map obeys a causal factorization property, the causally ordered composition of the individual instruments coincides with the composite instrument; in particular, the instruments may be combined in either order if the coupling regions are causally disjoint. This is the central consistency property of the proposed framework. The general concepts and results are illustrated by an example in which both “system” and “probe” are quantized linear scalar fields, coupled by a quadratic interaction term with compact spacetime support. System observables induced by simple probe observables are calculated exactly, for sufficiently weak coupling, and compared with first order perturbation theory.

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. *

Quantum fields are well known to violate the weak energy condition of general relativity: the renormalised energy density at any given point is unbounded from below as a function of the quantum state. By contrast, for the scalar and electromagnetic fields it has been shown that weighted averages of the energy density along timelike curves satisfy 'quantum weak energy inequalities' (QWEIs) which constitute lower bounds on these quantities. Previously, Dirac QWEIs have been obtained only for massless fields in two-dimensional spacetimes. In this paper we establish QWEIs for the Dirac and Majorana fields of mass m ≥ 0 on general four-dimensional globally hyperbolic spacetimes, averaging along arbitrary smooth timelike curves with respect to any of a large class of smooth compactly supported positive weights. Our proof makes essential use of the microlocal characterisation of the class of Hadamard states, for which the energy density may be defined by point-splitting.

Quantum weak energy inequalities (QWEI) provide state-independent lower bounds on averages of the renormalized energy density of a quantum field. We derive QWEIs for the electromagnetic and massive spin-one fields in globally hyperbolic space–times 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 averaging along static trajectories in ultrastatic space–times; as specific examples we consider Minkowski space (in which case the topological restrictions may be dispensed with) and the static Einstein universe. A significant part of the paper is devoted to the definition and properties of Hadamard states of spin-one fields in curved space–times, particularly with regard to their microlocal behavior.

Linearized Einstein gravity (with possibly nonzero cosmological constant) is quantized in the framework of algebraic quantum field theory by analogy with Dimock's treatment of electromagnetism [Rev. Math. Phys. 4 (1992) 223-233]. To achieve this, the classical theory is developed in a full, rigorous and systematic fashion, with particular attention given to the circumstances under which the symplectic product is weakly non-degenerate and to the related question of whether the space of solutions is separated by the classical observables on which the quantum theory is modelled.