Quantum Energy Inequalities Research Articles

Absolute Quantum Energy Inequalities in Curved Spacetime

Quantum Energy Inequalities (QEIs) are results which limit the extent to which the smeared renormalized 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 utilizes the Sobolev wave-front set to give a complete characterization 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.

Balanced homodyne detectors in quantum field theory

Within the dipole approximation we describe the interaction of a photodiode with the quantum electric field. The diode is modelled by an electron in a bound state which upon interaction, treated perturbatively in the paper, can get excited to one of the scattering states. We furthermore analyze a balanced homodyne detector (BHD) with a local oscillator (LO) consisting of two photodiodes illuminated by a monochromatic coherent state. We show, that to the leading order the BHD's output measures the expectation value of the quantum electric field, in the state without the LO, restricted to the frequency of the LO. The square of the output measures the two-point function of the quantum field. This shows that the BHDs provide tools for measurements of sub-vacuum (negative) expectation values of the squares quantum fields and thus for test of Quantum Energy Inequality - like bounds, or other QFT effects under the influence of external conditions.

Quantum energy inequalities for the non-minimally coupled scalar field

In this paper, we discuss local averages of the energy density for the non-minimally coupled scalar quantum field, extending a previous investigation of the classical field. By an explicit example, we show that such averages are unbounded from below on the class of Hadamard states. This contrasts with the minimally coupled field, which obeys a state-independent lower bound known as a quantum energy inequality (QEI). Nonetheless, we derive a generalized QEI for the non-minimally coupled scalar field, in which the lower bound is permitted to be state-dependent. This result applies to general globally hyperbolic curved spacetimes for coupling constants in the range 0 < ξ ⩽ 1/4. We analyse the state dependence of our QEI in four-dimensional Minkowski space and show that it is a non-trivial restriction on the averaged energy density in the sense that the lower bound is of lower order, in energetic terms, than the averaged energy density itself.

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.

An absolute quantum energy inequality for the Dirac field in curved spacetime

Quantum weak energy inequalities (QWEIs) are results which limit the extent to which the smeared renormalized energy density of a quantum field can be negative. On globally hyperbolic spacetimes the massive quantum Dirac field is known to obey a QWEI in terms of a reference state chosen arbitrarily from the class of Hadamard states; however, there exist spacetimes of interest on which state-dependent bounds cannot be evaluated. In this paper we prove the first QWEI for the massive quantum Dirac field on four-dimensional globally hyperbolic spacetime in which the bound depends only on the local geometry; such a QWEI is known as an absolute QWEI.

Mass dependence of quantum energy inequality bounds

In a recent paper [C. J. Fewster and M. J. Pfenning, 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 space-time 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 behavior 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 space-time. In addition, we generalize some of our technical results to general Lp spaces and draw comparisons with a similar result in the literature.

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 renormalized 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 that 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.

Quantum Inequality Bounds for Free Rarita–Schwinger Field inFlat Spacetime

Although quantum field theory allows the local energy densitynegative, it also places severe restrictions on the negativeenergy. One of the restrictions is the quantum energyinequality (QEI), in which the energy density is averaged overtime, or space, or over space and time. By now temporal QEIs havebeen established for various quantum fields, but less work hasbeen done for the spacetime quantum energy inequality. In thispaper we deal with the free Rarita–Schwinger field and present aquantum inequality bound on the energy density averaged over spaceand time. Comparison with the QEI for the Rarita–Schwinger fieldshows that the lower bound is the same with the QEI. At the sametime, we find the quantum inequality for the Rarita–Schwingerfield is weaker than those for the scalar and Dirac fields. Thisfact gives further support to the conjecture that the morefreedom the field has, the more easily the field displays negativeenergy density and the weaker the quantum inequality becomes.

Bounds on the energy densities of ground states on static spacetimes of compact objects

In this paper we investigate quantum fields propagating on given, static, spherically symmetric spacetimes, which are isometric to a part of the Schwarzschild spacetime. Without specifying the internal geometry we show, that there exist bounds on the energy densities of ground states of a quantum scalar field on such spacetimes. The bounds (from above and below) come from the so-called Quantum Energy Inequalities, and are centered around the energy density of the Boulware state (the ground state for Schwarzschild spacetime). The specific value of the bound from below depends critically on the distance $\ell$ from the horizon, where the spacetimes of compact objects cease to be isometric to the Schwarzschild spacetime. In the limit of small $\ell$ we prove, that the energy densities of ground states cannot be below the Boulware level.

Restrictions on negative energy density for the Dirac field in flat spacetime

This paper investigates the quantum Dirac field in n+1-dimensional flat spacetime and derives a lower bound in the form of quantum inequality on the energy density averaged against spacetime sampling functions. The state-independent quantum inequality derived in the present paper is similar to the temporal quantum energy inequality and it is stronger for massive field than for massless one. It also presents the concrete results of the quantum inequality in 2 and 4-dimensional spacetimes.

Lower Bounds on Negative Energy Densities for the ScalarField in Flat Spacetime

We obtain a lower bound on the spacetime-weighted average of the energydensity for the scalar field in four-dimensional flat spacetime. Thebound takes the form of a quantum inequality. The inequality does notrely on the quantum state and its form is only related to the weights,namely the spacetime sampling functions which are assumed to be smooth,positive and compactly supported. It is found that the inequality isjust equal to the temporal quantum energy inequality. When thecharacteristic length of the temporal sampling function tends to zero,the lower bound becomes divergent. This is consistent with the fact thatthe spatial restriction on negative energy density does not exist infour-dimensional spacetime.

Topological censorship and chronology protection

AbstractOver the past two decades, substantial efforts have been made to understand the way in which physics enforces the ordinary topology and causal structure that we observe on observed scales – from subnuclear to cosmological. We review the status of topological censorship and the topology of event horizons; chronology protection in classical and semiclassical gravity; and related progress in establishing quantum energy inequalities.

Is general relativity ‘essentially understood’?

Over the past two decades, substantial efforts have been made to understand the way in which physics enforces the ordinary topology and causal structure that we observe on observed scales – from subnuclear to cosmological. We review the status of topological censorship and the topology of event horizons; chronology protection in classical and semiclassical gravity; and related progress in establishing quantum energy inequalities.

P-Nuclearity in a New Perspective

In this Letter 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 ε-entropy and operator spaces, which might permit connections to be drawn between phase space criteria and quantum energy inequalities.

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 space-time 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.

Quantum energy inequalities in two dimensions

Quantumenergy inequalities (QEIs) were established by Flanagan for the massless scalarfield on two-dimensional Lorentzian spacetimes globally conformal to Minkowskispace. We extend his result to all two-dimensional globally hyperbolic Lorentzianspacetimes and use it to show that flat spacetime QEIs give a good approximationto the curved spacetime results on sampling time scales short in comparisonwith natural geometric scales. This is relevant to the application of QEIsto constrain exotic spacetime metrics.