Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory

Abstract

Summary. We give a brief review of quantum energy inequalities (QEIs) and then discuss two lines of work which suggest that QEIs are closely related to various natural properties of quantum field theory which may all be regarded as stability conditions. The first is based on joint work with Verch, and draws connections between microscopic stability (microlocal spectrum condition), mesoscopic stability (QEIs) and macroscopic stability (passivity). The second direction considers QEIs for a countable number of massive scalar fields, and links the existence and scaling properties of QEIs to the spectrum of masses. The upshot is that the existence of a suitable QEI with polynomial scaling is a sufficient condition for the model to satisfy the Buchholz-Wichmann nuclearity criterion. We briefly discuss on-going work with Ojima and Porrmann which seeks to gain a deeper understanding of this relationship.

Quantum energy inequalities and local covariance II: categorical formulation

“…It is tempting to speculate that this link might run more deeply, and might indeed suggest an approach to quantum inequalities via the phase space properties of quantum field theory, in which the 'gain of derivatives' can be explicitly linked to nontriviality of the QI. Further evidence and comments in this direction may be found in [17,20]. It also supports the contention that our definition of quantum inequalities is natural from a mathematical viewpoint.…”

“…In some models, however, local averages of the expectation value of the stress-energy tensor can be bounded from below as functions of the state, and these bounds constitute Quantum Energy Inequalities (QEIs). Inequalities of this type have been obtained for free fields in flat and curved spacetimes (see, e.g., [16,44,17] for discussion and references, and [26,45] for recent results) and for two-dimensional conformal field theories in Minkowski space [18]. (A second type of QEI, to be discussed in the next section, has been proved in many more cases.)…”

“…Accordingly, we see that the right-hand side of (17) is contained in the left (as σ(Φ) is closed). Conversely, if λ does not belong to the right-hand side of (17), then there exists ǫ > 0 for which the disc {µ ∈ C : |µ − λ| < ǫ} lies in the resolvent set of every Φ M (f ) (initially for M ∈ M, but hence for all M ∈ Man). Setting…”

“…QEIs are the remnants in QFT of the classical energy conditions of general relativity (see [16,44,17] for recent reviews) and usually take the form of state-independent lower bounds on suitable averages of the stress-energy tensor. In [24], it was shown that known examples of QEIs can be formulated in a covariant fashion, and that this could be used to obtain a priori bounds on ground state energy densities in the Casimir effect and similar situations (see also [40]).…”

Violation of the Spatial Quantum Inequality and the Quantum Interest Conjecture: A New Approach

“…There are a number of papers that claim to show that there are limits on this negative energy density Flanagan, 1997;Fewster et al, 1998;Ford, 2010;Bostelann et al, 2009;Fewster et al, 2008;Eveson et al, 2007;Fewster, 2005;Ford et al, 1998;Pfenning et al, 1998). These limits are called the quantum inequalities.…”