Quantum energy inequalities and local covariance II: categorical formulation

Abstract

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.

Quantum strong energy inequalities

“…It is therefore important to explain in what way our state-dependent bounds are nontrivial. The strategy we adopt follows [20,28] in which a state-dependent lower bound of the schematic form…”

Locally Covariant Quantum Field Theory with External Sources

“…The connection thus employed in [25] is much more rigid than that of [9], which does not integrate to a unique parallel transport between fibres (see [9, p. 222]). 11 With these thoughts in mind, the approach in 10 The correct dualization would be to the vector dual bundle of the tangent bundle regarded in the category of affine bundles, but this would lead to the following logical problem: instead of replacing the problem of quantizing affine Poisson spaces by quantizing linear Poisson spaces (which Fedosov's method does, as explained above), the choice of affine connection in [25] replaces the problem of quantizing affine Poisson spaces by quantizing affine Poisson spaces in the fibres. 11 Unique parallel transport depends on theorems on existence and uniqueness of solutions to differential equations that do not necessarily apply to bundles with infinite-dimensional fibres (beyond the Banach case).…”

“…11 With these thoughts in mind, the approach in 10 The correct dualization would be to the vector dual bundle of the tangent bundle regarded in the category of affine bundles, but this would lead to the following logical problem: instead of replacing the problem of quantizing affine Poisson spaces by quantizing linear Poisson spaces (which Fedosov's method does, as explained above), the choice of affine connection in [25] replaces the problem of quantizing affine Poisson spaces by quantizing affine Poisson spaces in the fibres. 11 Unique parallel transport depends on theorems on existence and uniqueness of solutions to differential equations that do not necessarily apply to bundles with infinite-dimensional fibres (beyond the Banach case). A related point is that [9] obtains a deformation of the [25] might be better described as 'Fedosov-inspired' rather than an application of Fedosov's method as such; nonetheless, this procedure does lead to the correct 'improved algebra'.…”

“…On the other hand, the locally covariant framework also has had an impact on applications of quantum field theory to, e.g. quantum energy inequalities [11,13] and cosmology [23,24,28].…”

Conformal Generally Covariant Quantum Field Theory: The Scalar Field and its Wick Products

“…Some interesting new applications have been developed following this line of thinking, we remind here the work of Buchholz and Schlemmer [BS07] and Schlemmer and Verch [SV08], where the authors deal consistently with expectation values of fields in different spacetimes. Another interesting use of similar ideas can be found in the derivation of local energy bounds in curved spacetime as performed by Fewster [Fe07]. The use of these concepts plays a central role in the development of a perturbative theory of quantum gravity as well, to this end we would like to remind the interesting paper of Brunetti and Fredenhagen [BF06].…”