Dynamical Locality of the Free Maxwell Field

Abstract

Abstract:We consider the non-interacting source-free Maxwell field, described both in terms of the vector potential and the field strength. Starting from the classical field theory on contractible globally hyperbolic spacetimes, we extend the classical field theory to general globally hyperbolic spacetimes in two ways to obtain a 'universal' theory and a 'reduced' theory. The quantum field theory in terms of the unital * -algebra of the smeared quantum field is then obtained by an application of a suitable quantisation functor. We show that the universal theories fail local covariance and dynamical locality owing to the possibility of having non-trivial radicals in the classical and non-trivial centres in the quantum case. The reduced theories are both locally covariant and dynamically local. These models provide new examples relevant to the discussion of how theories should be formulated so as to describe the same physics in all spacetimes.

Algebraic Quantum Field Theory in Curved Spacetimes

“…where T M M M is the renormalised stress-energy tensor of the theory. 11 Similar computations for other models [136,9,46,64,59] support the view that, in general, the functional derivative of the relative Cauchy evolution may be interpreted as a stress-energy tensor. One is therefore led to regard the relative Cauchy evolution as a proxy for the action.…”

“…Note that this conclusion required no discussion of how free electromagnetism should be formulated in spacetimes other than Minkowski, beyond the requirements of local covariance and the timeslice property. This explains why 'topological observables' are absent from quantizations of Maxwell theory obeying these properties [59]. To restore them, one must relax the assumption of local covariance to permit noninjective maps [37,137,59,7].…”

“…There are some cases known in which dynamical locality fails, which appears to be always related to the presence of broken global gauge symmetries or topologically stabilised charges: the free Klein-Gordon field with m = 0, ξ = 0 in dimensions n ≥ 2, owing to the rigid gauge symmetry φ → φ + const [70]; the free massless current in 2-dimensions allowing disconnected spacetimes [70]; and the free Maxwell field in dimension n = 4, in a 'universal formulation' [37,59], owing to the presence of topological electric and magnetic charges in spacetimes with nontrivial second de Rham cohomology, which are eliminated in the reduced theory mentioned above. As already mentioned in Sect.…”

Abelian Duality on Globally Hyperbolic Spacetimes

“…Let us start by recalling a well-known result for the Cauchy problem of the Faraday tensor, see e.g. [DL12,FL14] and also [BF09, Chapter 3, Corollary 5] for details on how to treat initial data of not necessarily compact support. For the related Cauchy problem of the gauge potential see [SDH14].…”

Linear Yang–Mills Theory as a Homotopy AQFT

“…descent) of AQFTs. Within the traditional formulation in terms of gauge-invariant on-shell observable algebras, quantum gauge theories have very poor local-to-global properties as witnessed for example by the observation in [DL12,FL16] that Fredenhagen's universal algebra (which is a certain local-to-global construction) for Abelian Yang-Mills theory fails to encode crucial gauge theoretic features such as Dirac's charge quantization and Aharonov-Bohm phases. It was later understood and emphasized in [BSS15] that the failure of (too naive versions of) local-to-global constructions is due to higher categorical structures in classical and quantum gauge theories, which are neglected (i.e.…”