An Analogue of the Coleman–Mandula Theorem for Quantum Field Theory in Curved Spacetimes

Abstract

Abstract:The Coleman-Mandula (CM) theorem states that the Poincaré and internal symmetries of a Minkowski spacetime quantum field theory cannot combine nontrivially in an extended symmetry group. We establish an analogous result for quantum field theory in curved spacetimes, assuming local covariance, the timeslice property, a local dynamical form of Lorentz invariance, and additivity. Unlike the CM theorem, our result is valid in dimensions n ≥ 2 and for free or interacting theories. It is formulated for theories defined on a category of all globally hyperbolic spacetimes equipped with a global coframe, on which the restricted Lorentz group acts, and makes use of a general analysis of symmetries induced by the action of a group G on the category of spacetimes. Such symmetries are shown to be canonically associated with a cohomology class in the second degree nonabelian cohomology of G with coefficients in the global gauge group of the theory. Our main result proves that the cohomology class is trivial if G is the universal cover S of the restricted Lorentz group. Among other consequences, it follows that the extended symmetry group is a direct product of the global gauge group and S, all fields transform in multiplets of S, fields of different spin do not mix under the extended group, and the occurrence of noninteger spin is controlled by the centre of the global gauge group. The general analysis is also applied to rigid scale covariance.

Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries

“…But our construction allows for more exotic possibilities, since q can be non-commuting with respect to the coordinates x: in case A. 2 we have a κ-Minkowski spacetime with a gauge coordinate that acts on the spacetime coordinates, [x, q] ⊆ x, and in case A.1 we have the same thing, this time on a commutative spacetime. Cases B are stranger yet: the q coordinate appears on the right-hand-side of the commutation relation of the x, in a term that reminds of the canonical noncommutative spacetimes (see below).…”

“…S O (2)]. In the commutative case, the CMT implies that the most general group of symmetries of an SO(2) QFT on a Minkowski background is the direct product I SO(1, 1) × SO (2).…”

“…the Lorentz subalgebra (and therefore admits a homogeneous quantum Minkowski spacetime as a quotient). The commutation rules for the spacetime, x, and gauge, q coordinates that we obtain belong to the following five cases: If we want to interpret our starting point, the algebra iso(1, 1) ⊕ so (2), as the algebra of symmetries of a U (1) gauge theory on Minkowski space, then it is natural to identify the coordinates {x μ , q} as functions on the (trivial) principal fibre bundle M × U (1) (where M is Minkowski space), which indeed is the geometrical construction underlying a U (1) gauge theory. In the literature, one can find a notion of quantum principal bundle [44,[56][57][58][59][60][61][62][63][64], but this notion seems to be too restrictive: the gauge group is assumed to close a quantum group on its own [which can only be trivial in the case of SO (2)], and the coordinate q on the gauge group is assumed to commute with the noncommutative coordinates of the base space.…”

“…Our Lie bialgebras will be based on a central extension of the Poincaré and (A)dS algebras in (1 + 1) dimensions, which means g = iso(1, 1) ⊕ so (2) in the Poincaré case and g = so(2, 1) ⊕ so (2) in the case of (A)dS. As isotropy subalgebras we can either choose the Lorentz subalgebra so (1,1) or the direct sum of the Lorentz and so(2) subalgebras, so(1, 1) ⊕ so (2). In the interpretation of G representing the symmetries of a gauge bundle on a homogeneous spacetime, the first case corresponds to quotenting down to the bundle, while the second case corresponds to quotienting out the gauge coordinate as well and projecting down to spacetime.…”

“…However, the application of standard QFT on Minkowski spacetime to quantum gravity encounters fundamental problems, and the assumption of local Poincaré (Lorentz) invariance might be at the core of the issue. It appears highly significant, in this context, that the ColemanMandula no-go theorem [1,2] implies that Poincaré symmetry cannot be generalized within the framework of QFT with Lie-group symmetries.…”