Renormalized Quantum Yang–mills Fields in Curved Spacetime

Abstract

We present a proof that quantum Yang-Mills theory can be consistently defined as a renormalized, perturbative quantum field theory on an arbitrary globally hyperbolic curved, Lorentzian spacetime. To this end, we construct the non-commutative algebra of observables, in the sense of formal power series, as well as a space of corresponding quantum states. The algebra contains all gauge invariant, renormalized, interacting quantum field operators (polynomials in the field strength and its derivatives), and all their relations such as commutation relations or operator product expansion. It can be viewed as a deformation quantization of the Poisson algebra of classical Yang-Mills theory equipped with the Peierls bracket. The algebra is constructed as the cohomology of an auxiliary algebra describing a gauge fixed theory with ghosts and anti-fields. A key technical difficulty is to establish a suitable hierarchy of Ward identities at the renormalized level that ensure conservation of the interacting BRST-current, and that the interacting BRST-charge is nilpotent. The algebra of physical interacting field observables is obtained as the cohomology of this charge. As a consequence of our constructions, we can prove that the operator product expansion closes on the space of gauge invariant operators. Similarly, the renormalization group flow is proved not to leave the space of gauge invariant operators. The key technical tool behind these arguments is a new universal Ward identity that is formulated at the algebraic level, and that is proven to be consistent with a local and covariant renormalization prescription. We also develop a new technique to accomplish this renormalization process, and in particular give a new expression for some of the renormalization constants in terms of cycles.2

Quantum fields in curved spacetime

“…This failure to be conserved is called an "anomaly". By deriving suitable "consistency conditions" 24 on this anomaly, one can show [47] that an arbitrary renormalization prescription, consistent with all of the properties listed in thm. 2, can always be modified so as to remove this anomaly 25 .…”

“…The observables of the resulting theory are equivalent to the gauge invariant observables of the original Yang-Mills theory. As has been shown in [47], one can implement the procedure consistently obtaining in the end an algebra of gauge invariant quantum observables B I in perturbative interacting Yang-Mills theory.…”

“…The point is that such an object must be defined so as to be compatible with renormalization [27,47]. Informally speaking, one would like to define Q as the (graded) commutator with a "BRST-charge", which itself is obtained from a corresponding conserved Noether current associated with the BRST-invariance ofL , defined in turn as an interacting field in the algebraB I in the manner described in the previous section.…”

Perturbative algebraic quantum field theory and the renormalization groups

“…Our method is related to differential renormalization [19]. Further useful renormalization procedures in x-space are dimensional regularization [3], different kinds of analytic renormalization [23,29] and in case of 2-point functions a method relying on the Källen-Lehmann representation [16].…”

Dimensional regularization in position space and a Forest Formula for Epstein-Glaser renormalization

“…Such a procedure was first proposed by Bollini and Giambiagi [BG96] and was also tested in several examples by [GKP07]. It can be viewed as a particular 'analytic regularization', as introduced by Speer in the context of BPHZ-renormalization long ago [Spe71], and applied to EG renormalization by Hollands [Hol08]. A different approach had been taken by Rosen and Wright [RW90]: they implement dimensional regularization in x-space by making replacements on the level of the position space Feynmann rules.…”