Hadamard States for the Linearized Yang–Mills Equation on Curved Spacetime

Abstract

Abstract. We construct Hadamard states for the Yang-Mills equation linearized around a smooth, space-compact background solution. We assume the spacetime is globally hyperbolic and its Cauchy surface is compact or equal R d .We first consider the case when the spacetime is ultra-static, but the background solution depends on time. By methods of pseudodifferential calculus we construct a parametrix for the associated vectorial Klein-Gordon equation. We then obtain Hadamard two-point functions in the gauge theory, acting on Cauchy data. A key role is played by classes of pseudodifferential operators that contain microlocal or spectral type low-energy cutoffs.The general problem is reduced to the ultra-static spacetime case using an extension of the deformation argument of Fulling, Narcowich and Wald.As an aside, we derive a correspondence between Hadamard states and parametrices for the Cauchy problem in ordinary quantum field theory.

Hadamard Property of the in and out States for Klein–Gordon Fields on Asymptotically Static Spacetimes

“…By repeating the arguments in [GW1,GW2] this can be solved modulo terms in C ∞ (R; W −∞ (Σ)). Concretely, supposing for the moment that a(t) ≥ c(t)1 for c(t) > 0, upon setting ǫ = a 1 2 , b = ǫ + b 0 one obtains the equations:…”

“…Furthermore, if the state is Hadamard then U (s, t)c ± (t) propagate singularities along N ± (see the discussion in [GW2]). In Sect.…”

Curving flat space-time by deformation quantization?

“…This stems from the fact that for such space-times, the strong energy nuclearity condition has been proven for the free massive Klein-Gordon field, [Ver93]. For further applications in AQFT see [Str00], [FS15], [GW15], [LS15] and [San16], to mention just a few.…”

Well‐posedness of the Laplacian on manifolds with boundary and bounded geometry

“…Our well-posedness result is of interest in itself, as a general result on analysis on non-compact manifolds, but also because it has applications to partial differential equations on singular and non-compact spaces, which are more and more often studied. Examples are provided by analysis on curved space-times in general relativity and conformal field theory, see [11,12,29,30,40,44,67] and the references therein. An important example of manifolds with bounded geometry is that of asymptotically hyperbolic manifolds, which play an increasingly important role in geometry and physics [5,15,24,34].…”