A Regularization Approach to Non-smooth Symplectic Geometry

Abstract

We introduce non-smooth symplectic forms on manifolds and describe corresponding Poisson structures on the algebra of Colombeau generalized functions. This is achieved by establishing an extension of the classical map of smooth functions to Hamiltonian vector fields to the setting of non-smooth geometry. For mildly singular symplectic forms, including the continuous non-differentiable case, we prove the existence of generalized Darboux coordinates in the sense of a local non-smooth pull-back to the canonical symplectic form on the cotangent bundle.Mathematics Subject Classification (2010): 53D05,46F30

Green operators in low regularity spacetimes and quantum field theory

“…The remainder of the section shows how to go from existence and uniqueness results on R n+1 to global results on a globally hyperbolic C 1,1 spacetime. Our approach to this closely follows Ringström [48] and the causality results for C 1,1 metrics [31] ensure that the existence proof remains similar to the smooth case. The next step is to define appropriate Green operators.…”

“…The proof will be based on the following two lemmas. The proof of Lemma 6.13 in the C 1,1 setting can be carried out following that of [1] with suitable modifications using results of low regularity causality theory [12,49,31]. In particular, the proof uses the facts that, the causal relation is closed, that if S, S t are Cauchy hypersurfaces of O and S is also a Cauchy hypersurface of M , then S t is a Cauchy hypersurface of M and the existence of Cauchy hypersurfaces in globally hyperbolic spacetimes.…”

“…Let q ∈ M and without loss of generality suppose that q ∈ I + (Σ). Then since our spacetime is globally hyperbolic we may find a p such that q ∈ I − (p) ∩ I + (Σ) := U where by C 1,1 causality theory U has compact closure [31]. Now suppose there exist two solutions u and ũ to the above initial value problem.…”

“…From this point of view the choice of C 1,1 metrics is natural since it allows one to deal with spacetimes where the curvature remains finite while allowing for discontinuities in the energy-momentum tensor at, for example, an interface or the surface of a star. From the mathematical point of view C 1,1 is the minimal condition which ensures existence and uniqueness of geodesics and for which the standard results from smooth causality theory go through more or less unchanged [31]. It also ensures that the solutions to the wave equation are in H 2 loc (M ) (see Appendix B for details of the function spaces we use) as shown in Theorem 4.7 below, which ensures we have enough regularity to define the quantisation functors we need.…”

“…[48,Theorem 2.6.4]). In Section 4 we introduce the notion of global hyperbolicity [49] and temporal functions for nonsmooth metrics [43] as well as the other results from C 1,1 causality theory that we require [31]. We use the fact that even for C 1,1 spacetimes the temporal function can be chosen to be smooth so we can write M as R × Σ, where Σ is a Cauchy surface, and define function spaces where we make a split between space and time.…”