The coupled Maxwell-Lorentz system describes feed-back action of electromagnetic fields in classical electrodynamics. When applied to point-charge sources (viewed as limiting cases of charged fluids) the resulting nonlinear weakly hyperbolic system lies beyond the scope of classical distribution theory. Using regularized derivatives in the framework of Colombeau algebras of generalized functions we analyze a two-dimensional analogue of the Maxwell-Lorentz system. After establishing existence and uniqueness of solutions in this setting we derive some results on distributional limits of solutions with δ-like initial values.

In this paper we develop the mathematics required in order to provide a description of the observables for quantum fields on low-regularity spacetimes. In particular we consider the case of a massless scalar field φ on a globally hyperbolic spacetime M with C 1,1 metric g. This first entails showing that the (classical) Cauchy problem for the wave equation is well-posed for initial data and sources in Sobolev spaces and then constructing low-regularity advanced and retarded Green operators as maps between suitable function spaces. In specifying the relevant function spaces we need to control the norms of both φ and g φ in order to ensure that g • G ± and G ± • g are the identity maps on those spaces. The causal propagator G = G + − G − is then used to define a symplectic form ω on a normed space V (M ) which is shown to be isomorphic to ker g . This enables one to provide a locally covariant description of the quantum fields in terms of the elements of quasi-local C * -algebras.

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We study particular classes of states on the Weyl algebra $$\mathcal {W}$$ W associated with a symplectic vector space S and on the von Neumann algebras generated in representations of $$\mathcal {W}$$ W . Applications in quantum physics require an implementation of constraint equations, e.g., due to gauge conditions, and can be based on the so-called Dirac states. The states can be characterized by nonlinear functions on S, and it turns out that those corresponding to non-trivial Dirac states are typically discontinuous. We discuss general aspects of this interplay between functions on S and states, but also develop an analysis for a particular example class of non-trivial Dirac states. In the last part, we focus on the specific situation with $$S = L^2(\mathbb {R}^n)$$ S = L 2 ( R n ) or test functions on $$\mathbb {R}^n$$ R n and relate properties of states on $$\mathcal {W}$$ W with those of generalized functions on $$\mathbb {R}^n$$ R n or with harmonic analysis aspects of corresponding Borel measures on Schwartz functions and on temperate distributions.