Reviews in Mathematical Physics

We consider the Klein–Gordon equation on a static spacetime and minimally coupled to a static electromagnetic potential. We show that it is essentially self-adjoint on $C_{\mathrm{\text{c}}}^{\infty }$. We discuss various distinguished inverses and bisolutions of the Klein–Gordon operator, focusing on the so-called Feynman propagator. We show that the Feynman propagator can be considered the boundary value of the resolvent of the Klein–Gordon operator, in the spirit of the limiting absorption principle known from the theory of Schrödinger operators. We also show that the Feynman propagator is the limit of the inverse of the Wick rotated Klein–Gordon operator.

We construct and study the stochastic force field generated by a Poisson distribution of sources at finite density, $x_1,x_2,\dots ,$ in ${ℝ}^3$ each of them yielding a long range potential $Q_i\mathrm{\Phi }(x-x_i)$ with possibly different charges $Q_i\in ℝ$. The potential $\mathrm{\Phi }$ is assumed to behave typically as $|x{|}^{-s}$ for large $\left|x\right|$, with $s>1/2$. We will denote the resulting random field as “generalized Holtsmark field”. We then consider the dynamics of one tagged particle in such random force fields, in several scaling limits where the mean free path is much larger than the average distance between the scatterers. We estimate the diffusive time scale and identify conditions for the vanishing of correlations. These results are used to obtain appropriate kinetic descriptions in terms of a linear Boltzmann or Landau evolution equation depending on the specific choices of the interaction potential.

We present a definition of a (super)-modular functor which includes certain interesting cases that previous definitions do not allow. We also introduce a notion of topological twisting of a modular functor, and construct formally a realization by a 2-dimensional topological field theory valued in twisted $K$-modules. We discuss, among other things, the $N=1$-supersymmetric minimal models from the point of view of this formalism.