Reviews in Mathematical Physics

The mathematical development of Yang–Mills theory is an extremely fruitful subject. The purpose of this paper is to give non-experts and researchers in interdisciplinary areas a quick overview of some history, key ideas and recent developments in this subject.

We prove that the Gärtner–Ellis generating function of probability distributions associated with KMS states of weakly interacting fermions on the lattice can be written as the limit of logarithms of Gaussian Berezin integrals. The covariances of the Gaussian integrals are shown to have a uniform Pfaffian bound and to be summable in general cases of interest, including systems that are not translation invariant. The Berezin integral representation can thus be used to obtain convergent expansions of the generating function in terms of powers of its parameter. The derivation and analysis of the expansions of logarithms of Berezin integrals are the subject of the second part of the present work. Such technical results are also useful, for instance, in the context of quantum information theory, in the computation of relative entropy densities associated with fermionic Gibbs states, and in the theory of quantum normal fluctuations for weakly interacting fermion systems.

We study formal and non-formal deformation quantizations of a family of manifolds that can be obtained by phase space reduction from ${ℂ}^{1+n}$ with the Wick star product in arbitrary signature. Two special cases of such manifolds are the complex projective space $ℂ{\mathbb{P}}^n$ and the complex hyperbolic disc ${𝔻}^n$. We generalize several older results to this setting: The construction of formal star products and their explicit description by bidifferential operators, the existence of a convergent subalgebra of “polynomial” functions, and its completion to an algebra of certain analytic functions that allow an easy characterization via their holomorphic extensions. Moreover, we find an isomorphism between the non-formal deformation quantizations for different signatures, linking, e.g., the star products on $ℂ{\mathbb{P}}^n$ and ${𝔻}^n$. More precisely, we describe an isomorphism between the (polynomial or analytic) function algebras that is compatible with Poisson brackets and the convergent star products. This isomorphism is essentially given by Wick rotation, i.e. holomorphic extension of analytic functions and restriction to a new domain. It is not compatible with the ${}^{\ast }$-involution of pointwise complex conjugation.

In this paper, we reduce the problem of quantization of the Yang–Mills field Hamiltonian to a problem for defining a probability measure on an infinite-dimensional space of gauge equivalence classes of connections on ${ℝ}^3$. We suggest a formally self-adjoint expression for the quantized Yang–Mills Hamiltonian as an operator on the corresponding Lebesgue $L^2$-space. In the case when the Yang–Mills field is associated to the abelian group $U(1)$, we define the probability measure which depends on two real parameters $m>0$ and $c\ne 0$. This yields a non-standard quantization of the Hamiltonian of the electromagnetic field, and the associated probability measure is Gaussian. The corresponding quantized Hamiltonian is a self-adjoint operator in a Fock space the spectrum of which is $\{0\}\cup [\frac{1}{2}m,\infty )$, i.e. it has a gap.

In this paper, we construct a conformal scattering theory for Dirac fields in the interior of a Reissner–Nordström-like black hole, between the black hole event horizon and the Cauchy horizon. The main result is a resolution of the characteristic Cauchy problem for the Dirac equation on the horizons by solving a system of wave equations and re-interpreting its solution as the components of the required Dirac solution.