Reviews in Mathematical Physics

A gauge-invariant mass term for nonabelian gauge fields in two dimensions can be expressed as the Wess–Zumino–Witten (WZW) action. Hard thermal loops in the gauge theory in four dimensions at finite temperatures generate a screening mass for some components of the gauge field. This can be expressed in terms of the WZW action using the bundle of complex structures (for Euclidean signature) or the bundle of lightcones over Minkowski space. We show that a dynamically generated mass term in three dimensions can be put within the same general framework using the bundle of Sasakian structures.

We prove localization and probabilistic bounds on the minimum level spacing for the Anderson tight-binding model on the lattice in any dimension, with single-site potential having a discrete distribution taking $N$ values, with $N$ large. These results hold for all energies under an assumption of weak hopping.

Bundles of C*-algebras can be used to represent limits of physical theories whose algebraic structure depends on the value of a parameter. The primary example is the $\hslash \to 0$ limit of the C*-algebras of physical quantities in quantum theories, represented in the framework of strict deformation quantization. In this paper, we understand such limiting procedures in terms of the extension of a bundle of C*-algebras to some limiting value of a parameter. We prove existence and uniqueness results for such extensions. Moreover, we show that such extensions are functorial for the C*-product, dynamical automorphisms, and the Lie bracket (in the $\hslash \to 0$ case) on the fiber C*-algebras.

In this work, we obtain the integrated density of states for the Schrödinger operators with decaying random potentials acting on ${\ell }^2({\mathbb{Z}}^d)$. We also study the asymptotic of the largest and smallest eigenvalues of its finite volume approximation.

We study the spectral and scattering theory of light transmission in a system consisting of two asymptotically periodic waveguides, also known as one-dimensional photonic crystals, coupled by a junction. Using analyticity techniques and commutator methods in a two-Hilbert spaces setting, we determine the nature of the spectrum and prove the existence and completeness of the wave operators of the system.

We consider a free massive quantized Klein–Gordon field in a spacetime $(M,g)$ containing a stationary bifurcate Killing horizon, i.e. a bifurcate Killing horizon whose Killing vector field is globally time-like in the right wedge ${{ℳ}}^{+}$ associated to the horizon.

We prove the existence of the Hartle–Hawking–Israel (HHI) vacuum state, which is a pure state on the whole spacetime whose restriction to ${{ℳ}}^{+}$ is a thermal state ${\omega }_{T_{\mathrm{\text{H}}}}$ for the time-like Killing field at Hawking temperature $T_{\mathrm{\text{H}}}=\kappa {(2\pi )}^{-1}$, where $\kappa$ is the surface gravity of the horizon.

We show that the HHI state is a Hadamard state and is the unique Hadamard state which is equal to the double $T_{\mathrm{\text{H}}}^{-1}$-KMS state in the double wedge ${{ℳ}}^{-}\cup {{ℳ}}^{+}$. We construct the HHI state by Wick rotation in Killing time coordinates, using the notion of the Calderón projector for elliptic boundary value problems.