Reviews in Mathematical Physics

We consider simple superalgebras which are a supersymmetric extension of the spin algebra in the cases where the number of odd generators does not exceed 64. All of them contain a super Poincaré algebra as a contraction and another as a subalgebra. Because of the contraction property, some of these algebras can be interpreted as de Sitter or anti de Sitter superalgebras. However, the number of odd generators present in the contraction is not always minimal due to the different splitting properties of the spinor representations under a subalgebra. We consider the general case, with arbitrary dimension and signature, and examine in detail particular examples with physical implications in dimensions d = 10 and d = 4.

We study the Floquet Hamiltonian -i∂_t + H + V(ωt), acting in L²([0,T],ℋ, dt), as depending on the parameter ω = 2π/T. We assume that the spectrum of H in ℋ is discrete, , but possibly degenerate, and that t ↦ V(t) ∈ ℬ(ℋ) is a 2π-periodic function with values in the space of Hermitian operators on ℋ. Let J > 0 and set . Suppose that for some σ > 0 it holds true that ∑_{hm > hn}M_mM_n (h_m - h_n)^-σ < ∞ where M_m is the multiplicity of h_m. We show that in that case there exist a suitable norm to measure the regularity of V, denoted ∊_V, and positive constants, ∊_⋆ and δ_⋆, with the property: if ∊_V < ∊_⋆ then there exists a measurable subset Ω_∞ ⊂ Ω₀ such that its Lebesgue measure fulfills |Ω_∞| ≥ |Ω₀| - δ_⋆ ∊_V and the Floquet Hamiltonian has a pure point spectrum for all ω∈Ω_∞.

We prove an adiabatic theorem for the evolution of spectral data under a weak additive perturbation in the context of a system without an intrinsic time scale. For continuous functions of the unperturbed Hamiltonian the convergence is in norm while for a larger class functions, including the spectral projections associated to embedded eigenvalues, the convergence is in the strong operator topology.

We consider the 2D Navier–Stokes system, perturbed by a random force, such that sufficiently many of its Fourier modes are excited (e.g. all of them are). We discuss the results on the existence and uniqueness of a stationary measure for this system, obtained in last years, homogeneity of the measures and some their limiting properties. Next we use these results to prove that solutions of the equations obey the central limit theorem and the strong law of large numbers.

We study a quantum moment map and propose an invariant for G-invariant star products on a G-transitive symplectic manifold. We start by describing a new method to construct a quantum moment map for G-invariant star products of Fedosov type. We use it to obtain an invariant that is invariant under G-equivalence. In the last section we give two simple examples of such invariants, which involve non-classical terms and provide new insights into the classification of G-invariant star products.