Reviews in Mathematical Physics

We examine the solutions of the semilinear wave equation, and, in particular, of the ${\phi }^q$ model of quantum field theory in the curved spacetime. More exactly, for $2<q<4$ we prove that the solution of the massless self-interacting scalar field equation in the Einstein–de Sitter universe has finite lifespan.

Wigner’s Theorem states that bijections of the set ${\mathcal{𝒫}}_1(H)$ of one-dimensional projections on a Hilbert space $H$ that preserve transition probabilities are induced by either a unitary or an anti-unitary operator on $H$ (which is uniquely determined up to a phase). Since elements of ${\mathcal{𝒫}}_1(H)$ define pure states on the C*-algebra $B(H)$ of all bounded operators on $H$ (though typically not producing all of them), this suggests possible generalizations to arbitrary C*-algebras. This paper is a detailed study of this problem, based on earlier results by R. V. Kadison (1965), F. W. Shultz (1982), K. Thomsen (1982), and others. Perhaps surprisingly, the sharpest known version of Wigner’s Theorem for C*-algebras (which is a variation on a result from Shultz, with considerably simplified proof) generalizes the equivalence between the hypotheses in the original theorem and those in an analogous result on (anti-)unitary implementability of Jordan automorphisms of $B(H)$, and does not yield (anti-)unitary implementability itself, far from it: abstract existence results that do give such implementability seem arbitrary and practically useless. As such, it would be fair to say that there is no Wigner Theorem for C*-algebras.

In the limit $\hslash \to 0$, we analyze a class of Schrödinger operators $H_{\hslash }={\hslash }^{2}L+\hslash W+V\cdot {\mathrm{\text{id}}}_{{ℰ}}$ acting on sections of a vector bundle ${ℰ}$ over a Riemannian manifold $M$ where $L$ is a Laplace type operator, $W$ is an endomorphism field and the potential energy $V$ has a non-degenerate minimum at some point $p\in M$. We construct quasimodes of WKB-type near $p$ for eigenfunctions associated with the low-lying eigenvalues of $H_{\hslash }$. These are obtained from eigenfunctions of the associated harmonic oscillator $H_{p,\hslash }$ at $p$, acting on smooth functions on the tangent space.

We extend Howland time-independent formalism to the case of completely positive and trace preserving dynamics of finite-dimensional open quantum systems governed by periodic, time-dependent Lindbladian in Weak Coupling Limit, expanding our result from previous papers. We propose the Bochner space of periodic, square integrable matrix-valued functions, as well as its tensor product representation, as the generalized space of states within the time-independent formalism. We examine some densely defined operators on this space, together with their Fourier-like expansions and address some problems related to their convergence by employing general results on Banach space-valued Fourier series, such as the generalized Carleson–Hunt theorem. We formulate Markovian dynamics in the generalized space of states by constructing appropriate time-independent Lindbladian in standard (Lindblad–Gorini–Kossakowski–Sudarshan) form, as well as one-parameter semigroup of bounded evolution maps. We show their similarity with Markovian generators and dynamical maps defined on matrix space, i.e. the generator still possesses a standard form (extended by closed perturbation) and the resulting semigroup is also completely positive, trace preserving and a contraction.

We consider ensembles of real symmetric band matrices with entries drawn from an infinite sequence of exchangeable random variables, as far as the symmetry of the matrices permits. In general, the entries of the upper triangular parts of these matrices are correlated and no smallness or sparseness of these correlations is assumed. It is shown that the eigenvalue distribution measures still converge to a semicircle but with random scaling. We also investigate the asymptotic behavior of the corresponding ${\ell }_2$-operator norms. The key to our analysis is a generalization of a classic result by de Finetti that allows to represent the underlying probability spaces as averages of Wigner band ensembles with entries that are not necessarily centered. Some of our results appear to be new even for such Wigner band matrices.