Том 306, № 1 (2019)

We use the Bogoliubov quasi-average approach to studying phase transitions in random matrix models related to a zero-dimensional version of the fermionic SYK model with replicas. We show that in the model with quartic interaction deformed by a quadratic term, there exist either two or four different phases with nonvanishing replica off-diagonal correlation functions.

In this paper we summarize the results obtained in some of our recent studies in the form of a series of theorems. We present new real bases of functions in L²(B_r) that are eigenfunctions of the p-adic pseudodifferential Vladimirov operator defined on a compact set B_r ⊂ ℚ_p of the field of p-adic numbers ℚ_p and on the whole ℚ_p. We demonstrate a relationship between the constructed basis of functions in L²(ℚ_p) and the basis of p-adic wavelets in L²(ℚ_p). A real orthonormal basis in the space L²(ℚ_p, u(x) d_px) of square integrable functions on ℚ_p with respect to the measure u(x) d_px is described. The functions of this basis are eigenfunctions of a pseudodifferential operator of general form with kernel depending on the p-adic norm and with measure u(x) d_px. As an application of this basis, we present a method for describing stationary Markov processes on the class of ultrametric spaces \(\mathbb{U}\) that are isomorphic and isometric to a measurable subset of the field of p-adic numbers ℚ_p of nonzero measure. This method allows one to reduce the study of such processes to the study of similar processes on ℚ_p and thus to apply conventional methods of p-adic mathematical physics in order to calculate their characteristics. As another application, we present a method for finding a general solution to the equation of p-adic random walk with the Vladimirov operator with general modified measure u(∣x∣_p) d_px and reaction source in ℤ_p.

We consider a generalization of Maxwell’s equations on a pseudo-Riemannian manifold M of arbitrary dimension in the presence of electric and magnetic charges and prove that if the cohomology groups H²(M) and H³(M) are trivial, then solving these equations reduces to solving the d’Alembert—Hodge equation.

Significant phenomenological success and nice theoretical properties of general relativity (GR) are well known. However, GR is not a complete theory of gravity. Hence, there are many attempts to modify GR. One of the current approaches to a more complete theory of gravity is a nonlocal modification of GR. The nonlocal gravity approach, which we consider here without matter, is based on the action \(S=(16 \pi G)^{-1} \int \sqrt{-g}(R-2 \Lambda+P(R) \mathcal{F}(\square) Q(R)) d^{4} x\), where R is the scalar curvature, Γ is the cosmological constant, P(R) and Q(R) are some differentiable functions of R, and \(\mathcal{F}(\square)=\sum\nolimits_{n=1}^{+\infty} f_{n} \square^{n}\) is an analytic function of the corresponding d’Alembert operator □. We present here a brief review of some general properties and cosmological solutions for some specific functions P(R) and Q(R).

In a previous paper, we developed an analysis in associative commutative algebras and in modules over them, which may be useful in problems of contemporary mathematical and theoretical physics. Here we work out similar methods in the noncommutative case.

The paper is devoted to characterization theorems for idempotent distributions on the p-adic number field ℚ_p in terms of functional relations for the characteristic functions of these distributions.

The well-known evolution equations of a solenoidal vector field with integral curves frozen into a continuous medium are presented in an invariant form in the four-dimensional spacetime. A fundamental 1-form (4-potential) is introduced, and the problem of variation of the action (integral of the 4-potential along smooth curves) is considered. The extremals of the action in the class of curves with fixed endpoints are described, and the conservation laws generated by symmetry groups are found. Under the assumption that the electric and magnetic fields are orthogonal, Maxwell’s equations are represented as evolution equations of a solenoidal vector field. The role of the velocity field is played by the normalized Poynting vector field.

A generalization of the Yang-Mills equations is proposed. It is shown that any solution of the Yang-Mills equations (in the Lorentz gauge) is also a solution of the new generalized equation. It is also shown that the generalized equation has solutions that do not satisfy the Yang-Mills equations.

We study sequences of compositions of independent identically distributed random one-parameter semigroups of linear transformations of a Hilbert space and the asymptotic properties of the distributions of such compositions when the number of terms in the composition tends to infinity. To study the expectation of such compositions, we apply the Feynman-Chernoff iterations obtained via Chernoff’s theorem. By the Feynman-Chernoff iterations we mean prelimit expressions from the Feynman formulas; the latter are representations of one-parameter semigroups or related objects in terms of the limit of integrals over Cartesian powers of an appropriate space, or some generalizations of such representations. In particular, we study the deviation of the values of compositions of independent random semigroups from their expectation and examine the validity for such compositions of analogs of the limit theorems of probability theory such as the law of large numbers. We obtain sufficient conditions under which any neighborhood of the expectation of a composition of n random semigroups contains the (random) value of this composition with probability tending to one as n → ∞ (it is this property that is viewed as the law of large numbers for compositions). We also present examples of sequences of independent random semigroups for which the law of large numbers for compositions fails.

The properties of the generalized Gelfand-Shilov spaces \(S_{{b_n}}^{{a_k}}\) are studied from the viewpoint of deformation quantization. We specify the conditions on the defining sequences (a_k) and (b_n) under which \(S_{{b_n}}^{{a_k}}\) is an algebra with respect to the twisted convolution and, as a consequence, its Fourier transformed space \(S_{{a_k}}^{{b_n}}\) is an algebra with respect to the Moyal star product. We also consider a general family of translation-invariant star products. We define and characterize the corresponding algebras of multipliers and prove the basic inclusion relations between these algebras and the duals of the spaces of ordinary pointwise and convolution multipliers. Analogous relations are proved for the projective counterpart of the Gelfand-Shilov spaces. A key role in our analysis is played by a theorem characterizing those spaces of type S for which the function exp(iQ(x)) is a pointwise multiplier for any real quadratic form Q.

One of the main methods for describing the dynamics of open quantum systems is the method of quantum master equations. These equations describe the dynamics of the reduced density operator of a system interacting with a reservoir. In this case, averaging is performed over the degrees of freedom of the reservoir, which does not allow one to describe the dynamics of reservoir observables. In this paper we show that applying the Zwanzig projection operator method, which is used in deriving quantum master equations, one can also derive dynamic equations for reservoir observables. As an example, we derive dynamic equations for the average number of quanta (photons, phonons) of a bosonic reservoir in the approximation of its weak coupling to the system in the case of the dipole interaction Hamiltonian.

We recall the basics of multiplier ideal sheaves and formulate our recent solution of Demailly’s strong openness conjecture on multiplier ideal sheaves and related results. Then we present some applications in complex geometry, including some new results related to the vanishing and finiteness theorems for analytic cohomology groups with multiplier ideal sheaves in the case of pseudo-effective line bundles over holomorphically convex manifolds, and to the generalized Siu lemma and pseudo-effectiveness of the twisted relative pluricanonical bundles and their direct images.