Theoretical and Mathematical Physics最新文献

We consider a conformal scalar field theory with the (lambda phi^4) self-coupling in Rindler and Minkowski coordinates at a finite-temperature with the Planckian distribution for exact modes. The solution of the one-loop Dyson–Schwinger equation is found through the order (lambda^{3/2}). The appearance of a thermal (Debye) mass is shown. Unlike the physical mass, the thermal mass gives a gap in the energy spectrum in the quantization in the Rindler coordinates. The difference between such calculations in Minkowski and Rindler coordinates for exact modes is discussed. It is also shown that states with a temperature lower than the Unruh temperature are unstable. It is proved that for the canonical Unruh temperature, the thermal mass is equal to zero. The contribution to the quantum average of the stress–energy tensor is also calculated, it remains traceless even in the presence of the thermal mass.

The incompatibility between quantum measurements (as mathematically represented by positive operator-valued measures, i.e., POVMs) is a key feature of quantum mechanics and is intrinsically related to the noncommutativity of operators. For both theoretical and practical considerations, it is desirable to quantify the degree of incompatibility between quantum measurements, and considerable effort has been devoted to this issue. In this paper, we provide a novel approach to measurement incompatibility by exploiting the Lüders channels derived from POVMs and employing the measurement disturbance to quantify incompatibility. This is achieved by constructing an approximately joint measurement for a pair of POVMs, which is an enlarged POVM with the correct marginal property for one of the two POVMs but not necessarily for the other. The degree of failure of the marginal property for the other POVM is a kind of measurement disturbance and can be naturally interpreted as a quantifier of the incompatibility between the two POVMs. We reveal basic properties of this quantifier of measurement incompatibility, identify its maximal value in some cases, compare it with several popular measures in the literature, and illustrate it with some typical examples. Some related open issues are also discussed.

We consider a stochastic Laplacian growth model within the framework of normal random matrices. In the limit of large matrix size, the support of eigenvalues forms a planar domain with a sharp boundary that evolves stochastically as the matrix size increases. We show that the most probable growth scenario is similar to deterministic Laplacian growth, while alternative scenarios illustrate the impact of fluctuations. We prove that the probability distribution function of fluctuations is given by the circular unitary ensemble introduced by Dyson in 1962. The partition function of fluctuations is shown to be universal, depending solely on the fluctuation intensity and the problem’s geometry, regardless of the initial domain shape.

Rational solutions of several nonautonomous quadrilateral equations in the ABS and ABS* list are obtained in a neat form of Casoratians, which mostly relies on a single (tau) function. The corresponding nonautonomous bilinear equations are listed in difference and differential–difference forms by introducing an auxiliary variable. Instead of bilinearizing quadrilateral equations, we present their related Bäcklund transformation systems, which directly reduce to bilinear equations by specific transformations. As an application, a result related to the discrete Painlevé equation is given.

We study periodic solutions of the generalized Camassa–Holm equation (CH-(gamma) equation). We show that the generalized CH-(gamma) equation is also an important theoretical model because it is a completely integrable system. We obtain representation for periodic solutions of the generalized CH-(gamma) equation in the framework of the inverse spectral problem for a weighted Sturm–Liouville operator.

We discuss the equivalence between the Date–Jimbo–Kashiwara–Miwa (DJKM) construction and the Kac–Wakimoto (KW) construction of (widehat{sl}_2)-integrable hierarchies within the framework of bilinear equations. The DJKM method has achieved remarkable success in constructing integrable hierarchies associated with classical A, B, C, D affine Lie algebras. In contrast, the KW method exhibits broader applicability, as it can be employed even for exceptional E, F, G affine Lie algebras. However, a significant drawback of the KW construction lies in the great difficulty of obtaining Lax equations for the corresponding integrable hierarchies. Conversely, in the DJKM construction, Lax structures for numerous integrable hierarchies can be derived. The derivation of Lax equations from bilinear equations in the KW construction remains an open problem. Consequently, demonstrating the equivalent DJKM construction for the integrable hierarchies obtained via the KW construction would be highly beneficial for obtaining the corresponding Lax structures. In this paper, we use the language of lattice vertex algebras to establish the equivalence between the DJKM and KW methods in the (widehat{sl}_2)-integrable hierarchy for principal and homogeneous representations.

A family of representations of the Lie algebra of the diffeomorphism group in (mathbb{R}^d) is studied. A method for constructing representations of this family is proposed. Equations for matrices describing the action of the Lie algebra on the representation space are obtained. It is shown that the developed formalism is suitable for describing representations under which fields of linear homogeneous geometric objects are transformed. The formalism is shown to allow describing representations for which the representation space vectors cannot be expressed in terms of fields of linear homogeneous geometric objects. An example of such a representation is studied.

For a semilinear parabolic partial differential equation, we consider an asymptotic solution that converges to a traveling wave at large times (t). The velocity of such a wave is time dependent, and we construct the asymptotics as (ttoinfty). We find that the asymptotics contains logarithms and cannot be constructed in the form of a power series.

We construct a Bargmann-type isomorphism defined by the one-particle part (H) of the Fock space (Gamma(H)) for an infinite-dimensional space (H) with involution. The formulas obtained also make sense in the case (dim H<infty) and are closely related to the Segal–Bargmann space. Central to the construction is the notion of a shift-invariant distribution in the case of an infinite-dimensional domain of test functions.

We present a rigorous construction of the Feynman integral on the compactified Einstein Universe using white noise calculus. Our construction of functional averaging can also be thought of as a solution to a problem posed by Bogoliubov and Shirkov.