Semiclassical approximation meets Keldysh–Schwinger diagrammatic technique: scalar $$\varphi ^4$$

We study the evolution of the non-equilibrium quantum fields from a highly excited initial state in two approaches: the standard Keldysh–Schwinger diagram technique and the semiclassical expansion. We demonstrate explicitly that these two approaches coincide if the coupling constant g and the Plank constant $$\hbar $$ ħ are simultaneously small. Also, we discuss loop diagrams of the perturbative approach, which are summed up by the leading order term of the semiclassical expansion. As an example, we consider shear viscosity for the scalar field theory at the leading semiclassical order. We introduce the new technique that unifies both semiclassical and diagrammatic approaches and open the possibility to perform the resummation of the semiclassical contributions.

A Korteweg–DeVries type model for helical soliton solutions for quantum and continuum phenomena

Quantum mechanical states are normally described by the Schrödinger equation, which generates real eigenvalues and quantizable solutions which form a basis for the estimation of quantum mechanical observables, such as momentum and kinetic energy. Studying transition in the realm of quantum physics and continuum physics is however more difficult and requires different models. We present here a new equation which bears similarities to the Korteweg–DeVries (KdV) equation and we generate a description of transitions in physics. We describe here the two- and three-dimensional form of the KdV like model dependent on the Plank constant [Formula: see text] and generate soliton solutions. The results suggest that transitions are represented by soliton solutions which arrange in a spiral-fashion. By helicity, we propose a conserved pattern of transition at all levels of physics, from quantum physics to macroscopic continuum physics.

Asymptotics of solution to the nonstationary Schrödinger equation

The Cauchy problem with a rapidly oscillating initial condition for the homogeneous Schr?dinger equation was studied in [5]. Continuing the research ideas of this work and [3], in this paper we construct the asymptotic solution to the following mixed problem for the nonstationary Schr?dinger equation: Lhu ? ih?tu + h2?2xu-b(x,t)u = f(x,t), (x,t) ? ??= (0,1) x (0,T], u|t=0 = g(x), u|x=0 = u|x=1 = 0, (1) where h > 0 is a Planck constant, u = u(x,t,h). b(x,t), f(x,t) ? C??(??), g(x) ? C? [0,1] are given functions. The similar problem was studied in [7, 8] when the Plank constant is absent in the first term of the equation and asymptotics of solution of any order with respect to a parameter was constructed. In this paper, we use a generalization of the method used in [7].

Polynomial Optimization and Quantifier Elimination in Quantum Mechanics

In this short notes, we present the formulation of a kind of quantum algorithm, applicable to the polynomial optimization. The present algorithm introduces the quantum effect by means of the coordinate variables, the momentum variables, and the commutators, and the Plank constant, which shall establish the correspondence between the quantum and the classical cases. The formalism is an extension to the polynomial optimization in the ``classical'' sense; the relations among quantum mechanical values are represented by a semi-algebraic set ( by the set of polynomials and sign conditions), and the optimum is searched in this semi-algebraic set. In the context of this formalism, the optimization of the Ising model [QUBO model] will be a special case. However, the computational procedure proposed here is different from the quantum annealing which is now in vogue. We demonstrate the exemplary calculation through quantifier elimination (which is a kind of symbolic computation applicable to optimization and other problems).