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Moscow University Physics Bulletin

Discription of the Dinamics of Fermions by the Influence Functional Method

А.А. Бирюков, Марк Александрович Шлеенков

Published Dec 1, 2025• 0 citations
Publisher

Abstract

Abstract We consider a system of interacting spinor and electromagnetic fields. We describe the quantum state of the fields by a statistical density matrix. We present the equation of evolution of the density matrix of the field system in the method of integration over trajectories. The evolution equation for the density matrix of the fermion field is obtained as an averaging of the functional of the influence of the electromagnetic field. An explicit form of the functional of the influence of the electromagnetic field is found. We present the equations of motion for the fermion field taking into account the influence of the vacuum of the electromagnetic field.

DOI
10.3103/s0027134925702601

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In the path integral approach we describe evolution of interacting electromagnetic and fermionic fields by the use of density matrix formalism. The equation for density matrix and transitions probability for fermionic field is obtained as average of electromagnetic field influence functional. We obtain a formula for electromagnetic field influence functional calculating for its various initial and final state. We derive electromagnetic field influence functional when its initial and final states are vacuum. We present Lagrangian for relativistic fermionic field under influence of electromagnetic field vacuum.
The description of interacting multilevel quantum system and quantum electromagnetic field in terms of statistical density matrix is considered in path integral approach. The statistical operator and probability transitions of quantum system are represented as mean of electromagnetic field influence functional. The influence functionals are defined for specific models of electromagnetic field. The explicit form for singleand multimode electromagnetic field is concretized in cases when initial state of the field is represented by vacuum and pure coherent.
The description of interacting multilevel quantum system and quantum electromagnetic field in terms of statistical density matrix is considered in path integral approach. The statistical operator and probability transitions of quantum system are represented as mean of electromagnetic field influence functional. The influence functionals are defined for specific models of electromagnetic field. The explicit form for single- and multimode electromagnetic field is concretized in cases when initial state of the field is represented by vacuum and pure coherent.
We study the dynamics of multi-level quantum systems under the influence of electromagnetic field.Basing on the path integral approach to the quantum density matrix and transitions probability we represent functional averaging electromagnetic field influence.We demonstrate that the transition probability can be written as a path integral of real functional (the product of exponent and cosine).The numerical integration algorithm of the transitions probability in energy representation was developed.The numerical simulations of the quantum system dynamics under electromagnetic field action describe the well-known Rabi oscillations, their fluctuations and multi-photon quantum transitions.
We describe multilevel quantum systems dynamics in electromagnetic fields using the influence functional approach in path integration formalism. We obtain equations for the statistical density matrix and the probability of transitions of quantum systems induced by the electromagnetic field. The explicit form of the quantized electromagnetic field influence functional is concretized. Spontaneous transition of the quantum systems, the Rabi oscillations and their fluctuations are obtained. The two#photon photoelectric effect is explained.
The paper studies entangled states of two qubits interacting with each other and with an electromagnetic field. The state of the qubits is determined by a statistical density matrix. The degree of entanglement of the state is characterized by the Peres-Gorodeckii (PG) parameter. The statistical density matrix and its evolution are determined in the energy representation within the framework of the path integral formalism. The obtained equations determine the dependence of the PG parameter on the parameters of qubit dipole-dipole interaction and the acting electromagnetic field. The results of numerical calculations are presented in graphs for the PG parameter. It is shown that it is possible to choose parameters corresponding to qubit states with a high degree of entanglement (0.99).