Dirac procedure and the Hamiltonian formalism for cosmological perturbations in a Bianchi I universe

We apply the Dirac procedure for constrained systems to the Arnowitt-Deser-Misner formalism linearized around the Bianchi I universe. We discuss and employ basic concepts such as Dirac observables, Dirac brackets, gauge-fixing conditions, reduced phase space, physical Hamiltonian, canonical isomorphism between different gauge-fixing surfaces and spacetime reconstruction. We relate this approach to the gauge-fixing procedure for non-perturbative canonical relativity. We discuss the issue of propagating a basis for the scalar-vector-tensor decomposition as, in an anisotropic universe, the wavefronts of plane waves undergo a non-trivial evolution. We show that the definition of a gravitational wave as a traceless-transverse mode of the metric perturbation needs to be revised. Moreover there exist coordinate systems in which a polarization mode of the gravitational wave is given entirely in terms of a scalar metric perturbation. We first develop the formalism for the universe with a single minimally coupled scalar field and then extend it to the multi-field case. The obtained fully canonical formalism will serve as a starting point for a complete quantization of the cosmological perturbations and the cosmological background.

Relativistic Wave Equations

We derive the most general relativistically covariant linear differential equations, having at most two derivatives, for scalar, spinor and vector fields. We introduce the corresponding Lagrangian and Hamiltonian formalisms and present the expansion of the solutions in terms of plane waves. In each case, we study the propagation properties of the corresponding Green functions. We start with the simplest example of the Klein–Gordon equation for a real field and generalise it to that of N real, or complex fields. As a next step we derive the Weyl, Majorana and Dirac equations for spinor fields. They are first order differential equations and we show how to adapt to them the canonical formalism. We end with the Proca and Maxwell equations for massive and massless spin-one fields and, in each case, we determine the physical degrees of freedom.

Koopman wavefunctions and Clebsch variables in Vlasov–Maxwell kinetic theory

Motivated by recent discussions on the possible role of quantum computation in plasma simulations, here, we present different approaches to Koopman's Hilbert-space formulation of classical mechanics in the context of Vlasov–Maxwell kinetic theory. The celebrated Koopman–von Neumann construction is provided with two different Hamiltonian structures: one is canonical and recovers the usual Clebsch representation of the Vlasov density, the other is non-canonical and appears to overcome certain issues emerging in the canonical formalism. Furthermore, the canonical structure is restored for a variant of the Koopman–von Neumann construction that carries a different phase dynamics. Going back to van Hove's prequantum theory, the corresponding Koopman–van Hove equation provides an alternative Clebsch representation which is then coupled to the electromagnetic fields. Finally, the role of gauge transformations in the new context is discussed in detail.

FORMATION OF UNIVERSAL COMPETENCIES IN SOLVING HAMILTON EQUATIONS IN THE COURSE OF THEORETICAL PHYSICS (MODULE “CLASSICAL MECHANICS”)

Исследуется подход к формированию универсальных компетенций в курсе «Теоретическая физика. Модуль: Классическая механика» для студентов бакалавриата на примере раздела, связанного с нахождением закона движения тела. Акцент ставится на подходе к решению уравнений Гамильтона как системы дифференциальных уравнений первого порядка. Модуль «Классическая механика» является начальным этапом изучения теоретической физики. В этом разделе рассматриваются различные подходы к исследованию динамики механических систем, такие как Ньютоновская механика, Лагранжева механика и канонический формализм Гамильтона. Эти подходы являются эквивалентными, но формализм Гамильтона имеет ряд преимуществ. Тематика работы актуальна для студентов педагогических вузов, чьи профессиональные задачи предполагают умение осуществлять поиск, критический анализ и синтез информации, применять системный подход для решения поставленных задач (УК-1), обобщать теоретический материал и применять его к конкретным задачам с конкретными методическими целями. Цель разработки – помочь студентам увидеть общее и различное в решении задач, рассматривая предложенные задачи с единой позиции. Методическая задача состоит в формировании компетенций группы УК-1 при решении предлагаемых заданий. At present, the competence-based approach is dominant in education, since it presupposes, first of all, not the self-valuable assimilation of knowledge by students, but the opportunity to use this knowledge in the learning process to solve urgent problems. The most important feature of modern education is its universality. There are universal competences in all modules of the educational program and in various activities. This work is devoted to the formation of universal competencies in the course “Theoretical Physics. Module: Classical Mechanics” for undergraduate students on the example of a section related to finding the law of body motion. The emphasis is on the approach to solving Hamilton’s equations as a system of first-order differential equations. The module “Classical Mechanics” is the initial stage of the study of theoretical physics. This section discusses various approaches to the study of the dynamics of mechanical systems, such as Newtonian mechanics, Lagrangian mechanics, and Hamilton’s canonical formalism. These approaches are equivalent, but Hamilton’s formalism has several advantages. The topic of the work is relevant for students of pedagogical universities, whose professional tasks involve the ability to search, critical analysis and synthesis of information, apply a systematic approach to solving the assigned tasks (Universal Competencies-1), generalize theoretical material and apply it to specific tasks with specific methodological goals. The purpose of the development is to help students see the similar and different points in problem solving, considering the proposed problems from a unified position. The methodological task is to form the competencies of the UC-1 group when solving the proposed tasks.

Nonlinear dynamics of a charged particle in a strong non-null knot wave background

In this paper, we study the dynamics of a charged particle interacting with the non-null electromagnetic knot wave background. We analyze the classical system in the Hamilton–Jacobi formalism and find the action, the linear momentum and the trajectory of the particle. Also, we calculate the effective mass and the emitted radiation along the knot wave. Next, we quantize the system in the classical strong knot wave background by using the strong-field QED canonical formalism. We explicitly construct the Furry picture and calculate the Volkov solutions of the Dirac equation. As an application, we discuss the one-photon Compton effect where we determine the general form of the S-matrix. Also, we discuss in detail the first partial amplitudes in the transition matrix in two simple backgrounds and show that there is a pair of states for which these amplitudes are identical.

Phases of a matrix model with non-pairwise index contractions

Recently a matrix model with non-pairwise index contractions has been studied in the context of the canonical tensor model, a tensor model for quantum gravity in the canonical formalism. This matrix model also appears in the same form with different ranges of parameters and variables, when the replica trick is applied to the spherical $p$-spin model ($p=3$) in spin glass theory. Previous studies of this matrix model suggested the presence of a continuous phase transition around $R\sim N^2/2$, where $N$ and $R$ designate its matrix size $N\times R$. This relation between $N$ and $R$ intriguingly agrees with a consistency condition of the tensor model in the leading order of $N$, suggesting that the tensor model is located near or on the continuous phase transition point and therefore its continuum limit is automatically taken in the $N\rightarrow \infty$ limit. In the previous work, however, the evidence for the phase transition was not satisfactory due to the slowdown of the Monte Carlo simulations. In this work, we provide a new setup for Monte Carlo simulations by integrating out the radial direction of the matrix. This new strategy considerably improves the efficiency, and allows us to clearly show the existence of the phase transition. We also present various characteristics of the phases, such as dynamically generated dimensions of configurations, cascade symmetry breaking and a parameter zero limit, and discuss their implications for the canonical tensor model.

From complex holomorphic systems to real systems

Symmetries in modern physics are a fundamental subject of high relevance that allows appreciating more deeply the physical structure of a theory. In this paper, we analyze a gauge symmetry that appears in complex holomorphic systems. We show that a complex system can be reduced to different real systems, using different gauge conditions, and the gauge transformations connect several real systems in the complex space. We prove that the space of solutions of one system is related using a gauge transformation to another one. Gauge transformations are, in some cases, canonical transformations. However, in other cases, these are more general transformations that change the symplectic structure, but there is still a map between systems. We establish a construction to extend the analysis to the quantum case using path integrals through the Batalin–Fradkin–Vilkovisky theorem and within the canonical formalism, where we show explicitly that solutions of the Schrödinger equation are gauge-related.

Canonical Description for Formulation of Embedding Gravity as a Field Theory in a Flat Spacetime

We consider the approach to gravity in which four-dimensional curved spacetime is represented by a surface in a flat Minkowski space of higher dimension. After a short overview of the ideas and results of such an approach we concentrate on the study of the so-called splitting gravity, a form of this description in which constant value surface of a set of scalar fields in the ambient flat space-time defines the embedded surface. We construct a form of action which is invariant w.r.t. all symmetries of this theory. We construct the canonical formalism for splitting gravity. The resulting theory turns out to be free of constraints. However, the Hamiltonian of this theory is an implicit function of canonical variables. Finally, we discuss the path integral quantization of such a theory.