We develop a complete Dirac's canonical analysis for an alternative action that yields Maxwell's four-dimensional equations of motion. We study in detail the full symmetries of the action by following all steps of Dirac's method in order to obtain a detailed description of symmetries. Our results indicate that such an action does not have the same symmetries than Maxwell theory, namely, the model is not a gauge theory and the number of physical degrees of freedom are different.
We present a new theorem concerning a sufficient condition for a symmetric operator acting in a complex Hilbert space to be essentially self-adjoint. By applying the theorem, we prove that the Dirac Maxwell Hamiltonian, which describes a quantum system of a Dirac particle and a radiation field minimally interacting with each other, is essentially self-adjoint. Our theorem covers the case where the Dirac particle is in the Coulomb type potential.
The Hamiltonian structure of the gauge-free gyrokinetic Vlasov-Maxwell equations is presented in terms of a Hamiltonian functional and a gyrokinetic Vlasov-Maxwell bracket. The bracket is used to show that the gyrokinetic angular-momentum conservation law can also be expressed in Hamiltonian form. The Jacobi property of the gyrokinetic Vlasov-Maxwell bracket is also demonstrated explicitly.
We study the Dirac-Maxwell model quantized in the Lorenz gauge. In this gauge, the space of quantum mechanical state vectors inevitably be an indefinite metric vector space so that the canonical commutation relation (CCR) is realized in a Lorentz covariant manner. In order to obtain a physical subspace in which no negative norm state exists, the method first proposed by Gupta and Bleuler is applied with mathematical rigor. It is proved that a suitably defined physical subspace has a positive semi-definit metric, and naturally induces a physical Hilbert space with a positive definite metric. Th…
The most common physical formalisms are the Lagrangian formalism and the Hamiltonian formalism. From the superficial point of view, they are one and the same, but rewritten in other terms. However, it seems that the Hamiltonian formalism has a richer structure and is more convenient for studying the electromagnetic field, especially in the formalization of its geometrization. Unfortunately for field problems, there is a whole set of Hamiltonian formalisms. The authors study the applicability of different variants of the Hamiltonian formalism to the problems of electrodynamics. In this paper we…
A Hamiltonian field theory for the macroscopic Maxwell equations with fully general polarization and magnetization is stated in the language of differential forms. The precise procedure for translating the vector calculus formulation into differential forms is discussed in detail. We choose to distinguish between straight and twisted differential forms so that all integrals be taken over densities (i.e. twisted top forms). This ensures that the duality pairings, which are stated as integrals over densities, are orientation independent. The relationship between functional differentiation with r…
It is shown that Maxwell's equations in media without source can be written as a contact Hamiltonian vector field restricted to a Legendre submanifold, where this submanifold is in a fiber space of a bundle and is generated by either electromagnetic energy functional or co-energy functional. Then, it turns out that Legendre duality for this system gives the induction oriented formulation of Maxwell's equations and field intensity oriented one. Also, information geometry of the Maxwell fields is introduced and discussed.
The Vlasov-Maxwell equations possess a Hamiltonian structure expressed in terms of a Hamiltonian functional and a functional bracket. In the present paper, the transformation ("lift") of the Vlasov-Maxwell bracket induced by the dynamical reduction of single-particle dynamics is investigated when the reduction is carried out by Lie-transform perturbation methods. The ultimate goal of this work is to derive explicit Hamiltonian formulations for the guiding-center and gyrokinetic Vlasov-Maxwell equations that have important applications in our understanding of turbulent magnetized plasmas. Here,…
We consider the spinless Pauli-Fierz Hamiltonian which describes a quantum system of non-relativistic identical particles coupled to the quantized electromagnetic field. We study the time evolution in a mean-field limit where the number $N$ of charged particles gets large while the coupling to the radiation field is rescaled by $1/\sqrt{N}$. At time zero we assume that almost all charged particles are in the same one-body state (a Bose-Einstein condensate) and we assume also the photons to be close to a coherent state. We show that at later times and in the limit $N \rightarrow \infty$ the cha…
We show that nonrelativsitic scaling of the collisionless Vlasov-Maxwell system implies the existence of a formal invariant slow manifold in the infinite-dimensional Vlasov-Maxwell phase space. Vlasov-Maxwell dynamics restricted to the slow manifold recovers the Vlasov-Poisson and Vlasov-Darwin models as low-order approximations, and provides higher-order corrections to the Vlasov-Darwin model more generally. The slow manifold may be interpreted to all orders in perturbation theory as a collection of formal Vlasov-Maxwell solutions that do not excite light waves, and are therefore "dark". We p…
The aim of this paper is to create a large geometrical background on the dual 1-jet space J^{1*}(T,M) for a multi-time Hamiltonian approach of the electromagnetic and gravitational physical fields. Our geometric-physical construction is achieved starting only from a given quadratic Hamiltonian function of polymomenta H, which naturally produces a canonical nonlinear connection N, a canonical Cartan N-linear connection CΓ(N) and their corresponding local distinguished (d-) torsions and curvatures. In such a context, we construct some geometrical electromagnetic-like and gravitational-like field…
We slightly extend prior results about the derivation of the Maxwell-Schrödinger equations from the bosonic Pauli-Fierz Hamiltonian. More concretely, we show that the findings from [SIAM J. Math. Anal. 52(5), 4900-4936 (2020)] about the coherence of the quantized electromagnetic field also hold for soft photons with small energies. This is achieved with the help of an estimate from [arXiv:2202.05015] which proves that the domain of the number of photon operator is invariant during the time evolution generated by the Pauli-Fierz Hamiltonian.
The Hamiltonian formalism is extremely elegant and convenient to mechanics problems. However, its application to the classical field theories is a difficult task. In fact, you can set one to one correspondence between the Lagrangian and Hamiltonian in the case of hyperregular Lagrangian. It is impossible to do the same in gauge-invariant field theories. In the case of irregular Lagrangian the Dirac Hamiltonian formalism with constraints is usually used, and this leads to a number of certain difficulties. The paper proposes a reformulation of the problem to the case of a field without sources. …
In this paper, we consider the quantum version of the hamiltonian model describing friction introduced in [BDB]. This model consists of a particle which interacts with a bosonic reservoir representing a homogeneous medium through which the particle moves. We show that if the particle is confined, then the Hamiltonian admits a ground state if and only if a suitable infrared condition is satisfied. The latter is violated in the case of linear friction, but satisfied when the friction force is proportional to a higher power of the particle speed.
We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with a modified Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincaré theory. Possible applications of these formulations…
A new Hamiltonian structure of the Maxwell-Bloch equations is described. In this setting the Maxwell-Bloch equations appear as a member of a family of generalized Maxwell-Bloch systems. The family is parameterized by compact semi-simple Lie groups, the original Maxwell-Bloch system being the member corresponding to SU(2). The Hamiltonian structure is then used in the construction of a new family of symmetries and the associated conserved quantities of the Maxwell-Bloch equations.
Riemannian geometry is a particular case of Hamiltonian mechanics: the orbits of the hamiltonian $H=\frac{1}{2}g^{ij}p_{i}p_{j}$ are the geodesics. Given a symplectic manifold (Γ,ω), a hamiltonian $H:Γ\to\mathbb{R}$ and a Lagrangian sub-manifold $M\subsetΓ$ we find a generalization of the notion of curvature. The particular case $H=\frac{1}{2}g^{ij}\left[p_{i}-A_{i}\right]\left[p_{j}-A_{j}\right]+ϕ$ of a particle moving in a gravitational, electromagnetic and scalar fields is studied in more detail. The integral of the generalized Ricci tensor w.r.t. the Boltzmann weight reduces to the action …
We prove the existence of a $L^2$-normalized solitary wave solution for the Maxwell-Dirac equations in (3+1)-Minkowski space. In addition, for the Coulomb-Dirac model, describing fermions with attractive Coulomb interactions in the mean-field limit, we prove the existence of the (positive) energy minimizer.
The analytical method of solving the boundary problems for a system of equations describing the behaviour of electrons and an electric field in the Maxwell plasma half-space is developed. Here the diffusion reflection of electrons from the surface as a boundary condition is supposed. The exact solution for boundary value problem of the Vlasov-Maxwell equations through the eigenfunctions of the discrete spectrum and the continuous spectrum is obtained. The method of decomposition by eigenfunctions of discrete and continuous spectrum is used. The exact expression for the surface impedance also i…
In the paper we construct an hierarchy of integrable Hamiltonian systems which describe the variation of n-wave envelopes in nonlinear dielectric medium. The exact solutions for some special Hamiltonians are given in terms of elliptic functions of the first kind.
Our goal is the discussion of the problem of mathematical interpretation of basic postulates (or `principles') of Quantum Mechanics: transitions to quantum stationary orbits, the wave-particle duality, and the probabilistic interpretation, in the context of semiclassical self-consistent Maxwell--Schrödinger equations. We discuss possible relations of these postulates to the theory of attractors of Hamiltonian nonlinear PDEs and to a new general mathematical conjecture on global attractors of G-invariant nonlinear Hamiltonian partial differential equations with a Lie symmetry group G. This conj…
Techniques for coordinate changes that depend on both dependent and independent variables are developed and applied to the Maxwell-Vlasov Hamiltonian theory. Particle coordinate changes with a new velocity variable dependent on the magnetic field, with spatial coordinates unchanged, are lifted to transform the noncanonical Poisson bracket and, thus, the field Hamiltonian structure of the Vlasov-Maxwell equation. Several examples are given including magnetic coordinates, where the velocity is decomposed into components parallel and perpendicular to the local magnetic field, and the case of sphe…
The Lagrangian and Hamiltonian formulations of electromagnetism are reviewed and the Maxwell equations are obtained from the Hamiltonian for a system of many electric charges. It is shown that three of the equations which were obtained from the Hamiltonian, namely the Lorentz force law and two Maxwell equations, can be obtained as well from a set of postulated Poisson brackets. It is shown how the results derived from these brackets can be used to reconstruct the original Lagrangian for the theory aided by some reasoning based on physical concepts.
This paper discusses several methods for describing the dynamics of open quantum systems, where the environment of the open system is infinite-dimensional. These are purifications, phase space forms, master equation and liouville equation forms. The main contribution is in using Feynman-Kac formalisms to describe the infinite-demsional components.
We consider a class of nonlinear Boltzmann equations describing return to thermal equilibrium in a gas of colliding particles suspended in a thermal medium. We study solutions in the space $L^{1}(\mathbb{R}^{3}\times \mathbb{T}^3).$ Special solutions of these equations, called "Maxwellians," are spatially homogenous static Maxwell velocity distributions at the temperature of the medium. We prove that, for dilute gases, the solutions corresponding to smooth initial conditions in a weighted $L^{1}$-space converge to a Maxwellian in $L^{1},$ exponentially fast in time.
The canonical reduction algorithm is applied to Maxwell and Yang-Mills equations considered as Hamiltonian systems on some fiber bundles with symplectic and connection structures. The minimum interaction principle proved to have geometric origin within the reduction method devised.
The covariant Hamilton-Jacobi formulation of Maxwell's equations is derived from the first-order (Palatini-like) Lagrangian using the analysis of constraints within the De~Donder-Weyl covariant Hamiltonian formalism and the corresponding polysymplectic reduction.
Considering one of the fundamental notions of Bers' theory of pseudoanalytic functions the generating pair via an intertwining relation we introduce its generalization for biquaternionic equations corresponding to different first-order systems of mathematical physics with variable coefficients. We show that the knowledge of a generating set of solutions of a system allows one to obtain its different form analogous to the complex equation describing pseudoanalytic functions of the second kind and opens the way for new results and applications of pseudoanalytic function theory. As one of the exa…
In a system of coupled harmonic oscillators, the interaction can be represented by a real, symmetric and positive definite interaction matrix. The quantization of a Hamiltonian describing such a system has been done in the canonical case. In this paper, we take a more general approach and look at the system as a Wigner quantum system. Hereby, one does not assume the canonical commutation relations, but instead one just requires the compatibility between the Hamilton and Heisenberg equations. Solutions of this problem are related to the Lie superalgebras gl(1|n) and osp(1|2n). We determine the …
Unique transformation properties under the hyperspherical inversion of a partial differential equation describing a stationary scalar wave in an $N$-dimensional ($N\geqslant2$) Maxwell fish-eye medium are exploited to construct a closed form of the Green's function for that equation. For those wave numbers for which the Green's function fails to exist, the generalized Green's function is derived. Prospective physical applications are mentioned.
By using the Hamilton-Jacobi [$HJ$] framework the higher-order Maxwell-Chern-Simons theory is analyzed. The complete set of $HJ$ Hamiltonians and a generalized $HJ$ differential are reported, from which all symmetries of the theory are identified. In addition, we complete our study by performing the higher order Gitman-Lyakhovich-Tyutin [$GLT$] framework and compare the results of both formalisms.
We present a relativistic model describing a thin disk surrounded by a halo in presence of an electromagnetic field. The model is obtained by solving the Einstein-Maxwell equations on a particular conformastatic spacetime background and by using the distributional approach for the energy-momentum tensor. A class of solutions is obtained in which the gravitational and electromagnetic potentials are completely determined by a harmonic function only. A particular solution is given that is asymptotically flat and singularity-free, and satisfies all the energy conditions.
We prove the existence of a ground state of the Maxwell--Schrödinger equations in one spatial dimension, describing a specified amount of free charge under the influence of a fixed charge. For one case (equal free and fixed charge, i.e., a neutral atom), we introduce a new type of quartic Banach space, in which the Hamiltonian is naturally coercive. We also show that for a point charge, for any ratio of charge such that the ground state exists, it is symmetric and decreasing.
It is shown that Milne models (a subclass of FLRW spacetimes with negative spatial curvature) are nonlinearly stable in the set of solutions to the Einstein-Vlasov-Maxwell system, describing universes with ensembles of collisionless self-gravitating, charged particles. The system contains various slowly decaying borderline terms in the mutually coupled equations describing the propagation of particles and Maxwell fields. The effects of those terms are controlled using a suitable hierarchy based on the energy density of the matter fields.
We make use of the conformal compactification of Minkowski spacetime $M^{\#}$ to explore a way of describing general, nonlinear Maxwell fields with conformal symmetry. We distinguish the inverse Minkowski spacetime $[M^{\#}]^{-1}$ obtained via conformal inversion, so as to discuss a doubled compactified spacetime on which Maxwell fields may be defined. Identifying $M^{\#}$ with the projective light cone in $(4+2)$-dimensional spacetime, we write two independent conformal-invariant functionals of the $6$-dimensional Maxwellian field strength tensors -- one bilinear, the other trilinear in the f…
The action for a class of three-dimensional dilaton-gravity theories, with an electromagnetic Maxwell field and a cosmological constant, can be recast in a Brans-Dicke-Maxwell type action, with its free $ω$ parameter. For a negative cosmological constant, these theories have static, electrically charged, and spherically symmetric black hole solutions. Those theories with well formulated asymptotics are studied through a Hamiltonian formalism, and their thermodynamical properties are found. The theories studied are general relativity, a dimensionally reduced cylindrical four-dimensional general…
In the recent paper [J.\ Phys.\ A 44 (2011) 065203], we have arrived at the closed-form expression for the Green's function for the partial differential operator describing propagation of a scalar wave in an $N$-dimensional ($N\geqslant2$) Maxwell fish-eye medium. The derivation has been based on unique transformation properties of the fish-eye wave equation under the hyperspherical inversion. In this communication, we arrive at the same expression for the fish-eye Green's function following a different route. The alternative derivation we present here exploits the fact that there is a close m…
The generalized Maxwell equations including an additional scalar field are considered in the first-order formalism. The gauge invariance of the Lagrangian and equations is broken resulting the appearance of a scalar field. We find the canonical and symmetrical Belinfante energy-momentum tensors. It is shown that the traces of the energy-momentum tensors are not equal to zero and the dilatation symmetry is broken in the theory considered. The matrix Hamiltonian form of equations is obtained after the exclusion of the nondynamical components. The canonical quantization is performed and the propa…
