Proca Equation Research Articles

Field equations for the massive vector boson from Dirac and Weinberg formalisms

This paper is a follow-up to an earlier paper that discussed the single-particle quantum mechanics of massless bose particles. In the present paper we extend the analysis to the massive vector (j = 1) bosons that occur in electroweak interactions. As in the previous paper we make a connection between a generalization of the Dirac equation and the equations obtained by Weinberg from S-matrix field theory. The starting point is the Bargmann–Wigner generalization of the Dirac equation. This leads to the Proca equations for a vector potential field, then to Maxwell’s equations, which we finally relate to Weinberg’s equations. We spend some time analyzing the quantity Tr(Ψ(x)*Ψ(x)), where Ψ(x) is the Bargmann–Wigner wave function (a symmetric four by four matrix). Using Lagrangian and Hamiltonian density equations, we show that the trace has the interpretation of being the Hamiltonian density for the vector potential field. We also use the Lagrangian analysis to construct a conserved current via Noether’s theorem.

Modified Proca equation and modified dispersion relation from a power-law Lagrangian functional

This work discusses some developments in classical gauge field theory which have grown out of the ideas of non-standard Lagrangians in the framework of the calculus of variations. A substantial part of this work is concerned with the modification of some basic equations of classical field theories starting from a power-law Lagrangian functional with time-dependent power and which belongs to the class of non-standard Lagrangians. Many equations principally the modified Proca equation and some modified dispersion relations for the consequent field equations are derived.

J/Ψ-nuclear bound states

$J/\Psi$-nuclear bound state energies are calculated for a range of nuclei by solving the Proca (Klein-Gordon) equation. Critical input for the calculations, namely the medium-modified $D$ and $D^*$ meson masses, as well as the nucleon density distributions in nuclei, are obtained from the quark-meson coupling model. The attractive potential originates from the $D$ and $D^*$ meson loops in the $J/\Psi$ self-energy in nuclear medium. It appears that $J/\Psi$-nuclear bound states should produce a clear experimental signature provided that the $J/\Psi$ meson is produced in recoilless kinematics.

Casimir effect of massive vector fields

We study the Casimir effect due to a massive vector field in a system of two parallel plates made of real materials, in an arbitrary magnetodielectric background. The plane waves satisfying the Proca equations are classified into transverse modes and longitudinal modes which have different dispersion relations. Transverse modes are further divided into type I and type II corresponding to TE and TM modes in the massless case. For general magnetodielectric media, we argue that the correct boundary conditions are the continuities of H{sub ||}, {phi}, A, and {partial_derivative}{sub x}A{sub x}, where x is the direction normal to the plates. Although there are type I transverse modes that satisfy all the boundary conditions, it is impossible to find type II transverse modes or longitudinal modes that satisfy all the boundary conditions. To circumvent this problem, type II transverse modes and longitudinal modes have to be considered together. We call the contribution to the Casimir energy from type I transverse modes TE contribution, and the contribution from the superposition of type II transverse modes and longitudinal modes TM contribution. Their massless limits give, respectively, the TE and TM contributions to the Casimir energy of a massless vector field. The limit wheremore » the plates become perfectly conducting is discussed in detail. For the special case where the background has a unity refractive index, it is shown that the TM contribution to the Casimir energy can be written as a sum of contributions from two different types of modes, corresponding to type II discrete modes and type III continuum modes discussed by Barton and Dombey [G. Barton and N. Dombey, Ann. Phys. (N.Y.) 162, 231 (1985).]. For general background, this splitting does not work. The limit where both plates become infinitely permeable and the limit where one plate becomes perfectly conducting and one plate becomes infinitely permeable are also investigated.« less

Upper bounds on the photon mass

The effects of a nonzero photon rest mass can be incorporated into electromagnetism in a simple way using the Proca equations. In this vein, two interesting implications regarding the possible existence of a massive photon in nature, i.e., tiny alterations in the known values of both the anomalous magnetic moment of the electron and the gravitational deflection of electromagnetic radiation, are utilized to set upper limits on its mass. The bounds obtained are not as stringent as those recently found; nonetheless, they are comparable to other existing bounds and bring new elements to the issue of restricting the photon mass.

Localizing gauge fields on a topological Abelian string and the Coulomb law

The confinement of electromagnetic field is studied in axial symmetrical, warped, 6D World Brane, using a recently proposed topological abelian string vortex solution as background. It was found, that the massless gauge field fluctuations follow 4D Maxwell equations in the Lorenz gauge. The massless zero mode is localized when the thickness of the string-vortex is less than 5% $\beta/4\pi e^{2}$ there are not others localized massless modes. There is also an infinite of non localized massive Fourier modes, that follow 4 dimensional Proca equations with a continuous spectrum. To compute the corrections to the Coulomb potential, a radial cutoff was introduced, in order to achieve a discrete mass spectrum. As main result, a $\frac{1}{R^{2}}$ correction was found for the 4D effective Coulomb law. Although, this correction is milder than the $% \frac{1}{R^{3}}$ expected for flat 5+1 dimensional electromagnetism, the result is in correspondence with the observed behavior of the Coulomb potential at nowadays measurable distances.

The Proca equations derived from first principles

We show how the Proca equations can be deduced from first principles in a way that is similar to the derivations of the Dirac relativistic electron equation and Maxwell equations.

Electrodynamics in superconductors explained by Proca equations

A fully consistent model to study electrodynamics for superconductors in the stationary and non-stationary regimes has been developed based on Proca equations and a massive photon. In particular, this approach has been applied to study the electric field penetration depth in superconductors. The model shows a deviation from the charge contribution to an internal electric field compared to previous approaches.

Field Equations of Arbitrary Spin in Space-time with Torsion

A two spinor lagrangian formulation of field equations for massive particle of arbitrary spin is proposed in a curved space-time with torsion. The interaction between fields and torsion is expressed by generalizing the situation of the Dirac equation. The resulting field equations are different (except for the spin-1/2 case) from those obtained by promoting the covariant derivatives of the torsion free equations to include torsion. The non linearity of the equations, that is induced by torsion, can be interpreted as a self-interaction of the particle. The spin-1 and spin-3/2 cases are studied with some details by translating into tensor form. There result the Proca and Rarita-Schwinger field equations with torsion, respectively.

CAUSAL PROPAGATION OF SPIN-CASCADES

We gauge the direct product of the Proca with the Dirac equation that describes the coupling to the electromagnetic field of the spin-cascade (1/2, 3/2) residing in the four-vector spinor ψμ and analyze propagation of its wave fronts in terms of the Courant–Hilbert criteria. We show that the differential equation under consideration is unconditionally hyperbolic and the propagation of its wave fronts unconditionally causal. In this way we prove that the irreducible spin-cascade embedded within ψμ is free from the Velo–Zwanziger problem that plagues the Rarita–Schwinger description of spin-3/2. The proof extends also to the direct product of two Proca equations and implies causal propagation of the spin-cascade (0, 1, 2) within an electromagnetic environment.

A positive-definite scalar product for free Proca particle

We implement recent results of pseudo-Hermitian quantum mechanics to the description of relativistic massive particle with spin-one. We derive a one-parameter family of Lorentz invariant positive-definite scalar products on the space of solutions of Proca equation.

"SQUARE ROOT" OF THE PROCA EQUATION: SPIN-3/2 FIELD EQUATION

New equations describing particles with spin-3/2 are derived. The nonlocal equation with the unique mass can be considered as "square root" of the Proca equation in the same sense as the Dirac equation is related to the Klein–Gordon–Fock equation. The local equation describes spin-3/2 particles with three mass states. The equations considered involve fields with spin-3/2 and spin-1/2, i.e. multispin 1/2, 3/2. The projection operators extracting states with definite energy, spin, and spin projections are obtained. All independent solutions of the local equation are expressed through projection matrices. The first order relativistic wave equation in the 20-dimensional matrix form, the relativistically invariant bilinear form and the corresponding Lagrangian are given. Two parameters characterizing nonminimal electromagnetic interactions of fermions are introduced, and the quantum-mechanical Hamiltonian is found. It is proved that there is only causal propagation of waves in the approach considered.

Variations of the Speed of Light with Frequency and ImpliedPhoton Mass

A general frequency-dependent dispersion relation of the speed of lightin different mediums (vacuum, insulator, plasma) is deduced based onthe Proca equations. Several recent astronomical observations of thepulsars are used to set the limits on the photon rest mass by thismethod and several upper bounds of larger than one order improvementthan previous similar results are obtained. Considering the dispersionof the massive photon, the possible upper limits on the photon restmass are also derived from the recently experimental results fortesting the constancy of the speed of light in special relativity.

Vector bosons in the expanding universe

We exactly solve the relativistic wave equation for vector bosons in the expanding universe and show that the current of the vector bosons in this background is rapidly oscillating in early time. Additionally, we derive the solutions of the Proca equation from the solutions of the Duffin-Kemmer-Petiau (DKP) equations in the same background and obtain the massless-particle, photon, solutions by taking the $m^{2}\rightarrow 0$ limit of these solutions. PACS. 03.65.Pm, 04.60.-m, 98.80.Cq

Quantum action principle and base operators for bosons and fermions

A new quantum action principle is formulated by expressing the Klein–Gordon, Dirac and Proca equations in the form of eigenvalue equations for quantum action operators. These operators are Hermitian and they become the base operators for bosons and fermions. The principle postulates that their only allowed eigenvalue is Planck's constant. Relationships between the quantum action operators and the Casimir operators of the Poincaré group and translation group are derived. It is shown that the obtained results supplement Wigner's original work on classification of the irreducible representations of the Poincaré group.

THE SUPERFIELD QUANTISATION OF A SUPERPARTICLE ACTION WITH AN EXTENDED LINE ELEMENT

A massive superparticle action based on the generalised line element in N = 1 global superspace is quantised canonically. A previous method of quantising this action, based on a Fock space analysis, showed that states existed in three supersymmetric multiplets, each of a different mass. The quantisation procedure presented uses the single first class constraint as an operator condition on a general N = 1 superwavefunction. The constraint produces coupled equations of motion for the component wavefunctions. Transformations of the component wavefunctions are derived that decouple the equations of motion and partition the resulting wavefunctions into three separate supermultiplets. Unlike previous quantisations of superparticle actions in N = 1 global superspace, the spinor wavefunctions satisfy the Dirac equation and the vector wavefunctions satisfy the Proca equation. The off-shell closure of the commutators of the supersymmetry transformations, that include mass parameters, are derived by the introduction of auxiliary wavefunctions. To avoid the ghosts arising in a previous Fock space quantisation an alternative conjugation is used in the definition of the current, based on a Krein space approach.

Experimental tests of Coulomb's Law and the photon rest mass

Coulomb's Law is a fundamental principle describing the electric force between isolated charges, and represents the first quantitative law achieved in electromagnetism. The degree of confidence with which the law is experimentally known to hold was investigated after the law was put forth by Coulomb in 1785. The electrodynamics for massive particles suggests that a photon with a finite rest mass will cause a deviation from the inverse square law. So, modern interpretations of the possible deviation from Coulomb's inverse square law are usually associated with the non-zero photon mass. In this article, we first give a historical review of the foundation of Coulomb's inverse square law. Then, the experimental searches for validity of Coulomb's Law, particularly in its inverse square nature, are generally introduced. Based on Proca's equations, the unique simplest relativistic generalization of Maxwell's equations, the link between the deviation from Coulomb's Law and the upper limit on the photon rest mass based on the concentric-spheres apparatus established in the classical experiment of Cavendish is reviewed. Up to now, all the experiments show no evidence for a positive value, and the experimental result was customarily expressed as an upper limit on the deviation or on the photon rest mass. As a representative method with the double mission of testing of the validity of Coulomb's Law and of the photon rest mass, possible improvements for this kind of experiment are discussed.

Dynamics of superhorizon photons during inflation with vacuum polarization

We study asymptotic dynamics of photons propagating in the polarized vacuum of a locally de Sitter Universe. The origin of the vacuum polarization is fluctuations of a massless, minimally coupled, scalar, which we model by the one-loop vacuum polarization tensor of scalar electrodynamics. We show that late time dynamics of the electric field on superhorizon scales approaches that of an Airy oscillator. The magnetic field amplitude, on the other hand, asymptotically approaches a nonvanishing constant (plus an exponentially small oscillatory component), which is suppressed with respect to the initial (vacuum) amplitude. This implies that the asymptotic photon dynamics is more intricate than that of a massive photon obeying the local Proca equation.

Dynamics of superhorizon photons during inflation with vacuum polarization

We study asymptotic dynamics of photons propagating in the polarized vacuum of a locally de Sitter Universe. The origin of the vacuum polarization is fluctuations of a massless, minimally coupled, scalar, which we model by the one-loop vacuum polarization tensor of scalar electrodynamics. We show that late time dynamics of the electric field on superhorizon scales approaches that of an Airy oscillator. The magnetic field amplitude, on the other hand, asymptotically approaches a nonvanishing constant (plus an exponentially small oscillatory component), which is suppressed with respect to the initial (vacuum) amplitude. This implies that the asymptotic photon dynamics is more intricate than that of a massive photon obeying the local Proca equation.

Coulomb interaction on spin-1 particles

Using the electro-weak theory, we find the lowest order perturbative correction to a spin-1 particle in an external Coulomb field. We show this leads to a correction of order (Zα)4 and is independent of the mass of the external field. Previous work with Duffin–Kemmer–Petiau (see Nedjadi and Barrett [J. Math. Phys. 35 (1994) 4517]) and the Proca equation has failed to produce this correction.