Free Field Equations Research Articles

Solutions of 1+1 relativistic field models

A class of classical relativistic nonlinear field models has been recently introduced by two of us [A. Degasperis and F. Tinebra, J. Math. Phys. 34, 2950 (1993)]; they are solvable by an explicit mapping to free field equations. Here we show the behavior of particular solutions of these models for both scalar and spinor fields. In the real spectrum we find brachion and tachion localized solutions, and we show the large-time behavior of wave packets, while in the complex spectrum we prove that the asymptotic states may contain solitons, antisolitons, and point-like charged particles (“pointons”). We report on several processes such as soliton–antisoliton pair creation and annihilation, and pointon decay in two solitons.

Electromagnetic interpretation of the massless spin-1 field equation in curved space-time

The source-free Maxwell equations associated to the massless spin-1 free field equation are considered in curved space-time. The gauge invariance of the theory is discussed by using as starting point the notion of the spinor potential. The structure of the electromagnetic field in the case of the Robertson-Walker space-time is discussed by using the solutions of the massless spin-1 equations previously determined. The flat-universe case of the standard cosmology is studied exactly and considered in some limiting physical situations.

Separation of the massless field equations for arbitrary spin in the Robertson–Walker space–time

The massless spin 2 free-field equation is studied in the Robertson–Walker space–time via the Newman–Penrose formalism and separated by using a Chandrasekhar–Teukolski method. The resulting temporal and angular equations are explicitly integrated. The radial equations are solved in the flat universe case. The closed universe case shows, in principle, the existence of a discrete spectrum of the energy of the massless particles. In the general case of spin greater than 2 the massless field equations are shown to be separable by induction. The separated equations admit the same recurrence structure and analog interpretation properties of the spin 2 and less than 2 cases.

Separation of the massless spin-1 equation in Robertson-Walker space-time

The massless spin-1 free field equation is studied via the Newman-Penrose formalism and separated by the Chandrasekhar-Teukolski method. The temporal and angular equations are explicitly integrated. The radial equations are solved in the flat-universe case. The closed-universe case shows, in principle, the existence of a discrete spectrum of the energy of the massless particles.

An energy based macro-element method via a coupled finite element and boundary integral formulation

An energy-based, systematic method for the coupling of the finite element (FE) method and the boundary integral equation (BIE) method is described in this paper. This method allows the use of the BIE method to represent those subdomains of a structure that are best suited to BIE method, and the use of the FE method to represent the rest of the structure. Different subdomains and their associated FE or BIE representations are coupled naturally through the total potential energy functional of the system. The associated discretized problem from the proposed method consists of a linear system of equations with a symmetric and blockwise banded matrix. As in regular FE methods, the derivation is started from the total potential energy principle. However, in those subdomains that are best suited to the BIE method, the exact solution of the associated free field equation is used to represent the true solution. The size and shape of each BIE subdomain, called “macro-element” in this paper, may be designed freely to meet various practical requirements concerning, for example, numerical efficiency, machine storage limitations, mesh generations, etc. Most significantly, unlike the existing BIE techniques, the present method does not seem to require special treatments for corner effects, thus reducing the computational complexity. Numerical experiments have been performed for generalized Poisson's equation as a prototype situation. The extension to 2D-3D elastic problems is straightforward.

Minimal coupling of electromagnetic field in Riemann-Cartan spacetime for perfect fluids

We minimally couple the electromagnetic field to gravity in Riemann-Cartan spacetime in the self-consistent formalism for perfect fluids by treating the internal energy of matter as a function of the electromagnetic field. The overall Lagrangian of the gravitational field, perfect fluid, and the electromagnetic field is constrained to be gauge invariant under gauge transformations of the vector potential. The theory preserves both charge conservation and particle number conservation, and gives the usual form of the free field equations.

On the solution of a generalized Stefan problem

We present a numerical treatment of a generalized two-dimensional Stefan problem which models the solidification or melting of a solid in a melt. The irregular domain is transformed into a rectangular domain by a coordinate transformation, and the resulting parabolic equations are discretized using fourth-order difference formulae. The free surface and field equations are solved using Crank–Nicolson procedures. The results are found to be in good agreement with the linear perturbation analysis of Chadam and Ortoleva (1983). Additionally, an improvement in the estimate of Baillon et al. (1984) for breakdown time in the one-dimensional problem and corroborating numerical results are presented.

Static vacuum solution of direct Poincar� gauge theory in ten dimensions with four external

A torsion-free solution of the free gauge field equations of direct Poincare gauge theory on a ten-dimensional Minkowski space is constructed. This solution exhibits nontrivial curvature two-forms, but shaves the metric structure down to that of a four-dimensional Minkowski space. Universality of this solution with respect to the choice of the free field Lagrangian is established.

On canonical irreducible quantum field theories describing bosons and fermions

For a relativistic irreducible quantum field theory (QFT) in which the bosons and fermions fulfill the canonical commutation relation (CCR) and the canonical anticommutation relation (CAR), respectively, it is shown that in n>3 space dimensions Bose and Fermi fields fulfill free field equations. Furthermore which interactions might be compatible with CCR and CAR in lower space dimensions are characterized. The possible candidates are interactions of type Q2(ψ,ψ°)φ+P4(φ) in n=3, Q2(ψ,ψ°)P2(φ)+P6(φ) in n=2, and Q4(ψ,ψ°)F(φ) in n=1 space dimensions, where Qm and Pm are polynomials of degree up to m.

Isogroups of differential ideals of vector-valued differential forms: Application to partial differential equations

An older geometric technique for the study of invariance groups of partial differential equations, originally proposed by one of the authors and F. B. Estabrook, is generalized and extended to problems involving exterior equations for vector-valued or Lie algebra-valued exterior differential forms. Use of the method is demonstrated in the study of the symmetry groups of the two-dimensional Dirac equation and the full Yang-Mills free-field equations in Minkowski spacetime.

Causal green function in relativistic quantum mechanics

The causal Green function or Feynman propagator for the free-field Klein-Gordon equation and related singular functions, defined as distributions, are related to the causal time-boundary data. Probability densities and amplitudes are defined in terms of the solutions of the Klein-Gordon equation for a complex scalar field interacting with an electromagnetic field. The convergence of the perturbation expansion of the solution of the Klein-Gordon equation for a charged scalar particle in an external field is shown for well-behaved electromagnetic potentials. Other relativistic wave equations are discussed briefly.

Back-reaction calculation of quantum fields in curved spacetime with arbitrary curvature coupling.

A scheme for solving the semiclassical back-reaction equations for free quantum fields in curved spacetime is proposed. The quantum fields can have arbitrary masses and arbitrary couplings to the curvature. All back-reaction effects including vacuum polarizations can be taken into account. The scheme has been applied successfully to the reheating calculations of the Starobinsky-type inflationary models. The damping of the oscillation in curvature and the reheating of the Universe are seen explicitly.

Local conformal-invariance of the wave equation for finite-component fields. I. The conditions for invariance, and fully-reducible fields

The conditions for local conformal-invariance of the wave equation are obtained for finite-component fields, of Types Ia and Ib [in the terminology of Mack and Salam, Ann. Phys. 53, 174 (1969).] These conditions generate a set of locally invariant free massless field equations and restrict the relevant representation of the Lie algebra [(k4⊕d)⊕sl(2,C)] in the index space of the field to belong to a certain class. Those fully-reducible representations which are in this class are described in full. The corresponding Type Ia field equations include only the massless scalar field equation, neutrino equations, Maxwell’s equations, and the Bargmann–Wigner equations for massless fields of arbitrary helicity, and no others. In particular, it is confirmed [Bracken, Lett. Nuovo Cimento 2, 574 (1971)] that not all Poincaré-invariant sets of massless Type Ia field equations are conformal-invariant, contrary to some often-quoted results of McLennan [Nuovo Cimento 3, 1360 (1956)], which are shown to be invalid. It is also shown that in the case of a potential, the wave equation is never conformal-invariant in the strong sense (excluding gauge transformations).

Gauge-invariant canonical quantisation of the electromagnetic field and duality transformations

The free electromagnetic field is canonically quantised in a gauge-invariant way by interpreting the Fourier coefficients of the magnetic induction field B as generalised coordinates, and the coefficients of the electric field E as their conjugate momenta. The usual commutation relations among the components of E and B are obtained. A canonical transformation, corresponding to a rotation in generalised phase space, is made on the Fourier coefficients. This transformation is shown to give a duality transformation on the electric and magnetic fields. The free-field Maxwell equations and the commutation relations are invariant under duality transformations. However, if interactions are introduced, the invariance under duality transformations is broken, and the original canonical theory should be used.

On the linearization of source free gauge field equations

A scheme for linearizing the source free Yang-Mills field equations is given. An infinite parameter invariance group of the gauge field equations is discovered.

Free-field equations of conformal relativity in riemannian formalism. - I

The free-field equations are reduced to a form couched in the simple Riemannian geometry of space-time angle. The solution of these for all the predicted particle species is now a straightforward job involving techniques familiar from usual relativistic free-field theory. We recall that the central prediction of this theory is that massive stable particles and resonances occur in families of someTJP and mass spectra of nonaccumulating type. This is the main datum of high-energy physics, cf. theN family, the δ family, etc. In fact, « bare », « elementary » resonances of this general type occur already as the massive solutions of the free-field equations.

Free-field equations of conformal relativity in riemannian formalism

We solve our field equations in their Riemannian form for some of the « lowest » particle species : the scalar, vector, spinor and delta (spin 3/2 ). These massive solutions come in mass families composed of ground state plus « elementary » resonances of oneTJ P with predicted bare mass spectra of quasi-uniform type. The suggestion is that these may give a direct explanation of the well-known scattering resonance families such as theN family, the Δ family, etc.

The linear integral equations associated with any antiexact solution of the Yang–Mills free field equations

Any system of r antiexact 1-forms, Aα, that solves the Yang–Mills free field equations associated with a semisimple r-parameter gauge group is shown to satisfy a system of linear inhomogeneous integral equations. These equations take the form Aα = Aαi dxi, Aαi = Rαi +Yαjiρ( Aρj), where Rαi dxi are r antiexact 1-forms, each of which solves Maxwell’s free field equations, and Yαjiρ are linear Riemann–Graves integral operators that are also linear in the structure constants of the gauge group. The standard iteration procedure for solving such systems may be looked upon as generating a sequence of corrections to the Rαi fields that account for the nonabelian character of the gauge group.

A maxwellian tetrad theory of gravity with preferred frames: A

A theory of gravity is proposed which seeks to mimic maxwellian electromagnetism whilst maintaining the principle of equivalence. The curls of tetrad potentials are taken as field strengths and Maxwell-like free-field equations are set up which contain a non-linear gravitational current term and which are generally covariant but not locally Lorentz covariant. The weak field approximation is solved for static metrical spherical symmetry and solutions constructed which agree with the GR predictions.

On the structure of the free field equations

The structure of the Poincare-covariant equations for free massive fields is analysed. It is assumed that the supplementary conditions follow from the field equations. By use of the notion of the commutant the general form of such equations is given. The necessary and sufficient conditions for their existence are found.