On Integration of the Nonlinear d’Alembert-Eikonal System and Conditional Symmetry of Nonlinear Wave Equations

Abstract

We study integrability of a system of nonlinear partial differential equations consisting of the nonlinear d’Alembert equation □u = F (u) and nonlinear eikonal equation u _xµ u _x µ = G(u) in the complex Minkowski space R(1, 3). A method suggested makes it possible to establish necessary and sufficient compatibility conditions and construct a general solution of the d’Alembert-eikonal system for all cases when it is compatible. The results obtained can be applied, in particular, to construct principally new (non-Lie, non-similarity) solutions of the non-linear d’Alembert, Dirac, and Yang-Mills equations. Solutions found in this way are shown to correspond to conditional symmetry of the equations enumerated above. Using the said approach, we study in detail conditional symmetry of the nonlinear wave equation □w = F ₀(w) in the four-dimensional Minkowski space. A number of new (non-Lie) reductions of the above equation are obtained giving rise to its new exact solutions which contain arbitrary functions.     

κ-Minkowski spacetime and the star product realizations

We investigate a Lie algebra-type κ-deformed Minkowski spacetime with undeformed Lorentz algebra and mutually commutative vector-like Dirac derivatives. There are infinitely many realizations of κ-Minkowski space. The coproduct and the star product corresponding to each of them are found. An explicit connection between realizations and orderings is established and the relation between the coproduct and the star product, provided through an exponential map, is proved. Utilizing the properties of the natural realization, we construct a scalar field theory on κ-deformed Minkowski space and show that it is equivalent to the scalar, nonlocal, relativistically invariant field theory on the ordinary Minkowski space. This result is universal and does not depend on the realizations, i.e. the orderings, used.     

Solutions of the Bethe–Salpeter Equation in Minkowski Space and Applications to Electromagnetic Form Factors

We present a new method for solving the two-body Bethe–Salpeter equation in Minkowski space. It is based on the Nakanishi integral representation of the Bethe–Salpeter amplitude and on subsequent projection of the equation on the light-front plane. The method is valid for any kernel given by the irreducible Feynman graphs and for systems of spinless particles or fermions. The Bethe–Salpeter amplitudes in Minkowski space are obtained. The electromagnetic form factors are computed and compared to the Euclidean results.     

Electromagnetic form factor via Bethe-Salpeter amplitude in Minkowski space

For a relativistic system of two scalar particles, we find the Bethe-Salpeter amplitude in Minkowski space and use it to compute the electromagnetic form factor. The comparison with Euclidean space calculation shows that the Wick rotation in the form factor integral induces errors which increase with the momentum transfer Q². At JLab domain (Q ² = 10 GeV^2/c²), they are about 30%. Static approximation results in an additional and more significant error. On the contrary, the form factor calculated in light-front dynamics is almost indistinguishable from the Minkowski space one.     

Madelung Representation for Complex Nonlinear D’Alembert Equations in n-Dimensional Minkowski Space

Abstract

The Madelung representation ψ = u exp(iv) is considered for the d’Alembert equation □ _n ψ?F (|ψ|)ψ = 0 to develop a technique for finding exact solutions. We classify the nonlinear function F for which the amplitude and phase of the d’Alembert equation are related to the solutions of the compatible d’Alembert–Hamiltonian system.

The equations are studied in n-dimensional Minkowski space.     

Quantum creation of the universe with a minimum scalar-field energy

The quantum creation of a closed Friedmann universe is studied on the basis of a Wheeler-DeWitt equation with two arguments — a scale factor and a scalar-field potential. In the quasiclassical approximation the wave function of the universe (WF) starts to evolve at a zero scalar field. A near-Planckian energy density of the field arises as a result of tunneling through a potential barrier. In our opinion, this variant of the scenario most closely resembles creation ex nihilo. The only parameter controlling quantum evolution is the mass of a quantum of the scalar field. In the paper by Khalatnikov and Schiller [JETP Lett. 57,1 (1993)], tunneling through the classically inaccessible region of the superpotential U(a,φ) is calculated by the instanton method. However, this method requires that the potential U(a,φ) satisfy special conditions in the space (a,φ). For this reason, in the present paper the tunneling calculation is performed by the method of characteristics for the quasiclassical approximation of the Wheeler-DeWitt equation under the barrier. The WKB theory, which has been well-developed for one-dimensional problems, is employed along each characteristic. It is shown that the corresponding turning points are also points where U(a, φ)=0. The total barrier penetrability is obtained by averaging over a bundle of characteristics. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 5, 305–308 (10 September 1996)     

On the state space of the dipole ghost

A particular representation of SO(4, 2) is identified with the state space of the free dipole ghost. This representation is then given an explicit realization as the solution space of a 4th-order wave equation on a spacetime locally isomorphic to Minkowski space. A discrete basis for this solution space is given, as well as an explicit expression for its SO(4, 2) invariant inner product. The connection between the modes of dipole field and those of the massless scalar field is clarified, and a recent conjecture concerning the restriction of the dipole representation to the Poincaré subgroup is confirmed. A particular coordinate transformation then reveals the theory of the dipole ghost in Minkowski space. Finally, it is shown that the solution space of the dipole equation is not unitarizable in a Poincaré invariant manner.     

An Integral Transform in de Sitter Space

An integral transform on (4 + 1) de Sitter space, which is a generalization of the Fourier transform for a Dirac particle in Minkowski space, is considered, and proofs of its relevant properties are provided. With this integral transform we demonstrate explicitly that the momentum space eigenvalue equation for the second order Casimir operator of the de Sitter group is equivalent to a wave equation in Minkowski momentum space, which describes the mass-spin relation of a mechanical system in Minkowski space, consisting of two equal mass, point-like constituents rotating uniformly at a distance R from their geometric center, where R is the radius of the de Sitter space. Applications to the relativistic rotator are considered. Contrary to our previous results, we find that the relativistic rotator does not go into a structureless relativistic mass point in the elementary limit obtained by contracting the de Sitter group into the Poincaré group. Our analysis can be carried over, with relatively minor modifications, to anti-de Sitter space, and similar results hold there. Additional physical consequences are also discussed.     

Common vacuum conservation amplitude in the theory of the radiation of mirrors in two-dimensional space-time and of charges in four-dimensional space-time

The changes in the action (and thus the vacuum conservation amplitudes) in the proper-time representation are found for an accelerated mirror interacting with scalar and spinor vacuum fields in 1+1 space. They are shown to coincide to within a factor of e ² with changes in the action of electric and scalar charges accelerated in 3+1 space. This coincidence is attributed to the fact that the Bose and Fermi pairs emitted by a mirror have the same spins 1 and 0 as do the photons and scalar quanta emitted by charges. It is shown that the propagation of virtual pairs in 1+1 space can be described by the causal Green’s function Δ_f(z,μ) of the wave equation for 3+1 space. This is because the pairs can have any positive mass and their propagation function is represented by an integral of the causal propagation function of a massive particle in 1+1 space over mass which coincides with Δ_f(z,μ). In this integral the lower limit μ is chosen small, but nonzero, to eliminate the infrared divergence. It is shown that the real and imaginary parts of the change in the action are related by dispersion relations, in which a mass parameter serves as the dispersion variable. They are a consequence of the same relations for Δ_f(z, μ). Therefore, the emergence of a real part in the change in the action is a direct consequence of causality, according to which Re Δ_f(z,μ)≠0 only for timelike and lightlike intervals. Zh. éksp. Teor. Fiz. 116, 1523–1538 (November 1999)     

Analytic Equation of State of a Quasi One-Dimensional Model Lipid Monolayer

The equation of state of a quasi one-dimensional model lipid monolayer is obtained in analytic form. The method used is the Laplace transform approach leading to a homogeneous Fredholm integral equation. Two cases are studied. The first considers a purely short range repulsive potential, when we recover the results previously obtained by Gianotti et al. (J. Phys. A.: Math. Gen. 25:2889 (1992)). The second incorporates the long range attractive Kac potential, and the equation of state is calculated in the van der Waals limit. This extends the approach originally developed by Kac et al. (J. Math. Phys. 4:216 (1963)).     

A vector product formulation of special relativity and electromagnetism

The vector product method developed in previous articles for space rotations and Lorentz transformations is extended to the cases of four-vectors, anti-symmetric tensors, and their transformations in Minkowski space. The electromagnetic fields are expressed in six-vector form using the notationH +iE, and this vector form is shown to be relativistically invariant. The wave equations of electromagnetism are derived using these vector products. The following three equations are deduced, which summarize electrodynamics in a compact form: (1) Maxwell's four equations expressed as one, (2) the scalar and vector potential wave equations combined into one relation, and (3) the wave equations for the electric and magnetic fields and the continuity equation combined together. Space inversion, time reversal, and magnetic monopoles are also treated.On leave of absence from the University of Tel Aviv.     

One-loop approximation of Møller scattering in generalized Krein-space quantization

It has been shown that the negative-norm states necessarily appear in a covariant quantization of the free minimally coupled scalar field in de Sitter spacetime. In this processes ultraviolet and infrared divergences have been automatically eliminated. A natural renormalization of the one-loop interacting quantum field in Minkowski spacetime (λφ ⁴) has been achieved through the consideration of the negative-norm states defined in Krein space. It has been shown that the combination of quantum field theory in Krein space together with consideration of quantum metric fluctuation, results in quantum field theory without any divergences. Pursuing this approach, we express Wick’s theorem and calculate Møller scattering in the one-loop approximation in generalized Krein space. The mathematical consequence of this method is the disappearance of the ultraviolet divergence in the one-loop approximation.     

Bound-State Solutions of the Klein-Gordon Equation for the Generalized <Emphasis Type="Italic">PT</Emphasis>-Symmetric Hulthén Potential

The one-dimensional Klein-Gordon equation is solved for the PT-symmetric generalized Hulthén potential in the scalar coupling scheme. The relativistic bound-state energy spectrum and the corresponding wave functions are obtained by using the Nikiforov-Uvarov method which is based on solving the second-order linear differential equations by reduction to a generalized equation of hypergeometric type. PACS numbers: 03.65.Fd, 03.65.Ge     

What Can We Learn from Nonminimally Coupled Scalar Field Cosmology?

A novel exploration of nonminimally coupled scalar field cosmology is proposedin the framework of spatially flat Friedmann—Robertson—Walker spaces forarbitrary scalar field potentials V() and values of the nonminimal couplingconstant . This approach is self-consistent in the sense that the equation of stateof the scalar field is not prescribed a priori, but is rather deduced together withthe solution of the field equations. The role of nonminimal coupling appears tobe essential. A dimensional reduction of the system of differential equations leadsto the result that chaos is absent in the dynamics of a spatially flat FRW universewith a single scalar field. The topology of the phase space is studied and revealsan unexpected involved structure: according to the form of the potential V()and the value of the nonminimal coupling constant , dynamically forbiddenregions may exist. Their boundaries play an important role in the topologicalorganization of the phase space of the dynamical system. New exact solutionssharing a universal character are presented; one of them describes a nonsingularuniverse that exhibits a graceful exit from, and entry into, inflation. This behaviordoes not require the presence of the cosmological constant. The relevance of thissolution and of the topological structure of the phase space with respect to anemergence of the universe from a primordial Minkowski vacuum, in an extendedsemiclassical context, is shown.     

Recursive method for calculation of tree-level elements of arbitrary order <Emphasis Type="Italic">S</Emphasis> matrix in scalar electrodynamics

The possibility of creating a recursive method for the calculation of tree-level elements of an S-matrix based on the functional integration method is considered, as exemplified by scalar electrodynamics. One of the advantages of this method is the gauge invariance of amplitudes provided without additional operations with them.     

Radiation damping in closed expanding universes

The dynamics of a coupled model (harmonic oscillator-relativistic scalar field) in Conformal Robertson-Walker (k = +1) spacetimes is investigated. The exact radiation-reaction equation of the source-including the retarded radiation terms due to the closed space geometry – is obtained and analyzed. A suitable family of Lyapunov functions is constructed to show that, if the spacetime expands monotonely, then the source's energy damps. A numerical simulation of this equation for expanding Universes, with and without Future Event Horizon, is performed.     

Rindler solutions and their physical interpretation

We show that the singular behavior of Rindler solutions near horizon testifies to the currents of particles from a region arbitrarily close to the horizon. Besides, the Rindler solutions in right Rindler sector of Minkowski space can be represented as a superposition of only positive-or only negative-frequency plane waves; these states require infinite energy for their creation and possess infinite charge in a finite space interval, containing the horizon. The positive-or negative-frequency representations of Rindler solutions analytically continued to the whole Minkowski space make up a complete set of states in this space, which have, however, the aforementioned singularities. These positive (negative)-frequency states are characterized by positive (negative) total charge, the charge of the same sign in right (left) Rindler sector and by quantum number κ. But in other Lorentz invariant sectors they do not possess positive (negative)-definite charge density and have negative (positive) charge in left (right) Rindler sector. Therefore these states describe both the particle (antiparticle) and pairs, the mean number of which is given by Planck function of κ. These peculiarities make the Rindler set of solutions nonequivalent to the plane wave set and the inference on the existence of thermal currents for a Rindler observer moving in empty Minkowski space is unfounded. Zh. éksp. Teor. Fiz. 114, 777–785 (September 1998) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.