Relativistic quantum theory without quantized fields

Peculiarities of symmetrical quantum systems are considered with the aid of the Mackey's induced representations theory. The four-dimensional coordinate representation of the relativistic quantum mechanics suggested by Stueckelberg in 1941 is rederived, reinterpreted and generalized for an arbitrary spin. Then it is applied to introduce the causal propagator as a particleantiparticle transition amplitude without consideration of a field equation. Finally the theory of relativistic quantum particles interaction is reformulated without an appeal to the concept of quantized fields.     

Gauge-Invariant Chiral Schwinger Model with Faddeevian Regularization: Stueckelberg Term, Operator Solution, and Hamiltonian and BRST Formulations

A chiral Schwinger model with the Faddeevian regularization à la Mitra is studied in one-space one-time dimension in the conventional form of dynamics (on the hyperplanes x ⁰ = constant) called the Instant-Form (IF) dynamics. The original IF theory is seen to be gauge-noninvariant (GNI). Corresponding to this GNI model, a gauge-invariant (GI) theory is constructed through the so-called Stueckelberg term. The operator solution and the Hamiltonian and BRST formulations of the resulting GI theory, obtained by the inclusion of the Stueckelberg term in the action of the original GNI theory, are then investigated with some specific gauge choices. The physical contents of the original GNI theory are also recovered from the newly constructed GI theory under a special gauge.     

Theories of Variable Mass Particles and Low Energy Nuclear Phenomena

Variable particle masses have sometimes been invoked to explain observed anomalies in low energy nuclear reactions (LENR). Such behavior has never been observed directly, and is not considered possible in theoretical nuclear physics. Nevertheless, there are covariant off-mass-shell theories of relativistic particle dynamics, based on works by Fock, Stueckelberg, Feynman, Greenberger, Horwitz, and others. We review some of these and we also consider virtual particles that arise in conventional Feynman diagrams in relativistic field theories. Effective Lagrangian models incorporating variable mass particle theories might be useful in describing anomalous nuclear reactions by combining mass shifts together with resonant tunneling and other effects. A detailed model for resonant fusion in a deuterium molecule with off-shell deuterons and electrons is presented as an example. Experimental means of observing such off-shell behavior directly, if it exists, is proposed and described. Brief explanations for elemental transmutation and formation of micro-craters are also given, and an alternative mechanism for the mass shift in the Widom–Larsen theory is presented. If variable mass theories were to find experimental support from LENR, then they would undoubtedly have important implications for the foundations of quantum mechanics, and practical applications may arise.     

Quantum Gravity Induced from Unconstrained Membranes

The theory of unconstrained membranes of arbitrary dimension is presented. Their relativistic dynamics is described by an action which is a generalization of the Stueckelberg point-particle action. In the quantum version of the theory, the evolution of a membrane's state is governed by the relativistic Schrödinger equation. Particular stationary solutions correspond to the conventional, constrained membranes. Contrary to the usual practice, our spacetime is identified, not with the embedding space (which brings the problem of compactification), but with a membrane of dimension 4 or higher. A 4-membrane is thus assumed to represent spacetime. The Einstein-Hilbert action emerges as an effective action after functionally integrating out the membrane's embedding functions.     

Functional methods in thermofield dynamics: A real-time perturbation theory for quantum statistical mechanics

A generating functional is constructed for real-time Green functions in quantum statistical mechanics in the context of thermofield dynamics. The KMS condition is taken as an axiom which together with field equations fixes the generating functional for causal Green functions in an interacting quantum field theory. This leads to Feynman rules for diagrammatic real-time perturbation theory.     

Path integral for a Dirac particle in a quantized plane wave field

The problem of a relativistic spinning particle interacting with a quantized electromagnetic plane wave is treated by employing path integral methods and introducing the notion of a coherent state action. The dynamics of the particle spin is described using the supersymmetric action proposed recently by Alexandrou et al. in the so-called global representation. It is shown that to obtain the relative causal Green function, two fermionic identities, related directly to the classical equations of motion, have to be incorporated. The Green function, as constructed, allows us to extract in a natural way the expressions of the corresponding energy spectrum and wave functions. Received: 26 July 2002 / Revised version: 11 September 2002 / Published online: 25 October 2002     

Fractional Dynamics of Relativistic Particle

Fractional dynamics of relativistic particle is discussed. Derivatives of fractional orders with respect to proper time describe long-term memory effects that correspond to intrinsic dissipative processes. Relativistic particle subjected to a non-potential four-force is considered as a nonholonomic system. The nonholonomic constraint in four-dimensional space-time represents the relativistic invariance by the equation for four-velocity u _μ u ^μ +c ²=0, where c is a speed of light in vacuum. In the general case, the fractional dynamics of relativistic particle is described as non-Hamiltonian and dissipative. Conditions for fractional relativistic particle to be a Hamiltonian system are considered.     

Gibbs ensembles in relativistic classical and quantum mechanics

Classical and quantum Gibbs ensembles are constructed for equilibrium statistical mechanics in the framework of an extension to many-body theory of a relativistic mechanics proposed by Stueckelberg. In addition to the usual chemical potential in the grand canonical ensemble, there is a new potential corresponding to the mass degree of freedom of relativistic systems. It is shown that in the nonrelativistic limit the relativistic ensembles we have obtained reduce to the usual ones, and mass fluctuations for the free-particle gas approach the fluctuations in N. The ultrarelativistic limit of the canonical ensemble for the free-particle gas differs from the corresponding limit of the ensemble proposed by Jüttner and Pauli. Due to the mass degree of freedom, the quantum counting of states is different from that of the nonrelativistic theory. If the mass distribution is sufficiently sharp, the thermodynamical effects of this multiplicity will not be large. There may, however, be detectable effects such as a shift in the Fermi level and the critical temperature for Bose-Einstein condensation, and some change in specific heats.     

On the stueckelberg formula for non-adiabatic transitions

The phase corrected Stueckelberg and Landau-Zener formulae for the non-adiabatic transition probability in a two state problem are compared. It is shown that validity criteria on the derivation of the former imply precise equivalence between the two formulations for the linear curve crossing problem; but the Stueckelberg approach is seen to have greater flexibility. A proposed uniformization of the Stueckelberg form for use at low velocities is shown to be incorrect.     

Quantization of superparametrized monopoles

Classical finite-energy solutions of the SU(2) Yang-Mills-Higgs system in four-dimensional space-time are embedded in the supersymmetric extension of the theory. Finite supertranslations are constructed and are used to obtain a family of solutions to the supersymmetric field equations, parametrized by fermionic Majorana spinor parameters. The quantum theory around arbitrary classical solutions, parametrized by arbitrary bosonic (global and local) as well as fermionic (global) parameters, is constructed and discussed.     

de Broglie's Pilot-Wave Theory for the Klein–Gordon Equation and Its Space-Time Pathologies

We illustrate, using a simple model, that in the usual formulation the time-component of the Klein–Gordon current is not generally positive definite even if one restricts allowed solutions to those with positive frequencies. Since in de Broglie's theory of particle trajectories the particle follows the current this leads to difficulties of interpretation, with the appearance of trajectories which are closed loops in space-time and velocities not limited from above. We show that at least this pathology can be avoided if one adapts in a covariant form the formulation of relativistic point particle dynamics proposed by Gitman and Tyutin.     

Canonical proper time formulation of relativistic particle dynamics

A canonical (contact) transformation is performed on the time variable (in extended phase space) to reexpress relativistic dynamics in terms of proper time, leaving the generalized coordinates and canonical momentum as functions of this time variable. The form of the energy functional conjugate to this time variable is seen to be similar to that of a nonrelativistic dynamics at all values of particle momenta. The formulation is explored for single- and multiparticle classical systems. The (form) invariance of the theory is determined by a group which results from a similarity action of the contact group on the Poincaré group. One advantage of this approach is that it overcomes the no-interaction difficulties introduced by standard methods.     

Inflation and Late Time Acceleration Designed by Stueckelberg Massive Photon

We present a mini review of the Stueckelberg mechanism, which was proposed to make the abelian gauge theories massive as an alternative to Higgs mechanism, within the framework of Minkowski as well as curved spacetimes. The higher the scale the tighter the bounds on the photon mass, which might be gained via the Stueckelberg mechanism, may be signalling that even an extremely small mass of the photon which cannot be measured directly could have far reaching effects in cosmology. We present a cosmological model where Stueckelberg fields, which consist of both scalar and vector fields, are non-minimally coupled to gravity and the universe could go through a decelerating expansion phase sandwiched by two different accelerated expansion phases. We discuss also the possible anisotropic extensions of the model.     

On the quantum electrodynamics of moving bodies

A new synthesis of the principles of relativity and quantum mechanics is developed by replacing the Poincaré group for the de Sitter one. The new relativistic quantum mechanics is an indefinite-mass theory which is reduced to the standard theory on the mass shell. The charge conjugation acquires a geometrical meaning and the Stueckelberg interpretation for antiparticles naturally arises in the formalism. So the idea of the Dirac sea in the second-quantized formalism proves to be superfluous. The off-shell theory is free from ultraviolet divergences, which only appear in the process of mass-shell reduction.     

Five-dimensional theory of interacting scalar,electromagnetic, and gravitational fields

An (n+1) factorization of an (n+1)-dimensional Riemann manifold is performed. For a space permitting a Killing vector, the (n+l)-dimensional Hubert variational principle reduces to the variational principle for the corresponding quantities in an n-dimensional space. Hence, setting n=4 and n=3, versions of a unified theory of gravitational, electromagnetic, and scalar fields and the steady-space theory of general relativity theory, respectively, are constructed. The five-dimensional variational principle for geodesics reduces to the four-dimensional leastaction principle for the test charged particle moving in gravitational, electromagnetic, and scalar fields.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 58–65, November, 1979.     

Boundary crisis and suppression of Fermi acceleration in a dissipative two-dimensional non-integrable time-dependent billiard

Some dynamical properties for a dissipative time-dependent oval-shaped billiard are studied. The system is described in terms of a four-dimensional nonlinear mapping. Dissipation is introduced via inelastic collisions of the particle with the boundary, thus implying that the particle has a fractional loss of energy upon collision. The dissipation causes profound modifications in the dynamics of the particle as well as in the phase space of the non-dissipative system. In particular, inelastic collisions can be assumed as an efficient mechanism to suppress Fermi acceleration of the particle. The dissipation also creates attractors in the system, including chaotic. We show that a slightly modification of the intensity of the damping coefficient yields a drastic and sudden destruction of the chaotic attractor, thus leading the system to experience a boundary crisis. We have characterized such a boundary crisis via a collision of the chaotic attractor with its own basin of attraction and confirmed that inelastic collisions do indeed suppress Fermi acceleration in two-dimensional time-dependent billiards.     

Bohmian Trajectories for Kerr–Newman Particles in Complex Space-Time

Complexified Liénard–Wiechert potentials simplify the mathematics of Kerr–Newman particles. Here we constrain them by fiat to move along Bohmian trajectories to see if anything interesting occurs, as their equations of motion are not known. A covariant theory due to Stueckelberg is used. This paper deviates from the traditional Bohmian interpretation of quantum mechanics since the electromagnetic interactions of Kerr–Newman particles are dictated by general relativity. A Gaussian wave function is used to produce the Bohmian trajectories, which are found to be multi-valued. A generalized analytic continuation is introduced which leads to an infinite number of trajectories. These include the entire set of Bohmian trajectories. This leads to multiple retarded times which come into play in complex space-time. If one weights these trajectories by their natural Bohmian weighting factors, then it is found that the particles do not radiate, that they are extended, and that they can have a finite electrostatic self energy, thus avoiding the usual divergence of the charged point particle. This effort does not in any way criticize or downplay the traditional Bohmian interpretation which does not assume the standard electromagnetic coupling to charged particles, but it suggests that a hybridization of Kerr–Newman particle theory with Bohmian mechanics might lead to interesting new physics, and maybe even the possibility of emergent quantum mechanics.     

Stueckelberg Oscillations in a Two‐State Two‐Path Model of a Conical Intersection

The two‐state two‐path model is introduced as a minimized model to describe the quantum dynamics of an electronic wave packet in the vicinity of a conical intersection. It involves two electronic potential energy surfaces each of which hosts a pair of quasi‐classical trajectories over which the wave packet is assumed to be delocalized. When both trajectories evolve dynamically either diabatically or adiabatically, the full wave packet dynamics shows only features of the dynamics around avoided level crossings in the vicinity of the conical intersection. When one trajectory evolves adiabatically whereas the other trajectory follows a diabatic evolution, quantum mechanical interference of the wave packet components on each path generates Stueckelberg oscillations in the transition probability. These are surprisingly robust against a dissipative environment and, thus, should be a marker for conical intersections.     

Large-Scale Two-Dimensional Quantum Fluctuations of Five-Dimensional and Four-Dimensional Space-Time Metrics

Large-scale two-dimensional quantum fluctuations of five-dimensional space-time metric are constructed and the effect of the fluctuations on the nested four-dimensional worlds is studied. In doing so, the fluctuations affect not all four-dimensional worlds but only a part of them. The energy-momentum tensor of four-dimensional space-time has a physical form both in the absence and in the presence of fluctuations; it means that the fluctuations can be realized by real matter. A spatial region occupied by the fluctuations constructed in this work can be infinitely large and the fluctuations can occur during a long period of time. Therefore, we refer to these fluctuations as large-scale fluctuations.     

Use of catastrophe theory to obtain a fundamental understanding of elementary particle stability

Using Arnold's Classification Theorem applied to a four-dimensional manifold, it is shown that there is only a finite number of ways in which energy can discontinuously change state. It is demonstrated that each of these energy flow pathways can be associated with a distinct elementary particle. The theory not only shows how the formation of particles from the stress-energy present in the space-time manifold can be predicted from first principles, but also that there must exist five fundamental forces in a universe in which discontinuous energy transitions are possible. Finally, the existence of a new, as yet undiscovered particle is predicted, which is associated with this new fifth force.