The foundations of relativity

In a previous paper a stochastic foundation was proposed for microphysics: the nonrelativistic and relativistic domains were shown to be connected with two different approximations of diffusion theory; the relativistic features (Lorentz contraction for the coordinate standard deviation, covariant diffusion equation) were not derived from the relativistic formalism introduced at the start, but emerged from diffusion theory itself. In the present paper these results are given a new presentation, which aims at elucidating not the foundations of quantum mechanics, but those of relativity. This leads to a discussion of points still controversial in the interpretation of relativity. In particular two problems appear in a new light: the character of time and length alterations, and the privileged role of the velocityc. Besides, the question of a possible limitation of relativity (and more generally of the laws of mechanics) in the domain of particle substructure is raised and supported by exemples drawn from the hydrodynamical model of a spinned particle. Suggestions are presented for the possibility of a deeper conceptual unification of special and general relativity.     

Causal independence

Causal independence of the simultaneous positions and momenta of two distinguishable particles in nonrelativistic physics and causal independence of events in two relatively spacelike regions of space-time in relativity are analyzed and discussed. This review paper formulates causal independence in a general and operational way and summarizes the inferences drawn from it in non-relativistic quantum mechanics, classical relativistic point mechanics, quantum field theory, and classical field theory. Special attention is given to the open question of the relationship between local independence and commutativity in quantum field theory.Work performed under the auspices of the U.S. Atomic Energy Commission.     

Quantum relativity theory and quantum space-time

A quantum relativity theory formulated in terms of Davis' quantum relativity principle is outlined. The first task in this theory as in classical relativity theory is to model space-time, the arena of natural processes. It is shown that the quantum space-time models of Banai introduced in another paper is formulated in terms of Davis' quantum relativity. The recently proposed classical relativistic quantum theory of Prugoveki and his corresponding classical relativistic quantum model of space-time open the way to introduce, in a consistent way, the quantum space-time model (the quantum substitute of Minkowski space) of Banai proposed in the paper mentioned. The goal of quantum mechanics of quantum relativistic particles living in this model of space-time is to predict the rest mass system properties of classically relativistic (massive) quantum particles (elementary particles). The main new aspect of this quantum mechanics is that provides a true mass eigenvalue problem, and that the excited mass states of quantum relativistic particles can be interpreted as elementary particles. The question of field theory over quantum relativistic model of space-time is also discussed. Finally it is suggested that quarks should be considered as quantum relativistic particles.Supported by the Hungarian Academy of Sciences.     

Quantum Space-Time: A Review

The review presents systematically the results of studies which develop an idea of quantum properties of space-time in the microworld or near exotic objects (black holes, magnetic monopoles and others). On the basis of this idea motion equations of nonrelativistic and relativistic particles are studied. It is shown that introducing concept of quantum space-time at small distances (or near superdense matter) leads to an additional force giving rise to appearance of spiral-like behaviour of a particle along its classical trajectory. Given method is generalized to nonrelativistic quantum mechanics and to motion of a particle in gravitational force. In the latter case, there appears to be an antigravitational effect in the motion of a particle leading to different value of free-fall time (at least for gravitational force of exotic objects) for particles with different masses. Gravitational consequences of quantum space-time and tensor structures of physical quantities are inveatigated in detail. From experimental data on testing relativity and anisotropy of inertia estimation L ≦ 10⁻²² cm on the value of the fundamental length is obtained.     

Consistency of relativistic particle theories

While direct-interaction particle theories are generally thought to be incompatible with relativity in classical physics, such relativistic theories in quantum mechanics have recently attracted attention. The reasons for rejecting these theories in classical physics are based on the consideration of world lines, while relativistic quantum mechanics has no covariant position operator so that the classical refuting argument cannot be adapted.This paper discusses the consistency of relativistic particle theories with a finite number of degrees of freedom, without recourse to the position operator. A particle is described by a sub-algebra of observables at one time. Homogeneous transformations, including accelerations, must preserve the identity of particles, and therefore leave the sub-algebras invariant. It is shown that with this assumption only non-interacting particle theories are compatible with the principle of relativity, in quantum as well as classical mechanics.     

Space-time trajectories of quantum particles

The concept of trajectory is extended theoretically from classical mechanics through nonrelativistic and relativistic quantum mechanics. Forced motion of the particle as might be caused by an electromagnetic field is included in the equations. A new interpretation of the electromagnetic potential and the gauge transformation is presented. Using this formal structure, the problem of collecting particles into packets using trajectories is studied for both quantum mechanics and classical mechanics. Quantum mechanical trajectories are found to be significantly more restricted than those allowed by classical physics. The uncertainty principle comes from the second-order nature of the field equation without recourse to statistical arguments. The trajectories of particles in a quantum state can be calculated explicitly from the wave function (also see article in Volume 20, Number 6).     

The reality and dimension of space and the complexity of quantum mechanics

The dimension (and signature) of space is a result of distances being real numbers and quantum mechanical state functions being complex ones; it is an inescapable consequence of quantum mechanics and group theory. So nonrelativistic quantum mechanics cannot be complete (it requiresad hoc additional assumptions) and consistent (nor can classical physics), leading to relativity, quantum mechanics, and field theory. Implications of the constraints of consistency and physical reasonableness and of group theory for the structure of these theories are considered. It appears that there are simple, perhaps unavoidable reasons for the laws of physics, the nature of the world they describe, and the space in which they act.     

Gauge invariance,lorentz covariance and the observer

We discuss a number of questions related to the role of the observer in classical and quantum theories of fields, in particular electrodynamics. We find the gauge-independent parts of the electromagnetic potential, which are classical observables, both in a non-covariant manner and in a Lorentz covariant, observer-dependent way. We present an analysis of the probabilistic interpretation of relativistic quantum mechanics, similar to that of the nonrelativistic theory, and discuss the gauge invariance of the corresponding probability amplitudes.     

Trace Dynamics and a non-commutative special relativity

Trace Dynamics is a classical dynamical theory of non-commuting matrices in which cyclic permutation inside a trace is used to define the derivative with respect to an operator. We use the methods of Trace Dynamics to construct a non-commutative special relativity. We define a line-element using the Trace over space–time coordinates which are assumed to be operators. The line-element is shown to be invariant under standard Lorentz transformations, and is used to construct a non-commutative relativistic dynamics. The eventual motivation for constructing such a non-commutative relativity is to relate the statistical thermodynamics of this classical theory to quantum mechanics.     

Current density in relativistic quantum mechanics

We extend to relativistic theories the concepts of probability density and probability current density of nonrelativistic quantum mechanics, together with the charge and current densities that are used as sources of the electromagnetic field in the semi-classical theory of radiation. There are some limitations in the procedure, especially in the case of several particles.     

A probabilistic analysis of the difficulties of unifying quantum mechanics with the theory of relativity

A procedure is given for the transformation of quantum mechanical operator equations into stochastic equations. The stochastic equations reveal a simple correlation between quantum mechanics and classical mechanics: Quantum mechanics operates with “optimal estimations,” classical mechanics is the limit of “complete information.” In this connection, Schrödinger's substitution relationsp _x → -i? ?/?x, etc, reveal themselves as exact mathematical transformation formulas. The stochastic version of quantum mechanical equations provides an explanation for the difficulties in correlating quantum mechanics and the theory of relativity: In physics “time” is always thought of as a numerical parameter; but in the present formalism of physics “time” is described by two formally totally different quantities. One of these two “times” is a numerical parameter and the other a random variable. This last concept of time shows all the properties required by the theory of relativity and is therefore to be considered as the relativistic time.     

On classical and quantum relativistic dynamics

A canonical formalism for the relativistic classical mechanics of many particles is proposed. The evolution equations for a charged particle in an electromagnetic field are obtained and the relativistic two-body problem with an invariant interaction is treated. Along the same line a quantum formalism for the spinless relativistic particle is obtained by means of imprimitivity systems according to Mackey theory. A quantum formalism for the spin-1/2 particle is constructed and a new definition of spin1/2 in relativity is proposed. An evolution equation for the spin-1/2 particle in an external electromagnetic field is given. The Bargmann Michel, and Telegdi equation follows from this formalism as a quasiclassical approximation. Finally, a new relativistic model for hydrogenlike atoms is proposed. The spectrum predicted is in agreement with Dirac's when radiative corrections have been added.     

Evolution equation of the optical tomogram for arbitrary quantum Hamiltonian and optical tomography of relativistic classical and quantum systems

The dynamic equation for the optical tomogram of nonrelativistic quantum system with an arbitrary Hamiltonian is obtained. The kinetic equation in the classical relativistic kinetics is discussed, and its optical tomography representation is obtained. Dynamic equations for the Wigner functions of relativistic spinless quantum particles in electromagnetic and scalar fields are obtained. Optical tomographic-distribution functions of weakly relativistic spinless quantum particles are introduced, and dynamic equations for these functions in weak electric and scalar fields are obtained.     

Relativistic Quantum Mechanics and the Bohmian Interpretation

No Heading Conventional relativistic quantum mechanics, based on the Klein-Gordon equation, does not possess a natural probabilistic interpretation in configuration space. The Bohmian interpretation, in which probabilities play a secondary role, provides a viable interpretation of relativistic quantum mechanics. We formulate the Bohmian interpretation of many-particle wave functions in a Lorentz-covariant way. In contrast with the nonrelativistic case, the relativistic Bohmian interpretation may lead to measurable predictions on particle positions even when the conventional interpretation does not lead to such predictions.     

Space-Time Grains: Roots of Special and Doubly Special Relativity

We show that the special relativistic dynamics when combined with quantum mechanics and the concept of superstatistics can be interpreted as arising from two interlocked non-relativistic stochastic processes that operate at different energy scales. This interpretation leads to Feynman amplitudes that are in the Euclidean regime identical to transition probability of a Brownian particle propagating through a granular space. Some kind of spacetime granularity could be therefore held responsible for the emergence at larger scales of various symmetries. For illustration we consider also the dynamics and the propagator of a spinless relativistic particle. Implications for doubly special relativity, quantum field theory, quantum gravity and cosmology are discussed.     

Relativistic quantum bit commitment in real-time

A relativistic quantum exchange protocol making it possible to implement a bit commitment scheme is realized. The protocol is based on the idea that in the relativistic case the propagation of a field into a region of space accessible for measurement requires, in contrast to the nonrelativistic case, a finite time that depends on the structure of the states. The protocol requires one classical and several quantum communication channels. It turns out that it is possible in principle to preserve the secret bit for as long a period of time desired and with probability as close to 1 as desired.     

Adiabatic theorem and spectral concentration

The spectral concentration of arbitrary order for the Stark effect is proved to exist for a large class of Hamiltonians appearing in nonrelativistic and relativistic quantum mechanics. The results are consequences of an abstract result about the spectral concentration for self-adjoint operators. A general form of the adiabatic theorem of quantum mechanics, generalizing an earlier result of the author as well as some results by Lenard, is also proved.     

Quantum mechanics as relativistic statistics. I: The two-particle case

It is shown that non-relativistic quantum mechanics can be treated as a kind of relativistic statistical theory, which describes the indeterministic motion of classical particles. The theory is relativistic in the sense that the relativistic notion of the state and two-time equations of motion are used. The principles and relations of quantum mechanics are obtained from the principles of statistics and those of classical mechanics.