Gravitational collapse in quantum mechanics with relativistic kinetic energy

It is proved that the quantum mechanical Hamiltonian $\text{H = Σ}{}_{\mathrm{i=1}}{}^{\mathrm{N}}\text{(p}{}^2\text{+ m}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{? κ Σ}{}_{\mathrm{i>j}}\text{|x}{}_{\mathrm{i}}\text{? x}{}_{\mathrm{j}}\text{|}{}^{\mathrm{?1}}$ for bosons (resp, fermions) is bounded from below if N ≤ c_bκ^?1 (resp. $\text{N ≤ c}{}_{\mathrm{f}}\text{κ}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$). H is unbounded from below if N ≥ c_b^lκ^?1 (resp. $\text{N ≥ c}{}_{\mathrm{f}}{}^{\mathrm{l}}\text{κ}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$). The constants c_b and c_b^l (resp. c_f and c_f^l) differ by about a factor 2 (resp. 4).     

Finite-dimensional relativistic quantum mechanics

A finite-dimensional relativistic quantum mechanics is developed by first quantizing Minkowski space. Two-dimensional space-time event observables are defined and quantum microscopic causality is studied. Three-dimensional colored even observables are introduced and second quantized on a representation space of the restricted Poincaré group. Creation, annihilation, and field operators are introduced and a finite-dimensional Dirac theory is presented.     

Gravitational effects in quantum mechanics

To date, both quantum theory and Einstein’s theory of general relativity have passed every experimental test in their respective regimes. Nevertheless, almost since their inception, there has been debate surrounding whether they should be unified, and by now, there exists strong theoretical arguments pointing to the necessity of quantising the gravitational field. In recent years, a number of experiments have been proposed which, if successful, should give insight into features at the Planck scale. Here, we review some of the motivations, from the perspective of semi-classical arguments, to expect new physical effects at the overlap of quantum theory and general relativity. We conclude with a short introduction to some of the proposals being made to facilitate empirical verification.     

Statistical mechanics of relativistic plasmas

Based on the Klimontovich method of construction of a relativistic statistical mechanics the binary correlation function of a relativistic plasma is considered. Up to the order e⁴ for spatially homogeneous systems the relativistic binary correlation function can be expressed by the relativistic dielectric tensor, where higher-order correlation functions are neglected.In the case of thermodynamic equilibrium the relativistic dielectric tensor is calculated. The corresponding binary correlation function is determined up to the first relativistic order. The result is compared with those of Trubnikov/Kosachev (1968) and Krizan (1969).     

On the relativistic invariance of a quantum theory based on beables

We discuss the question of the relativistic invariance of a quantum theory based on beables, and we suggest the general outlines of one possible form of such a theory.     

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.     

Advances in relativistic molecular quantum mechanics

A quantum mechanical equation HΨ=EΨ$H\Psi =E\Psi$ is composed of three components, viz., Hamiltonian H$H$, wave function Ψ$\Psi$, and property E(λ)$E\left(\lambda \right)$, each of which is confronted with fundamental issues in the relativistic regime, e.g., (1) What is the most appropriate relativistic many-body Hamiltonian? How to solve the resulting equation? (2) How does the relativistic wave function behave at the coalescence of two electrons? How to do relativistic explicit correlation? (3) How to formulate relativistic properties properly?, to name just a few. It is shown here that the charge-conjugated contraction of Fermion operators, dictated by the charge conjugation symmetry, allows for a bottom-up construction of a relativistic Hamiltonian that is in line with the principles of quantum electrodynamics (QED). Various approximate but accurate forms of the Hamiltonian can be obtained based entirely on physical arguments. In particular, the exact two-component Hamiltonians can be formulated in a general way to cast electric and magnetic fields, as well as electron self-energy and vacuum polarization, into a unified framework. While such algebraic two-component Hamiltonians are incompatible with explicit correlation, four-component relativistic explicitly correlated approaches can indeed be made fully parallel to the nonrelativistic counterparts by virtue of the ‘extended no-pair projection’ and the coalescence conditions. These findings open up new avenues for future developments of relativistic molecular quantum mechanics. In particular, ‘molecular QED’ will soon become an active and exciting field.     

Quantum mechanics of relativistic spinless particles

A relativistic one-particle, quantum theory for spin-zero particles is constructed uponL ²(x, ct), resulting in a positive definite spacetime probability density. A generalized Schrödinger equation having a Hermitian HamiltonianH onL ²(x, ct) for an arbitrary four-vector potential is derived. In this formalism the rest mass is an observable and a scalar particle is described by a wave packet that is a superposition of mass states. The requirements of macroscopic causality are shown to be satisfied by the most probable trajectory of a free tardyon and a nontrivial framework for charged and neutral particles is provided. The Klein paradox is resolved and a link to the free particle field operators of quantum field theory is established. A charged particle interacting with a static magnetic field is discussed as an example of the formalism.     

Orthomodular lattice in relativistic non-quantum mechanics

In 1975–1980, W. Cegla and A. Z. Jadczyk studied the causality structure of space-time: two points of Minkowski space-timeM are causally independent iff they are different and spacelike or lightlike separated, and the measurable causally closed subsets ofM form an orthomodular lattice. We show that this lattice enables us to model, by a formalism close to the one of orthodox quantum mechanics, a definite experiment in relativistic non-quantum mechanics: the counting of identical point bodies by one or several radars in some particular regions of the space-time.     

Dirac's aether in relativistic quantum mechanics

The introduction by Dirac of a new aether model based on a stochastic covariant distribution of subquantum motions (corresponding to a vacuum state alive with fluctuations and randomness) is discussed with respect to the present experimental and theoretical discussion of nonlocality in EPR situations. It is shown (1) that one can deduce the de Broglie waves as real collective Markov processes on the top of Dirac's aether; (2) that the quantum potential associated with this aether's modification, by the presence of EPR photon pairs, yields a relativistic causal action at a distance which interprets the superluminal correlations recently established by Aspect et al.; (3) that the existence of the Einstein-de Broglie photon model (deduced from Dirac's aether) implies experimental predictions which conflict with the Copenhagen interpretation in certain specific testable interference experiments.     

Stochastic microcausality in relativistic quantum mechanics

A recently formulated concept of stochastic localizability is shown to be consistent with a concept of stochastic microcausality, which avoids the conclusions of Hegerfeldt's no-go theorem as to the inconsistency of sharp localizability of quantum particles and Einstein causality. The proposed localizability on quantum space-time is shown to lead to strict asymptotic causality. For finite time evolutions, upper bounds on propagation to the exterior of stochastic light cones are derived which show that the resulting probabilities are too small to be actually observable in a realistic context.Supported by an NSERC Fellowship.Suported in part by NSERC research grant No. A5206.     

Direct interactions in relativistic statistical mechanics

Directly interacting particles are considered in the multitime formalism of predictive relativistic mechanics. When the equations of motion leave a phase-space volume invariant, it turns out that the phase average of any first integral, covariantly defined as a flux across a 7n-dimensional surface, is conserved. The Hamiltonian case is discussed, a class of simple models is exhibited, and a tentative definition of equilibrium is proposed. Invited paper, dedicated to Prof. Lawrence P. Horwitz on the occasion of his 65th birthday, October 14, 1995.     

Jaumann transport in relativistic continuum mechanics

We define the Jaumann derivative of a tensor field in relativity by a formal generalization of a stress rate in viscoelasticity. A tensor field is said to be Jaumann transported iff its Jaumann derivative vanishes. It is found that the gravitational potentials are Jaumann transported identically. The concept of a complete rotation tensor has been introduced to study the Jaumann derivative with respect to a null vector field. This provides a characterization of the integrability of a hypersurface orthogonal congruence. A perfect fluid collapsing by neutrino emission and undergoing Jaumann transport with respect to the neutrino flow is found to be compatible with that of a catastrophic collapse. The circumstances leading to the existence of ghost neutrinos are cited. The degeneracy of the Kerr-Newman black hole into the Reissner-Nordstrom black hole is expressed in terms of the Jaumann propagation.     

Direct interactions in relativistic statistical mechanics

Directly interacting particles are considered in the multitime formalism of predictive relativistic mechanics. When the equations of motion leave a phase-space volume invariant, it turns out that the phase average of any first integral, covariantly defined as a flux across a 7n-dimensional surface, is conserved. The Hamiltonian case is discussed, a class of simple models is exhibited, and a tentative definition of equilibrium is proposed. Invited paper, dedicated to Prof. Lawrence P. Horwitz on the occasion of his 65th birthday, October 14, 1995.     

牛顿力学形式和相对论力学的协变性

指出若将能量（包括动能）、动量都理解为相对论中的能量和动量，则牛顿力学中的功能原理、动能定理、动量定理、牛顿运动定律及力对物体所作的功率、能量－动量守恒定律及其守恒条件在相对论中都是协变的，并给出了它们的协变形式。