Covariant description of Hamiltonian form for field dynamics

Hamiltonian form of field dynamics is developed on a space-like hypersurface in space-time. A covariant Poisson bracket on the space-like hypersurface is defined and it plays a key role to describe every algebraic relation into a covariant form. It is shown that the Poisson bracket has the same symplectic structure that was brought in the covariant symplectic approach. An identity invariant under the canonical transformations is obtained. The identity follows a canonical equation in which the interaction Hamiltonian density generates a deformation of the space-like hypersurface. The equation just corresponds to the Yang-Feldman equation in the Heisenberg pictures in quantum field theory. By converting the covariant Poisson bracket on the space-like hypersurface to four-dimensional commutator, we can pass over to quantum field theory in the Heisenberg picture without spoiling the explicit relativistic covariance. As an example the canonical QCD is displayed in a covariant way on a space-like hypersurface.     

Invariance, Symmetry and Meaning

The role of the concept of invariance in physics and geometry is analyzed, with attention to the closely connected concepts of symmetry and objective meaning. The question of why the fundamental equations of physical theories are not invariant, but only covariant, is examined in some detail. The last part of the paper focuses on the surprising example of entropy as a complete invariant in ergodic theory for any two ergodic processes that are isomorphic in the measure-theoretic sense.     

CONNECTION THEORY OF FIBRE BUNDLE AND PROLONGATION STRUCTURES OF NONLINEAR EVOLUTION EQUATIONS

It is shown that the proper geometrical framework for the nonlinear evolution equations (NEEs) and the soliton equations Should be the fibre bundle theory, the principal bundle and its associated bundle and their connection theory. Based upon the requirement of covariance of the geometrical quantities, a covariant generic geometry theory for the prolongation strutures of the NEEs is proposed and the fundamental equations for the prolongation structures are presented. From the fundamental equations it immediately follows that the comections corresponding to these NEEs always flat but with torsion and the covariant formulae satisfied by the conservation quantities associated with these NEEs are obtained. The prolongation structure of the MKdV equation, as an example, is concretely worked out by means of the covariant theory of the prolongation structure presented in this paper.     

Exact Solutions of Covariant Wave Equations with a Multipole Source Term on Curved Spacetimes

A formalism is presented for calculating exactsolutions of covariant inhomogeneous scalar and tensorwave equations whose source terms are arbitrary ordermultipoles on a curved background spacetime. The developed formalism is based on the theory ofthe higher-order fundamental solutions for wave equationwhich are the distributions that satisfy theinhomogeneous wave equation with the corresponding order covariant derivatives of the Dirac deltafunction on the right-hand side. Like the classicalGreen's function for a scalar wave equation, thehigher-order fundamental solutions contain a direct termwhich has support on the light cone as well as a tailterm which has support inside the light cone. Knowinghow to compute the fundamental solutions of arbitraryorder, one can find exact multipole solutions of wave equations on curved spacetimes. Wepresent complete recurrent algorithms for calculatingthe arbitrary-order fundamental solutions and the exactmultipole solutions in a form convenient for practical computations. As an example we apply thealgorithm to a massless scalar wave field on aparticular Robertson-Walker spacetime.     

Covariant Kolmogorov equation and entropy current for the relativistic Ornstein-Uhlenbeck process

The relativistic Ornstein-Uhlenbeck Process (ROUP), which is a toy-model of relativistic irreversible phenomena, is studied statistically in an explicitly covariant manner. An 8-dimensional phase space is introduced (four dimensions for space-time coordinates, and four dimensions for the 4-momentum coordinates), on which `extended' probability distributions are defined (the usual probability distribution is recovered as their restriction to the mass shell). An explicitly covariant Kolmogorov equation is derived for these `extended' probability distributions. The whole formalism is used to introduce a 4-current of conditional entropy and prove that the 4-divergence of this 4-current is always positive. This constitutes an H-theorem for the ROUP. Received 30 May 2001     

Covariant Relativistic Statistical Mechanics of Many Particles

In this paper the quantum covariant relativistic dynamics of many bodies is reconsidered. It is emphasized that this is an event dynamics. The events are quantum statistically correlated by the global parameter τ. The derivation of an event Boltzmann equation emphasizes this. It is shown that this Boltzmann equation may be viewed as exact in a dilute event limit ignoring three event correlations. A quantum entropy principle is obtained for the marginal Wigner distribution function. By means of event linking (concatenations) particle properties such as the equation of state may be obtained. We further reconsider the generalized quantum equilibrium ensemble theory and the free event case of the Fermi-Dirac and Bose-Einstein distributions, and some consequences. The ultra-relativistic limit differs from the non-covariant theory and is a test of this point of view.     

Lorentz-covariant dissipative lagrangian systems

The concept of dissipative hamiltonian system is converted to Lorentz-covariant form, with evolution generated jointly by two scalar functionals, the lagrangian action and the global entropy. A bracket formulation yields the local covariant laws of energy-momentum conservation and of entropy production. The formalism is illustrated by a derivation of the covariant Landau kinetic equation.     

Gauge Covariant Fermion Propagator in the Presence of Arbitrary External Gauge Field and Its Schwinger--Dyson Equation

Gauge covariance for Green＇s functions of a gauge theory through a fermion propagator in the presence of arbitrary external gauge field is proven and a formalism of gauge and Lorentz covariant Schwinger-Dyson equation for the fermion propagator with external gauge field is built up within ladder approximation.     

Thermodynamics and relativistic kinetic theory for <Emphasis Type="Italic">q</Emphasis>-generalized Bose–Einstein and Fermi–Dirac systems

The thermodynamics and covariant kinetic theory are elaborately investigated in a non-extensive environment considering the non-extensive generalization of Bose–Einstein (BE) and Fermi–Dirac (FD) statistics. Starting with Tsallis’ entropy formula, the fundamental principles of thermostatistics are established for a grand canonical system having q-generalized BE/FD degrees of freedom. Many particle kinetic theory is set up in terms of the relativistic transport equation with q-generalized Uehling–Uhlenbeck collision term. The conservation laws are realized in terms of appropriate moments of the transport equation. The thermodynamic quantities are obtained in a weak non-extensive environment for a massive pion–nucleon and a massless quark–gluon system with non-zero baryon chemical potential. In order to get an estimate of the impact of non-extensivity on the system dynamics, the q-modified Debye mass and hence the q-modified effective coupling are estimated for a quark–gluon system.     

Entropy of the FRW cosmology based on the brick wall method

The brick wall method in calculations of the entropy of black holes can be applied to the FRW cosmology in order to study the statistical entropy. An appropriate cutoff satisfying the covariant entropy bound can be chosen so that the entropy has a definite bound. Among the entropy for each of cosmological eras, the vacuum energy-dominated era turns out to give the maximal entropy which is in fact compatible with assumptions from the brick wall method.     

General relativistic Boltzmann equation, I: Covariant treatment

This series of two articles aims at dissipating the rather dense haze existing in the present literature around the General Relativistic Boltzmann equation. In this first article, the general relativistic one-particle distribution function in phase space is defined as an average of delta functions. Thereupon, the general relativistic Boltzmann equation, to be obeyed by this function, is derived. The use of either contravariant or covariant momenta leads to different, but equivalent, forms of the equation.The results of the present article are covariant, but not manifestly covariant. The transition to a manifestly covariant treatment, on the basis of off-shell momenta, is given in the second article.     

Geometrization of the electromagnetic field and dark matter

Fundamental concepts, symmetries and dynamic equations of the theory of dark matter are derived from the simple relation: everything in the concept of space and the concept of space in everything. It is shown that the electromagnetic field is the singlet state of the dark matter field and, hence, the last may be considered as a generalized electromagnetic field (shortly gef) and a simple solution is given to the old problem of connecting the electromagnetic field with geometric properties of the physical manifold itself. It is shown that gauge fixing renders the generalized electromagnetic field effectively massive while the Maxwell electromagnetic field remains massless. To learn more about interactions between matter and dark matter on the microscopic level (and to recognize the fundamental role of internal symmetry in this case), the general covariant Dirac equation is derived and its natural generalization is considered. The experiment is suggested to test the formulated theory.     

A Covariant Hierarchy of Kinetic Equations for Classical Particles and Fields

It is shown that several problems, that are met when constructing a covariant statistical theory, can be circumverted demanding that only the resulting equation for the macroscopic entities are covariant whereas only mathematical meaning is ascribed to concepts as phase space and ensemble density. Accordingly a hierarchy of equations for averaged particle distributions and fields and their correlations is derived.     

De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory

A quantization of field theory based on the De Donder-Weyl (DW) covariant Hamiltonian formulation is discussed. A hypercomplex extension of quantum mechanics, in which the space-time Clifford algebra replaces that of the complex numbers, appears as a result of quantization of Poisson brackets on differential forms which were put forward for the DW theory earlier. The proposed covariant hypercomplex Schrödinger equation is shown to lead in the classical limit to the DW Hamilton-Jacobi equation and to obey the Ehrenfest principle in the sense that the DW canonical field equations are satisfied for expectation values of properly chosen operators.     

Minimal gravitational coupling in the Newtonian theory and the covariant Schrödinger equation

The role of the Bargmann group (11-dimensional extended Galilei group) in nonrelativistic gravitation theory is investigated. The generalized Newtonian gravitation theory (Newton-Cartan theory) achieves the status of a gauge theory about as much as general relativity and couples minimally to a complex scalar field leading to a four-dimensionally covariant Schrödinger equation. Matter current and stress-energy tensor follow correctly from the Lagrangian. This theory on curved Newtonian space-time is also shown to be a limit of the Einstein-Klein-Gordon theory.Partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. A8059.     

Four‐dimensional couplings among BF and matter theories from BRST cohomology

The local and manifestly covariant Lagrangian interactions in four spacetime dimensions that can be added to a “free” model that describes a generic matter theory and an abelian BF theory are constructed by means of deforming the solution to the master equation on behalf of specific cohomological techniques.     

Covariant canonical quantization

We present a manifestly covariant quantization procedure based on the de Donder–Weyl Hamiltonian formulation of classical field theory. This procedure agrees with conventional canonical quantization only if the parameter space is d=1 dimensional time. In d>1 quantization requires a fundamental length scale, and any bosonic field generates a spinorial wave function, leading to the purely quantum-theoretical emergence of spinors as a byproduct. We provide a probabilistic interpretation of the wave functions for the fields, and we apply the formalism to a number of simple examples. These show that covariant canonical quantization produces both the Klein–Gordon and the Dirac equation, while also predicting the existence of discrete towers of identically charged fermions with different masses. Covariant canonical quantization can thus be understood as a “first” or pre-quantization within the framework of conventional QFT. PACS 04.62.+v; 11.10.Ef; 12.10.Kt     

General relativistic Boltzmann equation, II: Manifestly covariant treatment

In a preceding article we presented a general relativistic treatment of the derivation of the Boltzmann equation. The four-momenta occurring in this formalism were all on-shell four-momenta, verifying the mass-shell restriction p²=m²c². Due to this restriction, the resulting Boltzmann equation, although covariant, turned out to be not manifestly covariant. In the present article we switch from mass-shell momenta to off-shell momenta, and thereby arrive at a Boltzmann equation that is manifestly covariant.     

Gravitational law and spinning electron equation in a De Sitter symmetric space

A gravitational law is proposed on a De Sitter covariant space. Dirac's De Sitter covariant spinning electron equation is generalized to the presence of gravitational fields. The resulting equation differs from the generally covariant Dirac equation by a mass renormalization. The last result is a generalization of that of Gürsey and Lee [2] in case of the homogenous De Sitter metric, and it gives a wider outlook on the significance of this result from the point of view of gauge theory.Dedicated to Achille Papapetrou on the occasion of his retirement.     

Relativistically covariant Bohm-Bub hidden-variable theory for spin measurement of a single particle

We present a simple first step toward a relativistically covariant generalization of the Bohm-Bub hidden-variable theory. The model is applicable to spin measurement on a single Dirac particle and describes the collapse of the state vector to a spin-up or spin-down state. The essential postulate is that the hidden-variable vector transforms in the same way as the state vector under a Lorentz transformation. This yields a covariant collapse equation, which reduces to the ordinary Bohm-Bub equation for an observer stationary with respect to the particle and shows a time dilated collapse for a moving observer. The model yields the correct quantum transition probabilities for all observers.Deceased.