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.     

Covariant Hamiltonian Field Theory: Path Integral Quantization

The Hamiltonian counterpart of classical Lagrangian field theory is covariant Hamiltonian field theory where momenta correspond to derivatives of fields with respect to all world coordinates. In particular, classical Lagrangian and covariant Hamiltonian field theories are equivalent in the case of a hyperregular Lagrangian, and they are quasi-equivalent if a Lagrangian is almost-regular. In order to quantize covariant Hamiltonian field theory, one usually attempts to construct and quantize a multisymplectic generalization of the Poisson bracket. In the present work, the path integral quantization of covariant Hamiltonian field theory is suggested. We use the fact that a covariant Hamiltonian field system is equivalent to a certain Lagrangian system on a phase space which is quantized in the framework of perturbative quantum field theory. We show that, in the case of almost-regular quadratic Lagrangians, path integral quantizations of associated Lagrangian and Hamiltonian field systems are equivalent.     

Covariant canonical quantization of fields and Bohmian mechanics

We propose a manifestly covariant canonical method of field quantization based on the classical De Donder-Weyl covariant canonical formulation of field theory. Owing to covariance, the space and time arguments of fields are treated on an equal footing. To achieve both covariance and consistency with standard non-covariant canonical quantization of fields in Minkowski spacetime, it is necessary to adopt a covariant Bohmian formulation of quantum field theory. A preferred foliation of spacetime emerges dynamically owing to a purely quantum effect. The application to a simple time-reparametrization invariant system and quantum gravity is discussed and compared with the conventional non-covariant Wheeler-DeWitt approach.Received: 11 October 2004, Published online: 6 July 2005PACS: 04.20.Fy, 04.60.Ds, 04.60.Gw, 04.60.-m     

Construction of a covariant spinor field theory

A covariant theory is constructed of a spinor field in a space which is represented by the local topological product of a space X_n and a space of values of a geometrical object η. The covariant nonlinear spinor field theory constructed preserves the principles of the theory of the unified field and is compatible with the theory of gauge fields.     

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.     

Quantum energy inequalities and local covariance II: categorical formulation

We formulate quantum energy inequalities (QEIs) in the framework of locally covariant quantum field theory developed by Brunetti, Fredenhagen and Verch, which is based on notions taken from category theory. This leads to a new viewpoint on the QEIs, and also to the identification of a new structural property of locally covariant quantum field theory, which we call local physical equivalence. Covariant formulations of the numerical range and spectrum of locally covariant fields are given and investigated, and a new algebra of fields is identified, in which fields are treated independently of their realisation on particular spacetimes and manifestly covariant versions of the functional calculus may be formulated.     

Covariant Field Equations in Supergravity

Covariance is a useful property for handling supergravity theories. In this paper, we prove a covariance property of supergravity field equations: under reasonable conditions, field equations of supergravity are covariant modulo other field equations. We prove that for any supergravity there exist such covariant equations of motion, other than the regular equations of motion, that are equivalent to the latter. The relations that we find between field equations and their covariant form can be used to obtain multiplets of field equations. In practice, the covariant field equations are easily found by simply covariantizing the ordinary field equations.     

Covariant derivative on non-linear fiber bundles

A gauge field is usually described as a connection on a principal bundle. It induces a covariant derivative on associated vector bundles, sections of which represent matter fields. In general, however, it is not possible to define a covariant derivative on non-linear fiber bundles, i.e. on those which are not vector bundles. We definelogarithmic covariant derivatives acting on two special non-linear fiber bundles — on the principal bundle and on the local gauge group bundle. The logarithmic derivatives map from sections of these bundles to the sections of the local gauge algebra bundle. Some properties of the logarithmic derivatives are formulated.     

A maxwellian tetrad theory of gravity with preferred frames: A

A theory of gravity is proposed which seeks to mimic maxwellian electromagnetism whilst maintaining the principle of equivalence. The curls of tetrad potentials are taken as field strengths and Maxwell-like free-field equations are set up which contain a non-linear gravitational current term and which are generally covariant but not locally Lorentz covariant. The weak field approximation is solved for static metrical spherical symmetry and solutions constructed which agree with the GR predictions.     

Covariant open string field theory on multiple Dp-branes

We study covariant open bosonic string field theories on multiple Dp-branes by using the deformed cubic string field theory, which is equivalent to string field theory in the proper-time gauge. Constructing the Fock space representations of the three-string vertex and the four-string vertex on multiple Dp-branes, we obtain the field theoretical effective action in the zero-slope limit. On multiple D0-branes, the effective action reduces to the Banks-Fishler-Shenker-Susskind(BFSS) matrix model. We also discuss the relation between open string field theory on multiple D-instantons in the zero-slope limit and the Ishibashi-Kawai-Kitazawa-Tsuchiya(IKKT) matrix model.The covariant open string field theory on multiple Dp-branes could be useful to study the non-perturbative properties of quantum field theories in(p+1)-dimensions in the framework of the string theory. The non-zero-slope corrections may be evaluated systematically by using covariant string field theory.     

Exact quantisation of the relativistic Hopfield model

We investigate the quantisation in the Heisenberg representation of a relativistically covariant version of the Hopfield model for dielectric media, which entails the interaction of the quantum electromagnetic field with the matter dipole fields, represented by a mesoscopic polarisation field. A full quantisation of the model is provided in a covariant gauge, with the aim of maintaining explicit relativistic covariance. Breaking of the Lorentz invariance due to the intrinsic presence in the model of a preferred reference frame is also taken into account. Relativistic covariance forces us to deal with the unphysical (scalar and longitudinal) components of the fields, furthermore it introduces, in a more tricky form, the well-known dipole ghost of standard QED in a covariant gauge. In order to correctly dispose of this contribution, we implement a generalised Lautrup trick. Furthermore, causality and the relation of the model with the Wightman axioms are also discussed.     

Vertex calculation for a composite particle

The classical treatment and the quantization of composite relativistic systems is given a manifestly covariant formulation in presence of constraints. A particular formulation of Feynman's quantum mechanics is used to treat the scattering of composite relativistic systems. A covariant harmonic oscillator model is employed to calculate vertices of interactions: the results are similar to the corresponding ones in the usual field theories, but the presence of some convergence factors gives hope that a theory with composite particles may be finite.     

CONFORMALLY COVARIANT THEORY OF SECOND RANK ANTISYMMETRIC TENSOR FIELD

The conformally covariant field equation on second rank antisymmetric tensor is derived and its conformally covariant energy-momentum tensor is also obtained.     

Covariant extensions and the nonsymmetric unified field

The problem of generally covariant extension of Lorentz invariant field equations, by means of covariant derivatives extracted from the nonsymmetric unified field, is considered. It is shown that the contracted curvature tensor can be expressed in terms of a covariant gauge derivative which contains the gauge derivative corresponding to minimal coupling, if the universal constantp, characterizing the nonsymmetric theory, is fixed in terms of Planck's constant and the elementary quantum of charge. By this choice the spinor representation of the linear connection becomes closely related to the spinor affinity used by Infeld and Van Der Waerden in their generally covariant formulation of Dirac's equation.     

A canonical relativistic approach to quantize electromagnetic field in the presence of moving magneto-dielectric media

A canonical relativistic formulation is introduced to quantize electromagnetic field in the presence of a polarizable and magnetizable moving medium. The medium is modeled by a continuum of the second rank antisymmetric tensors in a phenomenological way. The covariant wave equation for the vector potential and the covariant constitutive equation of the medium are obtained as the Euler-Lagrange equations using the Lagrangian of the total system. A fourth rank tensor which couples the electromagnetic field and the medium is introduced. The susceptibility tensor of the medium is obtained in terms of this coupling tensor. The noise polarization tensor is calculated in terms of both the coupling tensor and the ladder operators of the tensors modeling the medium.     

A Spin-Statistics Theorem for Quantum Fields¶on Curved Spacetime Manifolds¶in a Generally Covariant Framework

A model-independent, locally generally covariant formulation of quantum field theory over four-dimensional, globally hyperbolic spacetimes will be given which generalizes similar, previous approaches. Here, a generally covariant quantum field theory is an assignment of quantum fields to globally hyperbolic spacetimes with spin-structure where each quantum field propagates on the spacetime to which it is assigned. Imposing very natural conditions such as local general covariance, existence of a causal dynamical law, fixed spinor- or tensor type for all quantum fields of the theory, and that the quantum field on Minkowski spacetime satisfies the usual conditions, it will be shown that a spin-statistics theorem holds: If for some of the spacetimes the corresponding quantum field obeys the “wrong” connection between spin and statistics, then all quantum fields of the theory, on each spacetime, are trivial. Received: 1 March 2001 / Accepted: 28 May 2001     

Tensor Distributions on Signature-changing Space-times

Irregularities in the metric tensor of a signature-changing space-time suggest that field equations on such space-times might be regarded as distributional. We review the formalism of tensor distributions on differentiable manifolds, and examine to what extent rigorous meaning can be given to field equations in the presence of signature-change, in particular those involving covariant derivatives. We find that, for both continuous and discontinuous signature-change, covariant differentiation can be defined on a class of tensor distributions wide enough to be physically interesting.     

Cosmology of general relativity without energy-momentum conservation

A modified version of the field equations of general relativity is obtained on relaxing the covariant energy-momentum conservation condition. This introduces a single arbitrary constant and does not appear to upset the successes of general relativity in or outside cosmology. The matter-dominated cosmological model, based on the generalized field equations, is discussed. It is shown to provide more room for consistency with the observational data.     

Covariant models of energy-information processes induced by the gravitational wave field in open hydro-gasdynamic systems

A covariant approach to the investigation of kinetic subsystems subjected to the action of the hydrodynamic component evolutionizing in the gravitational wave field is formulated. The list of model forces which are a covariant generalization of the well-known classical forces as well as the forces of purely gravitational origin is compiled.V. I. Ul'yanov-Lenin State University, Kazan'. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 79–84, June, 1992.