Toward the Born-Weyl Quantization of Fields

Elements of the quantization in field theorybased on the covariant polymomentum Hamiltonianformalism, a possibility of which was originallydiscussed in 1934 by Born and Weyl, are developed. Theapproach is based on a recently proposed Poisson bracketon differential forms. A covariant analogue of theSchrodinger equation for a hypercomplex wave function isput forward. A possible relation to the functional Schrodinger picture in quantum field theory isoutlined.     

Geometric quantization and quantum field theory in curved space-time

The Kostant-Souriau geometric quantization theory is applied to the problem of constructing a generally covariant quantum field theory. The occupation number formalism for a scalar field is introduced as a semiclassical approximation which is valid in low curvature regions of space-time and which depends on making a particular choice of polarization in the classical phase space of a single massive particle. The application of the formalism to particle creation problems is outlined.     

Toward a Finite-Dimensional Formulation of Quantum Field Theory

Rules of quantization and equations of motion for a finite-dimensional formulation of quantum field theory are proposed which fulfill the following properties: (a) Both the rules of quantization and the equations of motion are covariant; (b) the equations of evolution are second order in derivatives and first order in derivatives of the spacetime coordinates; and (c) these rules of quantization and equations of motion lead to the usual (canonical) rules of quantization and the (Schrödinger) equation of motion of quantum mechanics in the particular case of mechanical systems. We also comment briefly on further steps to fully develop a satisfactory quantum field theory and the difficuties which may be encountered when doing so.     

Precanonical quantization of Yang-Mills fields and the functional Schrödinger representation

Precanonical quantization of pure Yang-Mills fields, which is based on the covariant De Donder-Weyl (DW) Hamiltonian formulation, and its connection with the functional Schrödinger representation in the temporal gauge are discussed. The mass gap problem is related to the finite-dimensional spectral problem for a generalized Clifford-valued magnetic Schrödinger operator which represents the DW Hamiltonian operator.     

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     

Strings, world-sheet covariant quantization and Bohmian mechanics

The covariant canonical method of quantization based on the De Donder–Weyl covariant canonical formalism is used to formulate a world-sheet covariant quantization of bosonic strings. To provide the consistency with the standard non-covariant canonical quantization, it is necessary to adopt a Bohmian deterministic hidden-variable equation of motion. In this way, string theory suggests a solution to the problem of measurement in quantum mechanics. PACS 11.25.-w; 04.60.Ds; 03.65.Ta     

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     

Unified Field Theory from Enlarged Transformation Group. The Consistent Hamiltonian

A theory has been presented previously in which the geometrical structure of a real four-dimensional space time manifold is expressed by a real orthonormal tetrad, and the group of diffeomorphisms is replaced by a larger group. The group enlargement was accomplished by including those transformations to anholonomic coordinates under which conservation laws are covariant statements. Field equations have been obtained from a variational principle which is invariant under the larger group. These field equations imply the validity of the Einstein equations of general relativity with a stress-energy tensor that is just what one expects for the electroweak field and associated currents. In this paper, as a first step toward quantization, a consistent Hamiltonian for the theory is obtained. Some concluding remarks are given concerning the need for further development of the theory. These remarks include discussion of a possible method for extending the theory to include the strong interaction.     

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.     

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.     

Gauge quantization for spin-32 fields

The free massless Rarita-Schwinger equation and a recently constructed interacting field theory known as supergravity are invariant under fermionic gauge transformations. Gauge field quantization techniques are applied in both cases. For the free field the Faddeev-Popov ansatz for the generating functional is justified by showing that it is equivalent to canonical quantization in a particular gauge. Propagators are obtained in several gauges and are shown to be ghost-free and causal. For supergravity the Faddeev-Popov ansatz is presented and the gauge fixing and determinant terms are discussed in detail in a Lorentz covariant gauge. The Slavnov-Taylor identity is obtained. It is argued that supergravity theory is free from the difficulty of acausal wave propagation of the type found by Velo and Zwanziger and that pole residues in tree approximation S-matrix elements are positive as required by unitarity.     

Langevin equation in field theory

A new approach to quantum field theory is developed based on the Langevin equation (stochastic quantization). Applications to conventional and gauge theories are discussed, as well as various extensions; the Langevin difference equation, the complex Langevin equation in Minkowski space, etc.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 66–76, March, 1986.     

Stochastic quantization of supersymmetric field theory

The Parisi-Wu stochastic quantization method is applied to supersymmetric field theory. The Langevin equation, which reproduces the Green functions of euclidean field theory, is written in terms of superfields. Supersymmetric U(1) theory under gauge fixing and the large N reduction in chiral SU(N) theory are discussed. Regularization based on the stochastic method is studied also.     

Covariant unconstrained superfield action for the linearized D = 10 super-Yang-Mills theory

A new type of D = 10 harmonic superspace with two generations of harmonics allows us to reduce the D = 10, N = 1 Brink-Schwarz (BS) superpaticle to a system whose constraints are all first class, functionally independent and Lorentz-covariant. Given these properties, the covariant BFV-BRST quantization of the system is straightforward. By second quantizing this system, we circumvent the no-go theorem which forbids the existence of a covariant off-shell unconstrained superfield action for the linearized D = 10 super-Yang-Mills theory.     

Two-component theory of fermions

A variation of the theory of fermions is proposed in which the fermions are described by two-component spinors obeying a relativistic equation of the second order. In order to make the probability density of the spin-1/2 particles positive definite, the rule is established that complex conjugation of functions and Hermitian conjugation of operators are accompanied by the operation of spatial reflection. A one-particle theory in Hamiltonian form, a Lagrangian formalism for a free two-component field, and the second quantization of the theory are derived. For calculations in quantum electrodynamics a Hamiltonian is proposed similar to the interaction Hamiltonian of spinless particles with the electromagnetic field but containing a spin-dependent part.Translated from Izvestiya VUZ. Fizika, No. 5, pp. 53–58, May, 1971.     

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.     

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.     

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.