General Coordinate Transformations Research Articles

SUPERSYMMETRIC EXTENSION OF LOCAL LORENTZ SYMMETRY IN THE VIERBEIN FORMALISM OF EINSTEIN GRAVITY

The supersymmetric extension of local Lorentz symmetry in the vierbein formalism is presented in detail. In contrast to supergravity, this new supersymmetry is purely internal, that is, totally irrelevant to general coordinate transformations. Because of its non-Poincaré nature, the degrees of freedom of bosons and fermions need not be equal and it is unnecessary to introduce gravitino. Instead, it is important to introduce a new type of nonlinear realization, called “ξ-field realization”. The Einstein Lagrangian density is trivially superinvariant and the Dirac theory can be supersymmetrized. A globally superinvariant gauge-fixing plus FP-ghost Lagrangian density is constructed in the BRS-invariant way. In the simplest case, canonical quantization is explicitly carried out and unitarity is proved. It is shown that the manifestly covariant canonical formalism of quantum gravity is extended to the supersymmetric form without spoiling its beauty in contrast to supergravity. This supersymmetric theory provides a natural link between quantum gravity and gauge theory; its implies that the chiral gauge group should be SO (N).

Path-integral measure for topological field theories

We discuss how the dependence of the path-integral measure on the metric affects the properties of topological quantum field theories. We show that the choice of an invariant measure (under general coordinate transformations) preserves the topological character of these theories. We also discuss how topological invariants should be computed within this approach.

Superfield formulation of superparticles

A superfield action for the relativistic massless superparticle as a spinning particle is presented in the new gauge. The symmetries in the relativistic superparticle theory, that is to say, the invariances under, the reparametrization and the local supersymmetry-transformation in the parameter space, are manifest in this formalism. It is clear how these two kinds of transformations descend from a unified origin to be called “super-reparametrization”, which is a restricted form of the general coordinate transformation in the superspace. The action is manifestly invariant under these transformation by its constructions The minimal coupling with electromagnetic field is also constructed in the superfield formalism, and a manifestly invariant superfield action for the interacting superparticle is presented in our gauge. The formalism is extended to the construction of an action for theN=2 superparticle.

A Note on \overlinePoincare Gauge Theory of Gravity: Additional Comments

In a series of papers/H) one (T. K.) of the present authors has proposed a theory, which has been named a Poincare gauge theory of gravity. In Refs. 3)~5), special relativistic limits and conserved quantities have been discussed. In considerations of generators in Refs. 3) and 5), however, the coordinate transformations have been restricted to those which agree with Poincare transformations ,at spatial infinity. In view of the invariance of our action under the general coordinate transformations, we give the following discussion in the present note:*) (i) The coordinate transformation (III· 7 ·1) discussed in Refs. 3) and 5) is extended to**)

Approach to general covariance in string space of BRST string field theory.

We show that Becchi-Rouet-Stora-Tyutin (BRST) quantized string field theory can be made invariant under string-dependent general coordinate transformations x'/sup ..mu..//sup sigma//sup =/ = x'/sup ..mu..//sup sigma//sup =/(x(sigma)), where ..mu.. = 0,1,...,D-1 and 0less than or equal tosigma/sub 1/less than or equal to..pi... This invariance is achieved by introducing a string-space metric tensor g/sub ..mu..//sub sigma//sub >1/,..nu..sigma/sub =/(x(sigma)) and following the usual steps of general relativity, with the following differences: A constraint, which we term ''tangent contravariance,'' must be imposed on the coordinate transformations and metric tensor. This restriction emerges from requiring that partial/sub sigma/x/sup ..mu../ transforms in string space as a contravariant vector. Further stringent constraints on the metric tensor, including Ricci flatness, arise from demanding nilpotence of the BRST charge.

General statistics

We discuss statistical systems with infinitely many degrees of freedom. In distinction to conventional statistical theories we do not make any a priori identification between operators of the system and given observables. In particular, these systems are not formulated in a preexisting spacetime. Space and time should arise as dynamical structures of the system. It is shown that all such systems are equivalent and the theory is therefore unique. The action loses its fundamental meaning. The choice of an action corresponds to a “coordinate choice” on how the system is parametrized. The symmetry of the action consists of infinite-dimensional volume-conserving general coordinate transformations. The expectation values of operators are finite by definition. The emphasis of the work to be done lies in the identification of structures between operators which are reflected by our observation of space, time, particles etc. String theories are contained in these structures.

A GENERAL COVARIANT DERIVATION OF ELECTRON ENERGY GAIN IN A LASER ACCELERATOR

In this paper, we derive an expression of energy gain in a laser electron accelerator by using a general covariant coordinate transformation. It is pointed out that even when the external gravitational field, is absent, the electrons will still be additionally accelerated by the influence of background medium. In the vacunm case, the obtained results agree with those obtained by the Lorentz transformation.

The relationship between the conformal and gravitational anomalies

It is shown that the vectorial part of the general coordinate transformation (GCT) anomaly cannot be eliminated by local counterterms without generating the conformal anomaly for the system with classical GCT and conformal symmetries. The vectorial part of the GCT anomaly is equivalent to the non-topological conformal anomaly similarly as the chiral part of the GCT anomaly is equivalent to the topological Lorentz anomaly. In two dimensions, the conformal anomaly is topological, and equivalent to the Lorentz anomaly up to the factor i for the left (or right) part contribution only.

Simulation of environmental flow problems in geometrically complex domains. Part I: A general coordinate transformation

A two-part presentation is given in which a combination of domain splitting, coordinate transformation, and an implicit algorithm is suggested as a versatile three-dimensional program for accurate simulation of geometrically complex environmental flow problems. In part 1, a numerical method for solving the time-dependent laminar Navier-Stokes equations posed in general curvilinear coordinates is presented. The issues of equation form, consistency, and accuracy are considered as they relate to numerical simulation of incompressible flows on curvilinear grid systems. The method is applied to flows in square and polar cavities and evaluated in terms of corresponding benchmark physical and numerical experiments.

The CSDR approach to covariant quantisation of vielbein gauge fields

The formalism of coset space dimensional reduction is extended to vielbein gauge fields, carrying both general coordinate and local Weyl transformations. Solutions of the CSDR constraints for BRST supersymmetry over a (D+2)-dimensional graded superspace (xmu , theta , theta ) are superfields whose components are the usual vielbein, ghost and multiplier fields for general coordinate and Weyl invariance, plus an additional conventional scalar field. The Einstein action in D+2 dimensions reduces in D dimensions to that of a Dicke-Brans-Jordan model with a dimensionless coupling constant=1/2.

Constraint algebra for interacting quantum systems

Abstract We consider relativistic constrained systems interacting with external fields. We provide physical arguments to support the idea that the quantum constraint algebra should be the same as in the free quantum case. For systems with ordering ambiguities this principle is essential to obtain a unique quantization. This is shown explicitly in the case of a relativistic spinning particle, where our assumption about the constraint algebra plus invariance under general coordinate transformations leads to a unique S -matrix.

The algebra of the classical Hamiltonian mechanics as the closure of two finite-dimensional algebras

It is shown that any generator of the infinite-dimensional canonical transformation group in the Hamiltonian formulation of classical mechanics can be obtained as a linear combination of repeated commutators of generators of two finite-dimensional subgroups. This result has some structural similarity with the Ogievetsky theorem (1973) concerning the algebra of the general coordinate transformation group in the theory of relativity.

Operator ordering and supersymmetry

We investigate supersymmetric quantum mechanics in curved space. In particular we discuss, in the framework of the canonical formulation, the role of supersymmetry (SUSY) and of invariance under general coordinate transformations (GCT) in solving the operator ordering ambiguity. It is shown that GCT invariance is instrumental to achieve such result. We also examine the connection with the Weyl ordering of operators and the relevance of our results for the path integral approach.

A Gauge Invariant Action Containing Both Open and Closed String Fields

An interesting feature of the covariant open string field theory is that the closed string appears as a bound state of the open string.),2) This suggests the existence in open string field theory of a larger gauge symmetry which includes general coordinate transformation. Motivated by this, Hata and N ojiri proposed a new transformation on the open string field) with closed string functional parameter in the formulation of string field theory proposed in Refs. 4) and 5). The structure of the transformation and its algebra are those of stringy general coordinate transformation known in the closed string field theory.5) The gauge invariant open string action is, however, invariant under only the global part of the transformation.. It seems to contradict the existence of the closed string pole in open string one-loop amplitude which is computed from the BRS invariant action. On the basis of a covariant string field theory constructed by Kyoto group,4),5) we propose a new covariant action which contains both open and closed string fields and· is invariant not only under the gauge transformation in the usual open string theory but also under stringy general coordinate transformation. Since the action contains only the interaction terms as in case of the pre-geometrical cubic action in Ref. 6) or Ref. 7), it is background independent. The kinetic term is generated through field condensation of the fields. We investigate the form of the action after the field condensation correspondIng to the flat background. The relation between the new action and the BRS invariant action of Ref. 4) is also discussed in this paper. Before constructing the new action, we define several operations on string fields. First we' define the transition from an open ·string field (j) to a closed string field (denoted by «(j)) and that from a closed string field A to an open string field [A] as follows:*) .

Gravitational Anomaly and Path Integral Jacobian of Anti-Symmetric Tensor Gauge Field

A path integral formula is presented for the covariant gravitational anomaly of the self-dual anti-symmetric tensor gauge field. With the help of this formula, it is shown that the path integral Jacobian of the self-dual field strength under the general coordinate transformation is twice as large as the gravitational anomaly. It is also shown perturbatively that a chiral bi-spinor field accompanied with appropriate Faddeev-Popov ghosts describes a system of two self-dual anti-symmetric tensor gauge fields.

Path integrals on curved manifolds

A general framework for treating path integrals on curved manifolds is presented. We also show how to perform general coordinate and space-time transformations in path integrals. The main result is that one has to subtract a quantum correctionΔV∼h2 from the classical Lagrangian ℒ, i.e. the correct effective Lagrangian to be used in the path integral is ℒeff = ℒ−ΔV. A general prescription for calculating the quantum correction ΔV is given. It is based on a canonical approach using Weyl-ordering and the Hamiltonian path integral defined by the midpoint prescription. The general framework is illustrated by several examples: Thed-dimensional rotator, i.e. the motion on the sphereSd−1, the path integral ind-dimensional polar coordinates, the exact treatment of the hydrogen atom inR2 andR3 by performing a Kustaanheimo-Stiefel transformation, the Langer transformation and the path integral for the Morse potential.

MOMENTUM-SUBTRACTION RENORMALIZATION TECHNIQUES IN CURVED SPACE-TIME

Momentum-subtraction techniques, specifically BPHZ and Zimmermann’s Normal Product algorithm, are introduced as useful tools in the study of quantum field theories in the presence of background fields. In a model of a self-interacting massive scalar field, conformally coupled to a general asymptotically-flat curved space-time with a trivial topology, momentum-subtractions are shown to respect invariance under general coordinate transformations. As an illustration, general expressions for the trace anomalies are derived, and checked by explicit evaluation of the purely gravitational contributions in the free field theory limit. Furthermore, the trace of the renormalized energy-momentum tensor is shown to vanish at the Gell-Mann Low eigenvalue as it should.

Gauge symmetries in (super)string field theory.

We study gauge symmetries of the cubic formulation of open-string field theory in light of the recently discovered associativity anomalies. It is shown that the gauge group includes arbitrary general coordinate transformations, in accord with the existence of closed-string configurations. In the case of the superstring, supersymmetry is shown to be a gauge symmetry.

New symmetry in covariant open-string field theory.

We construct a new symmetry transformation in a covariant formulation of open-string field theory. It has a closed-string functional as its transformation parameter and consists of open-closed and open-open-closed string vertices. The structure of the transformation and its algebra are those of the (stringy) general coordinate transformation known in the closed-string field theory. However, because of the constraint on the transformation parameter necessary for the action invariance, the present transformation is restricted to represent only the global part of it.

Stochastic quantization and random surface approach to Polyakov string theory.

Polyakov string theories are quantized by stochastic methods. Langevin equations for a string coordinate and two-dimensional metric are invariant under general coordinate transformation and Weyl scaling. In these methods, the conformal anomaly cancels in 26 dimensions and scattering amplitudes of N tachyons in the tree level are computed. As a first step of analytic computation in the random surface approach of a Polyakov string, we evaluate the tachyonic mass and the tachyonic scattering amplitude for right triangles.