General Coordinate Transformations Research Articles

Treatment of constraints in the stochastic quantization method and covariantized Langevin equation

We study the treatment of the constraints in the stochastic quantization method. We improve the treatment of the stochastic consistency condition proposed by Namiki et al. by suitably taking into account the Ito calculus. Then we obtain an improved Langevi equation and the Fokker-Planck equation which naturally leads to the correct path integral quantization of the constrained system as the stochastic equilibrium state. This treatment is applied to an O( N) non-linear α model and it is shown that singular terms appearing in the improved Langevin equation cancel out the σ n(O) divergences in one loop order. We also ascertain that the above Langevin equation, rewritten in terms of idependent variables, is actually equivalent to the one in the general-coordinate transformation covariant and vielbein-rotation invariant formalish.

On GCT-Invariant Operator Ordering Rule11The project supported by National Natural Science Foundation of China and the Grant LWTZ-1298 of Academia Sinica

With an observation that the general coordinate transformation (GCT) invariant operator ordering rule proposed by DeWitt is valid only for some cases, a careful classification for classical observables that can be related through the well-defined GCT is presented and a generalized GCT-invariant ordering rule is derived.

Internal Translation Group, Extended Cartan-Maurer Condition and Soldering Forms

Considering the internal translation group in spin space, the concept of connection 1-form is somewhat extended by loosening the Cartan-Maurer condition. Soldering forms are then interpreted in terms of «extended connections» of the group. The general coordinate transformations are induced from the internal translations by imposing the extended Cartan-Maurer condition

Heat Kernel Expansion at Finite Temperature in Curved Space

The method of heat kernel expansion at finite temperature in curved space is proposed in this letter. The general formulas are derived which are covariant under the general coordinate transformations. The thermal De Witt-Seeley-Gilkey coefficients E2 (x,β) and E4 (x,β) are calculated.

Global gravitational anomaly-free topological field theory

We demonstrate the absence of global gravitational anomalies for a small class of three-manifolds in which topological field theory is defined, by establishing the absence of disconnected general coordinate transformations. Specifically, we prove the invariance of the Chern-Simons-Witten effective action under the mapping class group, that is, the group of equivalence classes of diffeomorphisms that cannot be smoothly deformed to the identity.

A comparative analysis of numerical techniques applied to a model of the otolith membrane

This paper describes three alternative finite difference methods applied to a model of the otolith membrane. The in-plane displacement of a thin elastic membrane is modeled by a second-order system of elliptic partial differential equations with a fixed boundary closely matching the shape of the otolith membrane. The first two finite difference methods impose a rectangular grid over the shape of the otolith membrane and use different boundary extrapolation schemes applied to boundary conditions. The third method uses an optimization technique to generate a non-uniform grid which conforms to the shape of the otolith membrane, and then generates a rectangular grid via general coordinate transformations. The coordinate transformations are applied to the partial differential equations and finite differences applied to the new rectangular grid with fixed boundary conditions. All three methods were used to simulate the membrane model on a digital computer. The numerical results are compared to evaluate the effectiveness of the coordinate transformation method.

Renormalization of quantum gravity with local GL(4,R) symmetry

Renormalizability of a quantum gravity model with the local GL(4,R) symmetry is investigated. The renormalization conditions based on the action which consists of curvature-squared terms only do not exclude the appearance of fourth-derivative terms in the metric as counterterms. With the addition of all possible, fourth-derivative terms in the metric to the most general curvature-squared Lagrangian, the renormalization becomes complete. The proof is done using the Slavnov-Taylor identities and the power counting argument. Here, a higher-derivative gauge fixing for the general coordinate transformation group is used. Unitarity of the model is not shown.

Gauge and general coordinate invariance in non-polynomial closed string field theory

An appropriate field configuration in non-polynomial closed string field theory is shown to correspond to a general off-shell field configuration in low-energy effective field theory. A set of string field theoretic symmetries that act on the fields in low-energy effective field theory as general coordinate transformation and antisymmetric tensor gauge transformation is identified. The analysis is carried out to first order in the fields; thus the symmetry transformations in string field theory reproduce the linear and the first non-linear terms in the gauge transformations in the low-energy effective field theory.

The derivation of Einstein equations from invariance principles

In a first-order Lagrangian theory of gravitation, the author proves rigorously that general coordinate transformations are Noetherian symmetries if and only if the Lagrangian is a linear combination of a cosmological term and of a Lagrangian giving Einstein equations. Only the pure gravity case is considered.

Existence of gauge field in any partially integrable systems

For any integrable and, more generally, any partially integrable model, the existence is shown of a Yang–Mills gauge field with zero curvature, whose gauge group consists of all general coordinate transformations. It has been suggested that this fact may have some connection with existence of an affine connection in the space with zero Riemann curvature tensor. A BRST-like quadratically nilpotent operator is also constructed for any theory with Poisson bracket structure whenever such a connection with zero Riemann curvature exists.

Requirements for correlation energy density functionals from coordinate transformations.

The behavior of the density-functional correlation energy under general linear coordinate transformations is considered. It is demonstrated that it is uniform and nonuniform scaling that yield new conditions which are fulfilled by the exact correlation energy functional, besides the well-known requirements of rotational, reflectional, and translational invariance. By considering various types of nonuniform scaling in the limit of the scale factor \ensuremath{\lambda} going to zero and infinity, a set of relations for the correlation energy is derived. These relations, which are easy to test on approximate functionals, are of the type ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}0}$(1/\ensuremath{\lambda})${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}\ensuremath{\lambda}}^{\mathit{x}\mathit{y}}$]=0 and ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}^{\mathit{x}}$]=0. In addition, a previous proof is simplified for ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}$]=const, which is bounded from below, and we identify the constant as a second-order energy. Here ${\mathit{n}}_{\ensuremath{\lambda}}$(r)=${\ensuremath{\lambda}}^{3}$n(\ensuremath{\lambda}x,\ensuremath{\lambda}y,\ensuremath{\lambda}z), ${\mathit{n}}_{\ensuremath{\lambda}}^{\mathit{x}}$(r)=\ensuremath{\lambda}n(\ensuremath{\lambda}x,y,z), and ${\mathit{n}}_{\ensuremath{\lambda}\ensuremath{\lambda}}^{\mathit{x}\mathit{y}}$(r)=${\ensuremath{\lambda}}^{2}$n(\ensuremath{\lambda}x,\ensuremath{\lambda}y,z) where n is the electron density. The limiting relations are not all met by any known approximate correlation energy functional. These relations apply to a correlation energy for addition to a Hartree-Fock calculation as well as to a correlation energy for use as part of a full density-functional calculation.

Gauge-covariant deformations, symmetries, and free parameters of string theory.

We examine the first-order deformations of the stress tensor of the free bosonic conformal field theory which preserve the two mutually commuting copies of the Virasoro algebra and involve just two derivatives. We find a much richer structure than canonical deformations, and, without using ghosts, exhibit a distinct deformation of the stress tensor for every solution to the linearized covariant equations of motion for the massless modes of the bosonic string. We are thus able to derive explicitly all general coordinate and two-form gauge transformations, about flat space-time, and we argue that all zero modes generate symmetries if we admit auxiliary fields. We also show that, to first order, there exists in addition a finite number of deformations, associated with world-sheet instantons, which may correspond to free parameters of string theory.

General covariance in pre-geometrical string field theory

We study the general coordinate transformation invariance of pre-geometrical string field theory (PGSFT), a closed string field theory described by a purely cubic action. General covariance is necessary if PGSFT is to be a formulation of string theory independent of the space-time background. The transformation of only the X μ (σ) coordinate cannot be a symmetry and we propose an improved transformation changing also the FP-ghost coordinate. We find that PGSFT has general covariance at least for (infinitesimal) coordinate change δ× μ = Ξ μ ( x) which is a polynomial in x μ .

Hard thermal loops in a gravitational field

We calculate the high-temperature (T4) limit of the 3-graviton vertex function, with a single loop of internal scalar particles in thermal equilibrium. We use the analytically continued imaginary-time formalism. We verify a particular case of the Ward identity connecting the 3- and 2-graviton functions. This confirms that there is covariance under general coordinate transformations (which reduce to the identity at infinity).

Vector and axial-vector decomposition of Einstein's gravitational action

Vector and axial-vector gravitational fields are introduced to express the Einstein action in the manner of electromagnetism. Their conformal scaling properties are examined, and the resemblance between the general coordinate and electromagnetic gauge transformation is elucidated. The chiral formulation of the gravitational action is constructed.

Zweibein Operator Formalism of Two-Dimensional Quantum Gravity

The unitary, manifestly covariant operator formalism of two-dimensional quantum gravity, presented previously, is extended to the zweibein formalism. All the two-dimensional (anti) commutation relations between primary fields are obtained in closed form. The four degrees of freedom of the zweibein are shown to be realized as q-number transformation functions of the general coordinate transformation, the local Lorentz transformation and the Weyl transformation. As the result, the explicit expression for the gravitgational extension of the Pauli-Jordan D-function is found in terms of the zweibein. Bosonized operator solutions known in solvable two-dimensional models are extended to the quantum-gravity case through the above q-number transformations

Representations of quantum general relativity using Ashtekar's variables

On the basis of the link class solutions for quantum general relativity found by Rovelli and Smolin in the loop representation, the author describes a possible procedure for assigning a Hilbert space structure to this set of solutions. On this Hilbert space operators can be defined by going back to the connection representation; this is possible consistently since the 'gauge choice' one makes by picking out specific loops in the link classes only corresponds to general coordinate transformations on the observables connected to the operators. In this way the loop transform of Rovelli and Smolin is restricted to the physical states of the theory. However, the results still contain products of delta functions at one point, and thus the description is not complete until a satisfying regularization procedure has been found. Nevertheless this formalism shows how physical observables of general relativity can be defined in a natural way as operators on the space of smooth link classes.

An Extended New General Relativity as a Reduction of $\overline{\mbox{Poincar\'{e}}}$ Gauge Theory of Gravity

An extended new general relativity (E. N. G. R.) is formulated as a reduction of Poincaré gauge theory (P, G. T.) of gravity whose gauge group is the covering group of the Poincaré group. Fundamental fields in the theory are a Higgs-type field ψ k , and the translational gauge potential Akµ and a matter field φ a . The components ekµ of the duals of the vierbein fields are given by ekµ = ∇ *µ ψ k = ψ k,µ + Akµ . Internal translation and the field ψ k , which the conventional new general relativity lacks, play fundamental roles in E. N. G. R. as well as in P, G. T. The restrictions on the Lagrangian density, identities and differential conservation laws following from the ( local internal translation ⊗ global SL (2, C ) ) ⊗ general coordinate transformations are given. For the specific gravitational Lagrangian LT = -(1/3κ) tklmtklm + (1/3κ) vkvk + a3akak , the generators of internal \overlinePoincaré and general affine coordinate transformations for an isolated system are examined. The results are quite parallel to those in P, G. T., but the former is not trivial reductions of the latter. The following is worth noticing also in E. N. G. R.: For the case in which {ψ k , Akµ , φ a } with ψ k ≃ e (0) kµyµ + ψ (0) k is employed as the set of independent variables, (a) the generators of ( internal translation ⊗SL (2, C )) give the energy-momentum and the total angular momentum for an isolated system, and (b) the generators of general affine coordinate transformations vanish and are trivial.

An Extended New General Relativity as a Reduction of \overlinePoincare Gauge Theory of Gravity: Generators of Internal and Coordinate Transformations

An extended new general relativity (E. N. G. R.) is formulated as a reduction of Poincare gauge theory (P. G. T.) of gravity whose gauge group is the covering group of the Poincare group. Fundamental fields in the theory are a Higgs-type field <p', and the translational gauge potential A'p and a matter field ,pa. The components e'p of the duals of the vierbein fields are given by e'p =f7t<p'=<p',p+A'p. Internal translation and the field <p',. which the conventional new general relativity lacks, play fundamental roles in E. N. G. R. as well as in P. G. T. The restrictions on the Lagrangian density, identities and differential conservation laws following from the (local internal translation ® global SL(2, e)) ® general coordinate transformations are given. For the specific gravitational Lagrangian LT = - (1/3K)t klm tklm +(1/3K)v'v. + a3a'a., the generators of internal Poincare and general affine coordinate transformations for an isolated system are examined. The results are quite parallel to those in P. G. T., but the former is not trivial reductions of the latter. The following is worth noticing also in E. N. G. R.: For the case in which {<p',A'p, ,pa} with <p''''''e(O)'pY P + <p(0)' is employed as the set of il).dependent variables, (a) the generators of (internal translation ® SL(2, e)) give the energy-momentum and the total angular momentum for an isolated system, and (b) the generators of ~eneral affine coordinate transformations vanish and are trivial.

On Behavior of Weyl's Gauge Field11The project supported in part by National Natural Science Foundation of China.

We consider a system, consisting of a metric tensor , a scalar field , a Weyl's gauge field and a scalar matter field , which is invariant under general coordinate transformation and Weyl's gauge transformation. Two kinds of identities and field equations are given and discussed. A special space-time with is considered in a gauge-independent manner. We point out that in a correct treatment where is not regarded as an independent variable, an auxiliary condition for Weyl's gauge field cannot be obtained. Therefore Weyl's gauge field can be treated as a usual field.