International Journal of Modern Physics A

We construct a discretized path integral for a one-variable minisuperspace model of gravity, which is made gauge-invariant by extending the method of Fradkin and Vilkovisky. Then we derive the Wheeler-DeWitt equation. The operator ordering of its kinetic term is fixed and turns out to be that of the Laplace-Beltrami operator.

Wave functions in the domain of observables such as the Hamiltonian are not always smooth functions on the classical configuration space Q. Rather, they are often best regarded as functions on a G bundle E_G over Q or as sections of an associated bundle. If H is a classical group which acts on Q, its quantum version H_G, which acts on E_G, is not always H, but an extension of H by G. A powerful and physically transparent construction of E_G and H_G, where G=U(1) and H₁(Q, Z)=0, has been developed using the path space . ( consists of paths on Q from a fixed point.) In this paper we show how to construct E_G and H_G when G is U(1) or U(1)×π₁(Q) and there is no restriction on H₁(Q, Z). The method is illustrated with concrete examples, such as a system of charges and monopoles. We argue also that is a sort of superbundle from which a large variety of bundles can be obtained by imposing suitable equivalence relations.

A survey is given of new results of the Painlevé test and nonlinear evolution equations where ordinary- and partial-differential equations are considered. We study the semiclassical Jaynes-Cumming model, the energy-eigenvalue-level-motion equation, the Kadomtsev-Petviashvili equation, the nonlinear Klein-Gordon equation and the self-dual Yang-Mills equation.

We derive a chirally split expression for the superdiffeomorphism anomaly in the two-dimensional superplane. The effective action from which this anomaly was obtained is related to the WZ-action associated with the superdiffeomorphism group. Furthermore, we show that the anomalous Ward identities generated by supercoordinate transformations encompass the wellknown OPE’s for the stress-energy supertensor. In conclusion, the extension to generic super Riemann surfaces is discussed.

A set of relative variables for the open string with P²>0 is found, which has Wigner covariance properties. It allows one to obtain global Lorentz-invariant abelianizations of the constraints, and then to find global Lorentz-invariant canonically conjugated gauge variables. By means of the multitemporal approach a noncanonical redundant set of Dirac observables is defined; these transform as spin-1 Wigner vectors and satisfy constraints of the type of σ models. The generalized Virasoro algebra, except for L₀, lives entirely in the gauge sector of the theory. The original canonical variables are expressed by means of the resulting generalized harmonic analysis in terms of these observables, and of the noncovariant center-of-mass ones. Quantization is not done, because a canonical basis of observables is still lacking, but the program to find them is expounded: instead of making a reduction from O(D–1)(P) to O(D–2), which would give a nonlinear realization of the Poincaré group like in the noncovariant approach, one has to find a Thomas-spin-adapted basis of observables which are Lorentz-invariant (except for three of them).

In this paper we investigate the implications of the topological nature of the (1, q) matrix models. We show that the algebraic structure is enough to derive the correlation functions of pure gravity. A straightforward generalization for several primaries is presented. We further find that the recursion relations in the multimatrix models may be derived from the contact algebra alone. We show that the contact algebra, using the ghost number assignment derived from the KdV hierarchy, is consistent only for contacts with the puncture operator and its descendents. We discuss the role of the contact algebra and its consistency in general.

Generalized systems of integrable nonlinear differential equations of the KdV type are considered from the point of view of self-dual Yang-Mills theory in space-times with signature (2, 2). We present a systematic method for embedding the rth flows of the SL(N) KdV hierarchy with N≥2 and r<N in the dimensionally reduced self-dual system using SL(N) as gauge group. We also find that for r>N the corresponding equations can be described in a similar fashion, provided that (in general) the rank of the gauge group increases accordingly. Certain connections of this formalism with W_N algebras are also discussed. Finally, we obtain a new class of nonlinear systems in two dimensions by introducing self-dual Ansätze associated with the algebras of Bershadsky and Polyakov.

We give a thoughtful presentation of proper orthocronous Lorentz transformations using the space-time (ℝ_1,3) and the Pauli (ℝ_3,0) algebras. We present in a closed finite form, a generic written as the exponential of elements of the associated Lie Algebra, together with a physical interpretation of the parameters. We call the result obtained the master equation and it is used to study several important topics of Special Relativity, e.g. exact product of two boosts and the Thomas precession. We also derive other results necessary for a dynamic interpretation of Lorentz transformations.

A two-dimensional Euclidean string world sheet realized as a conformal immersion in ℝⁿ is mapped into the Grassmannian G_2,n through the generalized Gauss map. In order for the Grassmannian to represent tangent planes to a given surface, n–2 integrability conditions must be satisfied by the G_2,n fields. These conditions are explicitly derived for arbitrary n by realizing G_2,n as a quadric in . Both the intrinsic and the extrinsic geometrical properties of the string world sheet are expressed in terms of the Kähler σ model fields.

The D-dimensional superstring theory is known to be invariant under the duality symmetry b ↔ α'/b, where is the constant radius of the N-dimensional internal space and α′ is the slope parameter, the (M+1)-dimensional physical space-time being flat, where D= M+N+1. It is shown, starting from the D-dimensional, vacuum Einstein theory, that this symmetry can be generalized to a curved space-time, with β(t)=–α(t) and , where t is comoving time, but it will only be an isometry of the (conformally transformed) physical space if this is characterized by a single radius function e^α(t). This would explain the homogeneity, isotropy and flatness of the Universe. The dimensionalities are uniquely determined by the dimensional reduction to be D=10, M=3, N=6. Further aspects of the duality symmetry are discussed and the notion of a negative dimension is introduced. For the closed type I superstring, density fluctuations sufficient to account for the presence of galaxies will be present, provided that G_μ≲10⁻⁶, where is the Newtonian gravitational constant, M_P is the Planck mass and μ=α′²/2πb⁶ is the effective mass per unit length of the string. This condition agrees with the estimate , obtained by Klein via quantization of the electric charge in the Kaluza-Klein theory. It is also in accord with the expectation that all the higher-dimensional superstring scales are of approximately the same order of magnitude.