Geometro-stochastic quantization of gravity. II

A second-quantized bundleE, called the quantum gravitational bundle, is constructed from graviton bundles in accordance with outlined general principles for geometro-stochastic second quantization, and quantum gravitational frame fields are introduced in it. The gravitational bundleE is the carrier of a semiclassical connection that can be used as a stepping stone in the construction of a second-quantized gravitational connection. The geometro-stochastic quantum propagations governed by such connections can be expressed in terms of path integrals over causal stochastic paths which embody stochastic parallel transport under free-fall conditions. Their epistemic implications for the quantum theory of measurement are discussed.1. Supported in part by the NSERC Research Grant No. A5206.     

Dynamical equation for scattering operator

A new dynamical equation for the scattering operator in Quantum Field Theory is derived. General properties of solutions of dynamical equation are discussed. A new method of constructing nonperturbative formula for the scattering operator is presented. Explicit construction of scattering operator in and models is carried out.Supported in part by the National Science Foundation under Grant GF-41958.     

The Boltzmann equation with a soft potential

The results of Part I are extended to include linear spatially periodic problems-solutions of the initial value are shown to exist and decay like . Then the full non-linear Boltzmann equation with a soft potential is solved for initial data close to equilibrium. The non-linearity is treated as a perturbation of the linear problem, and the equation is solved by iteration.Supported by the National Science Foundation under Grant Nos. MCS78-09525 and MCS76-07039 and by the United States Army under Contract No. DAAG29-75-C-0024     

Curvature of determinant bundles for degenerate families

We calculate the (1, 1) curvature of the Beilinson Schechtman connection for the determinant bundle associated to a family of Riemann surfaces with ordinary singularities. As consequences we obtain generalizations of theorems of Bismut and Bost.Supported in part by NSF Grant No DMS-9201022.Supported in part by NSC Grant No 83-0208-M-002-039, Republic of China.     

Gauge choice in Witten's energy expression

Witten's equation can be interpreted as a gauge fixing condition for classical supergravity. We rigorously prove the existence of asymptotically constant solutions of the more general gauge condition for almost all endomorphismsA of the spin bundle. Each gives an expression for the gravitational energy similar to Witten's. These include the choice , which yields the particularly elegant energy expression first noticed by Deser.Partially supported by N. S. F. Grant MCS-82-02018 at Harvard University     

Parallel transport in the determinant line bundle: The non-zero index case

For a product family of Weyl operators of possibly non-zero index on a compact manifoldX, we express parallel transport in the determinant line bundle in terms of the spectral asymmetry of a Dirac operator on ×X. This generalizes the results of [7], where we dealt only with invertible operators.Supported in part by NSF Grant No. PHY 8605978 and the Robert A. Welch FoundationSupported in part by NSF Grant No. PHY 8215249     

Geometro-stochastic quantization of a theory for extended elementary objects

The geometro-stochastic quantization of a gauge theory based on the (4,1)-de Sitter group is presented. The theory contains an intrinsic elementary length parameter R of geometric origin taken to be of a size typical for hadron physics. Use is made of a soldered Hilbert bundle over curved spacetime carrying a phase space representation of SO(4, 1) with the Lorentz subgroup related to a vierbein formulation of gravitation. The typical fiber of is a resolution kernel Hilbert space constructed in terms of generalized coherent states related to the principal series of unitary irreducible representations of SO(4, 1), namely de Sitter horospherical waves for spinless particles characterized by the parameter . The framework is, finally, extended to a quantum field-theoretical formalism by using bundles with Fock space fibers constructed from .Supported in part by NSERC Research Grant No. A5206.     

Quantum group structure in the Fock space resolutions of\widehat{sl}(n) representations

We describe a complex of Wakimoto-type Fock space modules for the affine Kac-Moody algebra . The intertwining operators that build the complex are obtained from contour integrals of so-called screening operators. We show that a quantum group structure underlies the algebra of screening operators. This observation greatly facilitates the explicit determination of the intertwiners. We conjecture that the complex provides a resolution of an irreducible highest weight module in terms of Fock spaces.Supported by the U.S. Department of Energy under Contract #DE-AC02-76ER03069.Supported by the NSF Grant #PHY-88-04561     

Born series for (2 cluster) → (2 cluster) scattering of two,three, and four particle Schrödinger operators

We investigate elastic and inelastic (2 cluster) (2 cluster) scattering for classes of two, three, and four body Schrödinger operators Formulas are derived for those generalized eigenfunctions ofH which correspond asymptotically in the past to two freely moving clusters. With these eigenfunctions, we establish a formula for the (2 cluster) (2 cluster)T-matrix and prove the convergence of a Born series for theT-matrix at high energy.Supported in part by the National Science Foundation under Grant PHY 78-08066     

Decay of correlations in the one dimensional Ising model withJ ij =|i−j|−2

A low temperature expansion is constructed for the one dimensional Ising model with Hamiltonian . It is shown that the two point function _i ; _j obeys upper and lower bounds of the formf()|i–j|^–2 for inverse temperature sufficiently large.Junior Fellow, Harvard University Society of Fellows. Supported in part by the National Science Foundation under Grant No. PHY79-16812.     

Statistical mechanics of quantum spin systems. II

In the first part of this paper we continue the general analysis of quantum spin systems. It is demonstrated, for a large class of interactions, that time-translations form a group of automorphisms of theC*-algebra of quasi-local observables and that the thermodynamic equilibrium states are invariant under this group. Further it is shown that the equilibrium states possess the Kubo-Martin-Schwinger analyticity and boundary condition properties. In the second part of the paper we give a general analysis of states which are invariant under space and time translations and also satisfy the KMS boundary condition. A discussion of these latter conditions and their connection with the decomposition of invariant states into ergodic states is given. Various properties pertinent to this discussion are derived.Supported in part by the Office of Naval Research Contract No. Nonr 1866 (5).     

Local rings of singularities andN=2 supersymmetric quantum mechanics

We investigate the Kähler structure arising inn-component,N=2 supersymmetric quantum mechanics. We defineL ²-cohomology groups of a modified and relate them to the corresponding spaces of harmonic forms. We prove that the cohomology is concentrated in the middle dimension, and is isomorphic to the direct sum of the local rings of the singularities of the superpotential. In the physics language, this means that the number of ground states is equal to the absolute value of the index of the supercharge, and each ground state contains exactlyn fermions.Supported in part by the Department of Energy under Grant DE-F602-88ER25065     

Phase transitions in anisotropic classical Heisenberg ferromagnets

The method of Peierls is used to prove the existence of a spontaneous magnetization for a spin system with nearest-neighbor interactions and Hamiltonian a (classical) unit vector at thei'th site, with ||<0.0298 and 0.0198 for a square lattice and simple cubic lattice, respectively.Research supported by the National Science Foundation Grant No. GP 11454.     

Quantum Riemann surfaces I. The unit disc

We construct a non-commutative ^*-algebra which is a quantum deformation of the algebra of continuous functions on the closed unit disc . is generated by the Toeplitz operators on a suitable Hilbert space of holomorphic functions onU.Supported in part by the National Science Foundation under grant DMS/PHY 88-16214     

Bäcklund transformations for the equationz_{uv} = 2z_u z_v /z + \bar z

The equation for the complex variablez is equivalent to the field equation for a SO(2, 1)-invariant nonlinear -model in two dimensions. A connection on a SU(2) bundle is given, such that the requirement of the vanishing curvature of the connection implies that equation. Bäcklund transformations, considered as gauge transformations of the connection, are explicity obtained.     

Universal Spin Structure

Any Dirac spin structure on a world manifold xis a subbundle of the composite spinor bundle , where is a bundle oftetrad gravitational fields. The bundle S admits generalcovariant transformations that enable us to discover theenergy-momentum conservation law in gauge gravitationtheory.     

On the twistor-spinors

Two interesting conformal invariants which are constant on the manifold are given for twistor-spinors on a spin manifold following the notion of a twistor-spinor associated to a twisted spin bundle. For a twisted spin bundle corresponding to a flat Hermitian vector bundle, the associated twistor-spinors admit the same conformal invariants.An analysis is made of the twistor-spinors given by , where f is a complex-valued function. There is only one case where is not a Killing spinor. An example is given of a compact spin manifold for which the situation is realized.     

Conformal tensor discontinuities in general relativity

The postulate is made that across a given hypersurfaceN the metric and its first derivatives are continuous. This postulate is used to derive conditions which must be satisfied by discontinuities in the Riemann tensor acrossN. These conditions imply that the conformal tensor jump is uniquely determined by the stress-energy tensor discontinuity ifN is non-null (and to within an additive term of type Null ifN is lightlike). Alternatively, and [R] determine ifN is non-null. These relationships between the conformal tensor and stress-energy tensor jumps are given explicitly in terms of a three-dimensional complex representation of the antisymmetric tensors. Application of these results to perfect-fluid discontinuities is made: is of type D across a fluid-vacuum boundary and across an internal, non-null shock front. is of type I (non-degenerate) in general across fluid interfaces across which no matter flows, except for special cases.The research reported herein was supported in part by the Atomic Energy Commission under contract number AT (11-1)-34, Project Agreement No. 125, while the author was at the University of California at Berkeley, and in part by National Science Foundation Grant GU-1598-University Science Development Program.     

Measurement of inclusive baryon production inB meson decays

Using the ARGUS detector at thee ⁺ e ^– storage ring DORIS II, we have studied nine decay modes of theD ⁰ meson. In addition to a higher-precision determination of the branching ratios for these modes, several new channels, the decays , and , have been observed for the first time.Supported in part by the Institute of Physics, University of Lund, Sweden     

Modified Weyl theory and extended elementary objects

To represent extension of objects in particle physics, a modified Weyl theory is used by gauging the curvature radius of the local fibers in a soldered bundle over space-time possessing a homogeneous space G/H of the (4, 1)-de Sitter group G as fiber. Objects with extension determined by a fundamental length parameter R₀ appear as islands D_(i) in space-time characterized by a geometry of the Cartan-Weyl type (i.e., involving torsion and modified Weyl degrees of freedom). Farther away from the domains D_(i), space-time is identified with the pseudo-Riemannian space of general relativity. Extension and symmetry breaking are described by a set of additional fields ( , given as a section on an associated bundle over space-time B with structural group = G D(1), where D(1) is the dilation group. Field equations for the quantities defining the underlying bundle geometry and for the fields are established involving matter source currents derived from a generalized spinor wave function. Einstein's equations for the metric are regarded as the part of the -gauge theory related to the Lorentz subgroup H of G exhibiting thereby the broken nature of the -symmetry for regions outside the domains D_(i).Talk presented at the International Conference on Field Theory and General Relativity held at Utah State University, Logan, Utah, June 26–July 2, 1988.