Models of the universe

In this paper a theory of models of the universe is proposed. We refer to such models ascosmological models, where a cosmological model is defined as an Einstein-inextendible Einstein spacetime. A cosmological model isabsolute if it is a Lorentz-inextendible Einstein spacetime,predictive if it is globally hyperbolic, andnon-predictive if it is nonglobally-hyperbolic. We discuss several features of these models in the study of cosmology. As an example, any compact Einstein spacetime is always a non-predictive absolute cosmological model, whereas a noncompact complete Einstein spacetime is an absolute cosmological model which may be either predictive or non-predictive. We discuss the important role played by maximal Einstein spacetimes. In particular, we examine the possible proper Lorentz-extensions of such spacetimes, and show that a spatially compact maximal Einstein spacetime is exclusively either a predictive cosmological model or a proper sub-spacetime of a non-predictive cosmological model. Provided that the Strong Cosmic Censorship conjecture is true, a generic spatially compact maximal Einstein spacetime must be a predictive cosmological model. It isconjectured that the Strong Cosmic Censorship conjecture isnot true, and converting a vice to a virtue it is argued that the failure of the Strong Cosmic Censorship conjecture would point to what may be general relativity's greatest prediction of all, namely,that general relativity predicts that general relativity cannot predict the entire history of the universe.     

Basic Brackets of a 2D Model for the Hodge Theory Without its Canonical Conjugate Momenta

We deduce the canonical brackets for a two (1+1)-dimensional (2D) free Abelian 1-form gauge theory by exploiting the beauty and strength of the continuous symmetries of a Becchi-Rouet-Stora-Tyutin (BRST) invariant Lagrangian density that respects, in totality, six continuous symmetries. These symmetries entail upon this model to become a field theoretic example of Hodge theory. Taken together, these symmetries enforce the existence of exactly the same canonical brackets amongst the creation and annihilation operators that are found to exist within the standard canonical quantization scheme. These creation and annihilation operators appear in the normal mode expansion of the basic fields of this theory. In other words, we provide an alternative to the canonical method of quantization for our present model of Hodge theory where the continuous internal symmetries play a decisive role. We conjecture that our method of quantization is valid for a class of field theories that are tractable physical examples for the Hodge theory. This statement is true in any arbitrary dimension of spacetime.     

Conformal transformations and conformal invariance in gravitation

Conformal transformations are frequently used tools in order to study relations between various theories of gravity and Einstein's general relativity theory. In this paper we discuss the rules of these transformations for geometric quantities as well as for the matter energy‐momentum tensor. We show the subtlety of the matter energy‐momentum conservation law which refers to the fact that the conformal transformation “creates” an extra matter term composed of the conformal factor which enters the conservation law. In an extreme case of the flat original spacetime the matter is “created” due to work done by the conformal transformation to bend the spacetime which was originally flat. We discuss how to construct the conformally invariant gravity theories and also find the conformal transformation rules for the curvature invariants R², R_abR^ab, R_abcdR^abcd and the Gauss‐Bonnet invariant in a spacetime of an arbitrary dimension. Finally, we present the conformal transformation rules in the fashion of the duality transformations of the superstring theory. In such a case the transitions between conformal frames reduce to a simple change of the sign of a redefined conformal factor.     

Area-preserving diffeomorphisms and supermembrane Lorentz invariance

The Lie algebra of area-preserving diffeomorphisms on closed membranes of arbitrary topology is investigated. On the basis of a harmonic decomposition we define the structure constants as well as two other tensors which appear in the supermembrane Lorentz generators. We derive certain identities between these tensors and analyze their validity when the areapreserving diffeomorphisms are approximated bySU(N). One of the additional tensors can then be identified with the invariant symmetric three-index tensor ofSU(N), while the second has no obvious analog. We prove that the Lorentz generators are classically conserved in the light-cone gauge for arbitrary membrane topology, as a consequence of these tensor identities. This formulation allows a systematic study of the violations of Lorentz invariance in theSU(N) approximation.     

Pseudo-supergravity extension of the bosonic string

We construct a “pseudo-supersymmetric” fermionic extension of the effective action of the bosonic string in arbitrary spacetime dimension D. The theory is invariant under pseudo-supersymmetry transformations up to the quadratic fermion order, which is sufficient in order to be able to derive Killing spinor equations in bosonic backgrounds, and hence to define BPS type solutions determined by a system of first-order equations. The pseudo-supersymmetric theory can be extended by coupling it to a Yang-Mills pseudo-supermultiplet. This also allows us to construct “α^′ corrections” involving quadratic curvature terms. An exponential dilaton potential term, associated with the conformal anomaly for a bosonic string outside its critical dimension, can also be pseudo-supersymmetrised.     

Exponential Times in the One-Dimensional Gross–Pitaevskii Equation with Multiple Well Potential

We present an algorithm for constructing the Wilson operator product expansion (OPE) for perturbative interacting quantum field theory in general Lorentzian curved spacetimes, to arbitrary orders in perturbation theory. The remainder in this expansion is shown to go to zero at short distances in the sense of expectation values in arbitrary Hadamard states. We also establish a number of general properties of the OPE coefficients: (a) they only depend (locally and covariantly) upon the spacetime metric and coupling constants, (b) they satisfy an associativity property, (c) they satisfy a renormalization group equation, (d) they satisfy a certain microlocal wave front set condition, (e) they possess a “scaling expansion”. The latter means that each OPE coefficient can be written as a sum of terms, each of which is the product of a curvature polynomial at a spacetime point, times a Lorentz invariant Minkowski distribution in the tangent space of that point. The algorithm is illustrated in an example.     

Three-Manifold Invariants and Their Relation with the Fundamental Group

We consider the 3-manifold invariant I(M) which is defined by means of the Chern–Simons quantum field theory and which coincides with the Reshetikhin–Turaev invariant. We present some arguments and numerical results supporting the conjecture that for nonvanishing I(M), the absolute value |I(M)| only depends on the fundamental group π₁ (M) of the manifold M. For lens spaces, the conjecture is proved when the gauge group is SU(2). In the case in which the gauge group is SU(3), we present numerical computations confirming the conjecture. Received: 15 November 1996 / Accepted: 17 June 1997     

Cubic interaction vertices for fermionic and bosonic arbitrary spin fields

Using the light-cone gauge approach to relativistic field dynamics, we study arbitrary spin fermionic and bosonic fields propagating in flat space of dimension greater than or equal to four. Generating functions of parity invariant cubic interaction vertices for totally symmetric and mixed-symmetry massive and massless fields are obtained. For the case of totally symmetric fields, we derive restrictions on the allowed values of spins and the number of derivatives. These restrictions provide a complete classification of parity invariant cubic interaction vertices for totally symmetric fermionic and bosonic fields. As an example of application of the light-cone formalism, we obtain simple expressions for the Yang–Mills and gravitational interactions of massive arbitrary spin fermionic fields. For some particular cases, using our light-cone cubic vertices, we discuss the corresponding manifestly Lorentz invariant and on-shell gauge invariant cubic vertices.     

String Theory and the Kauffman Polynomial

We propose a new, precise integrality conjecture for the colored Kauffman polynomial of knots and links inspired by large N dualities and the structure of topological string theory on orientifolds. According to this conjecture, the natural knot invariant in an unoriented theory involves both the colored Kauffman polynomial and the colored HOMFLY polynomial for composite representations, i.e. it involves the full HOMFLY skein of the annulus. The conjecture sheds new light on the relationship between the Kauffman and the HOMFLY polynomials, and it implies for example Rudolph’s theorem. We provide various non-trivial tests of the conjecture and we sketch the string theory arguments that lead to it.     

Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles

This paper is devoted to the rigorous proof of the universality conjecture of random matrix theory, according to which the limiting eigenvalue statistics ofn×n random matrices within spectral intervals ofO(n ^–1) is determined by the type of matrix (real symmetric, Hermitian, or quaternion real) and by the density of states. We prove this conjecture for a certain class of the Hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function (the potential in the quantum field theory) satisfying some regularity conditions.     

Local Wick Polynomials and Time Ordered Products¶of Quantum Fields in Curved Spacetime

In order to have well defined rules for the perturbative calculation of quantities of interest in an interacting quantum field theory in curved spacetime, it is necessary to construct Wick polynomials and their time ordered products for the noninteracting theory. A construction of these quantities has recently been given by Brunetti, Fredenhagen, and K?hler, and by Brunetti and Fredenhagen, but they did not impose any “locality” or “covariance” condition in their constructions. As a consequence, their construction of time ordered products contained ambiguities involving arbitrary functions of spacetime point rather than arbitrary parameters. In this paper, we construct an “extended Wick polynomial algebra”– large enough to contain the Wick polynomials and their time ordered products – by generalizing a construction of Dütsch and Fredenhagen to curved spacetime. We then define the notion of a local, covariant quantum field, and seek a definition of local Wick polynomials and their time ordered products as local, covariant quantum fields. We introduce a new notion of the scaling behavior of a local, covariant quantum field, and impose scaling requirements on our local Wick polynomials and their time ordered products as well as certain additional requirements – such as commutation relations with the free field and appropriate continuity properties under variations of the spacetime metric. For a given polynomial order in powers of the field, we prove that these conditions uniquely determine the local Wick polynomials and their time ordered products up to a finite number of parameters. (These parameters correspond to the usual renormalization ambiguities occurring in Minkowski spacetime together with additional parameters corresponding to the coupling of the field to curvature.) We also prove existence of local Wick polynomials. However, the issue of existence of local time ordered products is deferred to a future investigation. Received: 27 March 2001 / Accepted: 6 June 2001     

Letter: Interpolating Between the Bose-Einstein and the Fermi-Dirac Distributions in Odd Dimensions

We consider the response of a uniformly accelerated monopole detector that is coupled to a superposition of an odd and an even power of a quantized, massless scalar field in flat spacetime in arbitrary dimensions. We show that, when the field is assumed to be in the Minkowski vacuum, the response of the detector is characterized by a Bose-Einstein factor in even spacetime dimensions, whereas a Bose-Einstein as well as a Fermi-Dirac factor appear in the detector response when the dimension of spacetime is odd. Moreover, we find that, it is possible to interpolate between the Bose-Einstein and the Fermi-Dirac distributions in odd spacetime dimensions by suitably adjusting the relative strengths of the detector's coupling to the odd and the even powers of the scalar field. We point out that the response of the detector is always thermal and we, finally, close by stressing the apparent nature of the appearance of the Fermi-Dirac factor in the detector response.     

Characteristic cohomology ofp-form gauge theories

The characteristic cohomologyH ^k _char(d) for an arbitrary set of freep-form gauge fields is explicitly worked out in all form degreesk < n — 1, wheren is the spacetime dimension. It is shown that this cohomology is finite-dimensional and completely generated by the forms dual to the field strengths. The gauge invariant characteristic cohomology is also computed. The results are extended to interactingp-form gauge theories with gauge invariant interactions. Implications for the BRST cohomology are mentioned.     

Gauge invariant finite size spectrum of the giant magnon

It is shown that the finite size corrections to the spectrum of the giant magnon solution of classical string theory, computed using the uniform light-cone gauge, are gauge invariant and have physical meaning. This is seen in two ways: from a general argument where the single magnon is made gauge invariant by putting it on an orbifold as a wrapped state obeying the level matching condition as well as all other constraints, and by an explicit calculation where it is shown that physical quantum numbers do not depend on the uniform light-cone gauge parameter. The resulting finite size effects are exponentially small in the R-charge and the exponent (but not the prefactor) agrees with gauge theory computations using the integrable Hubbard model.     

Cutoff anti-de Sitter space/conformal-field-theory duality and the quest for braneworld black holes

Significant evidence is presented in favor of the holographic conjecture that "4D black holes localized on the brane found by solving the classical bulk equations in AdS5 are quantum corrected black holes and not classical ones." The quantum correction to the Newtonian potential is computed using a numerical computation of in Schwarzschild spacetime for matter fields in the zero-temperature Boulware vacuum state. For the conformally invariant scalar field the leading order term is equivalent to that previously obtained in the weak-field approximation using Feynman diagrams and which has been shown to be equivalent, via the anti-de Sitter space/conformal-field-theory (AdS/CFT) duality, to the analogous calculation in Randall-Sundrum braneworlds. The 4D backreaction equations are used to make a prediction about the existence and the possible spacetime structure of macroscopic static braneworld black holes.     

Infrared properties of two-loop,initial state spectator interactions in the Drell-Yan process

We use the light-cone axial gauge of proper-time ordered perturbation theory and study the soft-IR properties of the two-loop virtuals' diagrams considered by Bodwin, Brodsky and Lepage for ππ → μ⁺μ^- + X. It is shown that although the systematic summation over all possible spectator interactions removes the outside soft-IR divergences in the non-overlapping ladder Glauber diagrams, unphysical inside soft-IR divergences persist. So, in the light-cone axial gauge the on-shell Glauber region is not a gauge invariant concept which can be physically isolated from radiative corrections which non-trivially involve other diagrammatic regions. Due to gauge invariance it can be potentially misleading in eikonal phenomenologies based on perturbative QCD to assume an ad hoc inside soft-IR cutoff in analyzing possible non-abelian effects in multiple scatterings involving spectators.     

Whiteheadian approach to quantum theory and the generalized Bell's theorem

The model of the world proposed by Whitehead provides a natural theoretical framework in which to imbed quantum theory. This model accords with the ontological ideas of Heisenberg, and also with Einstein's view that physical theories should refer nominally to the objective physical situation, rather than our knowledge of that system. Whitehead imposed on his model the relativistic requirement that what happens in any given spacetime region be determined only by what has happened in its absolute past, i.e., in the backward light-cone drawn from that region. This requirement must be modified, for it is inconsistent with the implications of quantum theory expressed by a generalized version of Bell's theorem. Revamping the causal spacetime structure of the Whitehead-Heisenberg ontology to bring it into accord with the generalized Bell's theorem creates the possibility of a nonlocal causal covariant theory that accords with the statistical prediction of quantum theory.Based on lectures given at the University of Texas, March–May 1977. This work was supported in part by the United States Energy Research and Development Agency, in part by the University of Texas, and in part by CERN.     

Short-distance and light-cone expansions for products of currents

Short-distance and light-cone expansions for T-ordered products of currents are derived in the framework of the arbitrary order of the perturbation theory.     

Classical Scattering in the Covariant Two-Body Coulomb Potential

The problem of two relativistically-moving pointlike particles of constant mass is undertaken in an arbitrary Lorentz frame using the classical Lagrangian mechanics of Stückelberg, Horwitz, and Piron. The particles are assumed to interact at events along their world lines at a common world time, an invariant dynamical parameter which is not in general synchronous with the particle proper time. The Lorentz-scalar interaction is assumed to be the Coulomb potential (i.e., the inverse square spacetime potential) of the spacetime event separation. The classical orbit equations are found in 1 + 1 spacetime dimensions in the hyperbolic angle coordinates for the reduced problem. The solutions to the reduced motion in these coordinates are the spacetime generalizations of the nonrelativistic Kepler solutions. and they introduce an invariant eccentricity which is a function of other known constants of the motion for the reduced problem. Solutions compatible with physical scattering are obtained by the assumption that the eccentricity is a given function of the ratio of the particle masses.     

Axiomatic Quantum Field Theory in Curved Spacetime

The usual formulations of quantum field theory in Minkowski spacetime make crucial use of features—such as Poincaré invariance and the existence of a preferred vacuum state—that are very special to Minkowski spacetime. In order to generalize the formulation of quantum field theory to arbitrary globally hyperbolic curved spacetimes, it is essential that the theory be formulated in an entirely local and covariant manner, without assuming the presence of a preferred state. We propose a new framework for quantum field theory, in which the existence of an Operator Product Expansion (OPE) is elevated to a fundamental status, and, in essence, all of the properties of the quantum field theory are determined by its OPE. We provide general axioms for the OPE coefficients of a quantum field theory. These include a local and covariance assumption (implying that the quantum field theory is constructed in a local and covariant manner from the spacetime metric and other background structure, such as time and space orientations), a microlocal spectrum condition, an “associativity” condition, and the requirement that the coefficient of the identity in the OPE of the product of a field with its adjoint have positive scaling degree. We prove curved spacetime versions of the spin-statistics theorem and the PCT theorem. Some potentially significant further implications of our new viewpoint on quantum field theory are discussed.