Group Averaging for de Sitter free fields

Perturbative gravity about global de Sitter space is subject to linearization-stability constraints. Such constraints imply that quantum states of matter fields couple consistently to gravity {\it only} if the matter state has vanishing de Sitter charges; i.e., only if the state is invariant under the symmetries of de Sitter space. As noted by Higuchi, the usual Fock spaces for matter fields contain no de Sitter-invariant states except the vacuum, though a new Hilbert space of de Sitter invariant states can be constructed via so-called group-averaging techniques. We study this construction for free scalar fields of arbitrary positive mass in any dimension, and for linear vector and tensor gauge fields in any dimension. Our main result is to show in each case that group averaging converges for states containing a sufficient number of particles. We consider general $N$-particle states with smooth wavefunctions, though we obtain somewhat stronger results when the wavefunctions are finite linear combinations of de Sitter harmonics. Along the way we obtain explicit expressions for general boost matrix elements in a familiar basis.Comment: 33 pages, 2 figure

On the Generality of Refined Algebraic Quantization

The Dirac quantization `procedure' for constrained systems is well known to have many subtleties and ambiguities. Within this ill-defined framework, we explore the generality of a particular interpretation of the Dirac procedure known as refined algebraic quantization. We find technical conditions under which refined algebraic quantization can reproduce the general implementation of the Dirac scheme for systems whose constraints form a Lie algebra with structure constants. The main result is that, under appropriate conditions, the choice of an inner product on the physical states is equivalent to the choice of a ``rigging map'' in refined algebraic quantization.Comment: 12 pages, no figures, ReVTeX, some changes in presentation, some references adde

Dirac Quantization of Parametrized Field Theory

Parametrized field theory (PFT) is free field theory on flat spacetime in a diffeomorphism invariant disguise. It describes field evolution on arbitrary foliations of the flat spacetime instead of only the usual flat ones, by treating the `embedding variables' which describe the foliation as dynamical variables to be varied in the action in addition to the scalar field. A formal Dirac quantization turns the constraints of PFT into functional Schrodinger equations which describe evolution of quantum states from an arbitrary Cauchy slice to an infinitesimally nearby one.This formal Schrodinger picture- based quantization is unitarily equivalent to the standard Heisenberg picture based Fock quantization of the free scalar field if scalar field evolution along arbitrary foliations is unitarily implemented on the Fock space. Torre and Varadarajan (TV) showed that for generic foliations emanating from a flat initial slice in spacetimes of dimension greater than 2, evolution is not unitarily implemented, thus implying an obstruction to Dirac quantization. We construct a Dirac quantization of PFT,unitarily equivalent to the standard Fock quantization, using techniques from Loop Quantum Gravity (LQG) which are powerful enough to super-cede the no- go implications of the TV results. The key features of our quantization include an LQG type representation for the embedding variables, embedding dependent Fock spaces for the scalar field, an anomaly free representation of (a generalization of) the finite transformations generated by the constraints and group averaging techniques. The difference between 2 and higher dimensions is that in the latter, only finite gauge transformations are defined in the quantum theory, not the infinitesimal ones.Comment: 33 page

Semiclassical States in Quantum Cosmology: Bianchi I Coherent States

We study coherent states for Bianchi type I cosmological models, as examples of semiclassical states for time-reparametrization invariant systems. This simple model allows us to study explicitly the relationship between exact semiclassical states in the kinematical Hilbert space and corresponding ones in the physical Hilbert space, which we construct here using the group averaging technique. We find that it is possible to construct good semiclassical physical states by such a procedure in this model; we also discuss the sense in which the original kinematical states may be a good approximation to the physical ones, and the situations in which this is the case. In addition, these models can be deparametrized in a natural way, and we study the effect of time evolution on an "intrinsic" coherent state in the reduced phase space, in order to estimate the time for this state to spread significantly.Comment: 21 pages, 1 figure; Version to be published in CQG; The discussion has been slightly reorganized, two references added, and some typos correcte

A Uniqueness Theorem for Constraint Quantization

This work addresses certain ambiguities in the Dirac approach to constrained systems. Specifically, we investigate the space of so-called ``rigging maps'' associated with Refined Algebraic Quantization, a particular realization of the Dirac scheme. Our main result is to provide a condition under which the rigging map is unique, in which case we also show that it is given by group averaging techniques. Our results comprise all cases where the gauge group is a finite-dimensional Lie group.Comment: 23 pages, RevTeX, further comments and references added (May 26. '99

Almost Ideal Clocks in Quantum Cosmology: A Brief Derivation of Time

A formalism for quantizing time reparametrization invariant dynamics is considered and applied to systems which contain an `almost ideal clock.' Previously, this formalism was successfully applied to the Bianchi models and, while it contains no fundamental notion of `time' or `evolution,' the approach does contain a notion of correlations. Using correlations with the almost ideal clock to introduce a notion of time, the work below derives the complete formalism of external time quantum mechanics. The limit of an ideal clock is found to be closely associated with the Klein-Gordon inner product and the Newton-Wigner formalism and, in addition, this limit is shown to fail for a clock that measures metric-defined proper time near a singularity in Bianchi models.Comment: 16 pages ReVTeX (35 preprint pages

Time-of-arrival formalism for the relativistic particle

A suitable operator for the time-of-arrival at a detector is defined for the free relativistic particle in 3+1 dimensions. For each detector position, there exists a subspace of detected states in the Hilbert space of solutions to the Klein Gordon equation. Orthogonality and completeness of the eigenfunctions of the time-of-arrival operator apply inside this subspace, opening up a standard probabilistic interpretation.Comment: 16 pages, no figures, uses LaTeX. The section "Interpretation" has been completely rewritten and some errors correcte

Asymptotic Flatness, Little String Theory, and Holography

We argue that any non-gravitational holographic dual to asymptotically flat string theory in $d$-dimensions naturally resides at spacelike infinity. Since spacelike infinity can be resovled as a $(d-1)$-dimensional timelike hyperboloid (i.e., as a copy of de Sitter space in $(d-1)$ dimensions), the dual theory is defined on a Lorentz signature spacetime. Conceptual issues regarding such a duality are clarified by comparison with linear dilaton boundary conditions, such as those dual to little string theory. We compute both time-ordered and Wightman boundary 2-point functions of operators dual to massive scalar fields in the asymptotically flat bulk.Comment: 27 pages, 2 figures. Explicit discussion added of using the Wightman function method to calculate time-ordered boundary 2-point functions. The resulting branch cuts are linked to the bulk spectrum of state

On the Stress Tensor of Kerr/CFT

The recently-conjectured Kerr/CFT correspondence posits a field theory dual to dynamics in the near-horizon region of an extreme Kerr black hole with certain boundary conditions. We construct a boundary stress tensor for this theory via covariant phase space techniques. The structure of the stress tensor indicates that any dual theory is a discrete light cone quantum theory, in agreement with recent arguments by Balasubramanian et al. The key technical step in our construction is the addition of an appropriate counter-term to the symplectic structure, which is necessary to make the theory fully covariant and to resolve a subtle problem involving the integrability of charges.Comment: 19 page

The causal boundary of wave-type spacetimes

A complete and systematic approach to compute the causal boundary of wave-type spacetimes is carried out. The case of a 1-dimensional boundary is specially analyzed and its critical appearance in pp-wave type spacetimes is emphasized. In particular, the corresponding results obtained in the framework of the AdS/CFT correspondence for holography on the boundary, are reinterpreted and very widely generalized. Technically, a recent new definition of causal boundary is used and stressed. Moreover, a set of mathematical tools is introduced (analytical functional approach, Sturm-Liouville theory, Fermat-type arrival time, Busemann-type functions).Comment: 41 pages, 1 table. Included 4 new figures, and some small modifications. To appear in JHE