Noncanonical Quantization of Gravity. I. Foundations of Affine Quantum Gravity

The nature of the classical canonical phase-space variables for gravity suggests that the associated quantum field operators should obey affine commutation relations rather than canonical commutation relations. Prior to the introduction of constraints, a primary kinematical representation is derived in the form of a reproducing kernel and its associated reproducing kernel Hilbert space. Constraints are introduced following the projection operator method which involves no gauge fixing, no complicated moduli space, nor any auxiliary fields. The result, which is only qualitatively sketched in the present paper, involves another reproducing kernel with which inner products are defined for the physical Hilbert space and which is obtained through a reduction of the original reproducing kernel. Several of the steps involved in this general analysis are illustrated by means of analogous steps applied to one-dimensional quantum mechanical models. These toy models help in motivating and understanding the analysis in the case of gravity.Comment: minor changes, LaTeX, 37 pages, no figure

Extended Hamiltonian systems in multisymplectic field theories

We consider Hamiltonian systems in first-order multisymplectic field theories. We review the properties of Hamiltonian systems in the so-called restricted multimomentum bundle, including the variational principle which leads to the Hamiltonian field equations. In an analogous way to how these systems are defined in the so-called extended (symplectic) formulation of non-autonomous mechanics, we introduce Hamiltonian systems in the extended multimomentum bundle. The geometric properties of these systems are studied, the Hamiltonian equations are analyzed using integrable multivector fields, the corresponding variational principle is also stated, and the relation between the extended and the restricted Hamiltonian systems is established. All these properties are also adapted to certain kinds of submanifolds of the multimomentum bundles in order to cover the case of almost-regular field theories.Comment: 36 pp. The introduction and the abstract have been rewritten. New references are added and some little mistakes are corrected. The title has been slightly modifie

A time-extended Hamiltonian formalism

A Poisson structure on the time-extended space R x M is shown to be appropriate for a Hamiltonian formalism in which time is no more a privileged variable and no a priori geometry is assumed on the space M of motions. Possible geometries induced on the spatial domain M are investigated. An abstract representation space for sl(2,R) algebra with a concrete physical realization by the Darboux-Halphen system is considered for demonstration. The Poisson bi-vector on R x M is shown to possess two intrinsic infinitesimal automorphisms one of which is known as the modular or curl vector field. Anchored to these two, an infinite hierarchy of automorphisms can be generated. Implications on the symmetry structure of Hamiltonian dynamical systems are discussed. As a generalization of the isomorphism between contact flows and their symplectifications, the relation between Hamiltonian flows on R x M and infinitesimal motions on M preserving a geometric structure therein is demonstrated for volume preserving diffeomorphisms in connection with three-dimensional motion of an incompressible fluid.Comment: 14 pages, late

The Unruh-deWitt Detector and the Vacuum in the General Boundary formalism

We discuss how to formulate a condition for choosing the vacuum state of a quantum scalar field on a timelike hyperplane in the general boundary formulation (GBF) using the coupling to an Unruh-DeWitt detector. We explicitly study the response of an Unruh-DeWitt detector for evanescent modes which occur naturally in quantum field theory in the presence of the equivalent of a dielectric boundary. We find that the physically correct vacuum state has to depend on the physical situation outside of the boundaries of the spacetime region considered. Thus it cannot be determined by general principles pertaining only to a subset of spacetime.Comment: Version as published in CQ

Dual giant gravitons in AdSm_m ×\times Yn^n (Sasaki-Einstein)

We consider BPS motion of dual giant gravitons on Ad$S_5\times Y^5$ where $Y^5$ represents a five-dimensional Sasaki-Einstein manifold. We find that the phase space for the BPS dual giant gravitons is symplectically isomorphic to the Calabi-Yau cone over $Y^5$, with the K\"{a}hler form identified with the symplectic form. The quantization of the dual giants therefore coincides with the K\"{a}hler quantization of the cone which leads to an explicit correspondence between holomorphic wavefunctions of dual giants and gauge-invariant operators of the boundary theory. We extend the discussion to dual giants in $AdS_4 \times Y^7$ where $Y^7$ is a seven-dimensional Sasaki-Einstein manifold; for special motions the phase space of the dual giants is symplectically isomorphic to the eight-dimensional Calabi-Yau cone.Comment: 14 pages. (v2) typo's corrected; factors of AdS radius reinstated for clarity; remarks about dual giant wavefunctions in T^{1,1} expanded and put in a new subsectio

Non-standard connections in classical mechanics

In the jet-bundle description of first-order classical field theories there are some elements, such as the lagrangian energy and the construction of the hamiltonian formalism, which require the prior choice of a connection. Bearing these facts in mind, we analyze the situation in the jet-bundle description of time-dependent classical mechanics. So we prove that this connection-dependence also occurs in this case, although it is usually hidden by the use of the ``natural'' connection given by the trivial bundle structure of the phase spaces in consideration. However, we also prove that this dependence is dynamically irrelevant, except where the dynamical variation of the energy is concerned. In addition, the relationship between first integrals and connections is shown for a large enough class of lagrangians.Comment: 17 pages, Latex fil

Fibre bundle formulation of nonrelativistic quantum mechanics: I. Introduction. The evolution transport

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one but it is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it a pure state of some quantum system is described by a state section (along paths) of a (Hilbert) fibre bundle. Its evolution is determined through the bundle (analogue of the) Schr\"odinger equation. Now the dynamical variables and the density operator are described via bundle morphisms (along paths). The mentioned quantities are connected by a number of relations derived in this work. The present first part of this investigation is devoted to the introduction of basic concepts on which the fibre bundle approach to quantum mechanics rests. We show that the evolution of pure quantum-mechanical states can be described as a suitable linear transport along paths, called evolution transport, of the state sections in the Hilbert fibre bundle of states of a considered quantum system.Comment: 26 standard (11pt, A4) LaTeX 2e pages. The packages AMS-LaTeX and amsfonts are required. Revised: new material, references, and comments are added. Minor style chages. Continuation of quan-ph/9803083. For continuation of the this series see http://www.inrne.bas.bg/mathmod/bozhome

Quantizing N=2 Multicenter Solutions

N=2 supergravity in four dimensions, or equivalently N=1 supergravity in five dimensions, has an interesting set of BPS solutions that each correspond to a number of charged centers. This set contains black holes, black rings and their bound states, as well as many smooth solutions. Moduli spaces of such solutions carry a natural symplectic form which we determine, and which allows us to study their quantization. By counting the resulting wavefunctions we come to an independent derivation of some of the wall-crossing formulae. Knowledge of the explicit form of these wavefunctions allows us to find quantum resolutions to some apparent classical paradoxes such as solutions with barely bound centers and those with an infinitely deep throat. We show that quantum effects seem to cap off the throat at a finite depth and we give an estimate for the corresponding mass gap in the dual CFT. This is an interesting example of a system where quantum effects cannot be neglected at macroscopic scales even though the curvature is everywhere small.Comment: 49 pages + appendice

Noncommutative solitons on Kahler manifolds

We construct a new class of scalar noncommutative multi-solitons on an arbitrary Kahler manifold by using Berezin's geometric approach to quantization and its generalization to deformation quantization. We analyze the stability condition which arises from the leading 1/hbar correction to the soliton energy and for homogeneous Kahler manifolds obtain that the stable solitons are given in terms of generalized coherent states. We apply this general formalism to a number of examples, which include the sphere, hyperbolic plane, torus and general symmetric bounded domains. As a general feature we notice that on homogeneous manifolds of positive curvature, solitons tend to attract each other, while if the curvature is negative they will repel each other. Applications of these results are discussed.Comment: 26 pages, 3 figures, harvmac; references adde