Topics in Mathematical Physics, General Relativity and Cosmology in Honor of Jerzy Plebanski

PREFACE

CONTENTS

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To arrive at the AdS/CFT correspondence, Maldacena (building upon pre-existing work, most notably that of Klebanov et al.) assumed that already at the string theory level there exists a duality between the two alternative descriptions of D-brane physics, and then proceeded to take a low-energy limit, which in particular decouples the branes from the bulk. In this paper, aimed primarily at non-string-theorists, we review some of the issues one encounters when trying to formulate a precise duality statement at the level of string theory (prior to taking any limits). We also survey some of the results obtained in the study of the correspondence in the intermediate-energy regime where excited string modes are negligible but the branes are still coupled to the bulk. In this simplified context, we explain in particular how symmetries have been used to determine the form of the D3-brane effective action relevant to the duality, go over the derivation of a recipe to compute correlation functions in this theory, and discuss some of the questions that remain open.

A summary of on how black holes grow in full, non-linear general relativity is presented. Specifically, a notion of dynamical horizons is introduced and expressions of fluxes of energy and angular momentum carried by gravitational waves across these horizons are obtained. Fluxes are local and the energy flux is positive. Change in the horizon area is related to these fluxes. The flux formulae also give rise to balance laws analogous to the ones obtained by Bondi and Sachs at null infinity and provide generalizations of the first and second laws of black hole mechanics.

It is shown that there exists a commuting diagram of mappings between dynamics of classical systems on one side and variational principles for geodesic lines in stationary spacetimes of general relativity on the other. The construction of the mappings is based on classical Routh's and Jacobi's reduction procedures and on corresponding inverse procedures which are reviewed in the paper.

This work deals with solutions of the Einstein-Born-Infeld (EBI) theory, that can represent black holes or soliton structures (EBIons). The horizon structure of EBI black hole is analyzed as well as it is tested in the isolated horizon framework recently proposed by Ashtekar, considering a fixed charge and varying BI parameter.

The field of neutron interferometry achieved one of its most significant successes with the detection of the influence of gravity in the quantum mechanical phase of a thermal neutron beam. From the latest experimental readouts in this context an intriguing discrepancy has been elicited. Indeed, theory and experiment dissent by one per cent, and though this fact could be a consequence of the mounting of the experimental device, it might also embody a difference between the way in which gravity behaves in classical and quantum mechanics. In this work the effects, upon the interference pattern, of space-time torsion will be analyzed heeding its coupling with the spin of the neutron beam. It will be proved that, even with this contribution, there is enough leeway for a further discussion of the validity of the equivalence principle in nonrelativistic quantum mechanics.

Some properties of Plebański squeezing operator and squeezed states created with time-dependent quadratic in position and momentum Hamiltonians are reviewed. New type of tomography of quantum states called squeeze tomography is discussed.

The creation of brane universes induced by a totally antisymmetric tensor living in a fixed background spacetime is presented, where a term involving the intrinsic curvature of the brane is considered. A canonical quantum mechanical approach employing Wheeler-DeWitt equation is done. The probability nucleation for the brane is calculated taking into account both an instanton method and a WKB approximation. Some cosmological implications arose from the model are presented.

We study deformation quantization on an infinite-dimensional Hilbert space W endowed with a Poisson structure. We make explicit the example of Moyal star-product and we show that it is well defined on a subalgebra of C^∞ (W) specified by conditions of Hilbert-Schmidt type.

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We study the theory of the (1/2, 0) ⊕ (0,1/2) representation in helicity basis. Helicity eigenstates are not the parity eigenstates. This is in accordance with the consideration of Berestetskiĭ, Lifshitz and Pitaevskiĭ. Relations to the Gelfand-Tsetlin-Sokolik-type quantum field theory are discussed. Finally, a new form of the parity operator is proposed. It commutes with the Hamiltonian.

We propose a natural classification scheme of the second order supersymmetry transformations in quantum mechanics. The possibilities of manipulating spectra offered by those transformations are also explored. In particular, it is shown that the difficulty of modifying the excited state levels can be now overcome, therefore enlarging the possibilities offered by the standard first order treatment. The results are illustrated taking as the initial system the standard harmonic oscillator.

Self-dual Einstein metrics which admit one (rotational) symmetry vector are determined by solutions of the sDiff(2) Toda equation, which has also been studied in a variety of other physical contexts. Non-trivial solutions are difficult to obtain, with considerable effort in that direction recently. Therefore much effort has been involved with determining solutions with symmetries, and also with a specific lack of symmetries. The contact symmetries have been known for some time, and form an infinite-dimensional Lie algebra over the jet bundle of the equation. Generalizations of those symmetries to include derivatives of arbitrary order are often referred to as higher- or generalized-symmetries. Those symmetries are described, with the unexpected result that their existence also requires prolongations to "potentials" for the original dependent variables for the equation: potentials which are generalizations of those already usually introduced for this equation. Those prolongations are described, and the prolongations of the commutators for the symmetry generators are created. The generators so created form an infinite-dimensional, Abelian, Lie algebra, defined over these prolongations.

We describe a natural relationship between all 3^rd order ODEs with a vanishing Wunschmann invariant, with all conformal Lorentzian metrics on 3-manifolds and Cartan's normal O(3,2) conformal connections. The generalization to pairs of second order PDEs and their relationship to Cartan's normal O(4,2) conformal connections on four dimensional manifolds is discussed.

In this paper we survey some of the relations between Plebański description of self-dual gravity through the heavenly equations and the physics (and mathematics) of Strings. In particular we focus on the correspondence between the infinite hierarchy in the ground ring structure of BRST operators and its associated Boyer-Plebański construction of infinite conserved quantities in self-dual gravity. We comment on “Mirror Symmetry” in these models and the large-N duality between topological gauge theories in two dimensions and topological gravity in four dimensions. Finally D-branes in this context are briefly outlined.

In this contribution a noncommutative deformation of topological BF theory is formulated. Noncom-mutative topological gravity and noncommutative self-dual gravity are surveyed as examples of this BF theory. In the procedure it is shown that self-duality enters as a central ingredient that facilitates the construction of invariant noncommutative actions. In the dynamical case, the procedure constitutes a proposal for constructing noncommutative full Einstein theory.

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Electromagnetic wavelets are a family of 3 × 3 matrix fields 𝕎_z (x^′) parameterized by complex spacetime points z = x+iy with y timelike. They are translates of a basic wavelet 𝕎(z) holomorphic in the future-oriented union of the forward and backward tubes. Applied to a polarization vector p = p_m − ip_e, 𝕎(z) gives an anti-selfdual solution 𝕎(z)p derived from a selfdual Hertz potential , where S is the Synge function acting as a Whittaker-like scalar Hertz potential. Resolutions of unity exist giving representations of sourceless electromagnetic fields as superpositions of wavelets. With the choice of a branch cut, S splits into a difference S⁺ (z) − S⁻ (z) of retarded and advanced pulsed beams whose limits as y → 0 give the propagators of the wave equation. This yields a similar splitting of the wavelets and leads to their complete physical interpretation as pulsed beams absorbed and emitted by a disk source D(y) representing the branch cut. The choice of y determines the beam's orientation, collimation and duration, giving beams as sharp with pulses as short as desired. The sources are computed as spacetime distributions of electric and magnetic dipoles supported on D(y). The wavelet representation of sourceless electromagnetic fields now splits into representations with advanced and retarded sources. These representations are the electromagnetic counterpart of relativistic coherent-state representations previously derived for massive Klein-Gordon and Dirac particles.

We describe the generalized κ-deformations of D = 4 relativistic symmetries with finite masslike deformation parameter κ and an arbitrary direction in κ-deformed Minkowski space being noncommutative. The corresponding bicovariant differential calculi on κ-deformed Minkowski spaces are considered. Two distinguished cases are discussed: 5D noncommutative differential calculus (κ-deformation in time-like or space-like direction), and 4D noncommutative differential calculus having the classical dimension (noncommutative κ-deformation in light-like direction). We introduce also left and right vector fields acting on functions of noncommutative Minkowski coordinates, and describe the noncommutative differential realizations of κ-deformed Poincaré algebra. The κ-deformed Klein-Gordon field on noncommutative Minkowski space with noncommutative time (standard κ-deformation) as well as noncommutative null line (light-like κ-deformation) are discussed. Following our earlier proposal (see Refs. 1, 2) we introduce an equivalent framework replacing the local noncommutative field theory by the nonlocal commutative description with suitable nonlocal star product multiplication rules. The modification of Pauli–Jordan commutator function is described and the κ-dependence of its light-cone behaviour in coordinate space is explicitely given. The problem with the κ-deformed energy-momentum conservation law is recalled.

Structure formation is described by a Lemaître-Tolman model such that the initial density perturbation within a homogeneous background has a smaller mass than the structure into which it will develop, and accretes more mass during evolution. It is proved that any two spherically symmetric density profiles specified on any two constant time slices can be joined by a Lemaître-Tolman evolution, and exact implicit formulae for the arbitrary functions that determine the resulting L-T model are obtained. Examples of the process are investigated numerically.

We review some results concerning the classification of orientifolds and branes by K-theory, as well as the role played by the Atiyah-Herzibruch Spectral Sequence relating cohomology and K-theory. The non-existencie of certain types of orientifold planes, their fractional charge and the topological obstruction to have non-zero K-theory charges, avoiding global gauge anomalies, are some of the physical consequences of this relation.

We describe the holographic correspondence between field theories and string/M theory, focusing on the relation between compactifications of string/M theory on Anti-de Sitter spaces and conformal field theories. We review the background for this correspondence and discuss its motivations and the evidence for its correctness. We describe the main results that have been derived from the correspondence in the regime that the field theory is approximated by classical or semiclassical gravity. We focus on the case of the supersymmetric gauge theory in four dimensions.

We analize the Becchi-Rouet-Stora-Tyutin (BRST) cohomology of the teleparallelism equivalent of gravity, for which the algebra of constraints depends only on torsion. In order to construct the Laplace-Beltrami BRST-invariant operator Δ_T, the co-BRST operator is employed. In the case of purely axial torsion, the resulting formulae resemble rather closely that of Yang-Mills.

Diffeomorphism covariant theories with dynamical background metric, like gravity coupled to matter fields in the way expressed by Einstein-Hilbert's action or relativistic strings described by Polyakov's action, have ‘on-shell’ vanishing energy-momentum tensor t_μν because t_μν is, essentially, the Eulerian derivative associated with the dynamical background metric and thus t_μν vanishes ‘on-shell.’ Therefore, the equations of motion for the dynamical background metric play a double role: as equations of motion themselves and as a reflection of the fact that t_μν = 0. Alternatively, the vanishing property of t_μν can be seen as a reflection of the so-called ‘problem of time’ present in diffeomorphism covariant theories in the sense that t_μν are written as linear combinations of first class constraints only.

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Robertson and Hadamard-Robertson theorems for an arbitrary ordered field are given. Then the Heisenberg-Robertson, Robertson-Schrödinger and trace uncertainty relations in deformation quantization are found. The concept of intelligent state in quantum mechanics is extended to the deformation quantization formalism.

The present work is a brief review of the progressive search of improper δ-functions which are of interest in quantum mechanics and in the problem of motion in General Relativity Theory.

A brief overview is given of some of the mathematical structures underlying perturbation quantum field theory and renormalization, which may be relevant to the understanding of the small structure of space-time.

I illustrate a simple hamiltonian formulation of general relativity, derived from the work of Esposito, Gionti and Stornaiolo, which is manifestly 4d generally covariant and is defined over a finite dimensional space. The spacetime coordinates drop out of the formalism, reflecting the fact that they are not related to observability. The formulation can be interpreted in terms of Toller's reference system transformations, and provides a physical interpretation for the spinnetwork to spinnetwork transition amplitudes computable in principle in loop quantum gravity and in the spin foam models.

In this paper we prove the equivalence between the coefficients of the Magnus expansion provided by Mielnik and Plebański in Ref. 17 and the ones given by us in Ref. 22. The Magnus expansion is central to get approximately the exponential solutions of non autonomous linear differential equations. We also introduce the mathematical framework that puts the work of Magnus¹⁶ in a precise form (a free Lie algebra structure generated by a continuous set of operators²⁴).

We present a new method for the construction of exact cylindrical wave solutions of the Einstein–Maxwell equations which is based on solving two complete singular integral equations in the complex plane of an auxiliary analytical parameter. We demonstrate that in the case of a non-singular symmetry axis the group transformation of internal symmetries used in the solution generation process is defined by the axis expression of the Ernst potential.

We consider the concept of POV-measures in geometric formulation of quantum mechanics. The so called pseudo-Kählerian functions are introduced to be the generalization of observables on .

Discussed are field-theoretic models with degrees of freedom described by the n-leg field in an n-dimensional “space-time” manifold. Lagrangians are generally-covariant and invariant under the internal group GL(n, R). It is shown that the resulting field equations have some correspondence with Einstein theory and possess homogeneous vacuum solutions given by semisimple Lie group spaces or their appropriate deformations. There exists a characteristic link with the generalized Born-Infeld type nonlinearity and relativistic mechanics of structured continua. In our model signature is not introduced by hands, but is given by integration constants for certain differential equations.

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The mathematical theory of deformations has proved to be a powerful tool in modeling physical reality. We start with a short historical and philosophical review of the context and concentrate this rapid presentation on a few interrelated directions where deformation theory is essential in bringing a new framework – which has then to be developed using adapted tools, some of which come from the deformation aspect. Minkowskian space-time can be deformed into Anti de Sitter, where massless particles become composite (also dynamically): this opens new perspectives in particle physics, at least at the electroweak level, including prediction of new mesons. Nonlinear group representations and covariant field equations, coming from interactions, can be viewed as some deformation of their linear (free) part: recognizing this fact can provide a good framework for treating problems in this area, in particular global solutions. Last but not least, (algebras associated with) classical mechanics (and field theory) on a Poisson phase space can be deformed to (algebras associated with) quantum mechanics (and quantum field theory). That is now a frontier domain in mathematics and theoretical physics called deformation quantization, with multiple ramifications, avatars and connections in both mathematics and physics. These include representation theory, quantum groups (when considering Hopf algebras instead of associative or Lie algebras), noncommutative geometry and manifolds, algebraic geometry, number theory, and of course what is regrouped under the name of M-theory. We shall here look at these from the unifying point of view of deformation theory and refer to a limited number of papers as a starting point for further study.

Making use of the complex extension of the space-time we obtain the local expression of the most general Lanczos potential of a space-time that admits a shear-free congruence of null geodesies, assuming that the Lanczos potential and the Ricci tensor are suitably aligned to the congruence. We also show that such a potential is the symmetric part of an object that defines a flat metric connection with torsion.

In this talk I present a framework to take spin foam models to the continuum. The framework is presented taking 3d quantum gravity as an example and completing the program there. Theories in the continuum are defined using a generalization of the small lattice spacing limit of Lattice Gauge Theory (LGT) and the projective techniques used in Canonical Loop Quantization (CLQ). The existence of the theory in the continuum depends on the existence of this limit expressing the independence of the normalized partition function on the lattice in the limit of “small lattice spacing.”

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