abstract

The relatively simple Fibre-Bundle geometry of a Yang–Mills gauge theory — mainly the clear distinction between base and fibre — made it possible, between 1953 and 1971, to construct a fully quantized version and prove that theory’s renormalizability; moreover, nonperturbative (topological) solutions were subsequently found in both the fully symmetric and the spontaneously broken modes (instantons, monopoles). Though originally constructed as a model formalism, it became in 1974 the mathematical mold holding the entire Standard Model (i.e. QCD and the Electroweak theory). On the other hand, between 1974 and 1984, Einstein’s theory was shown to be perturbatively nonrenormalizable. Since 1974, the search for Quantum Gravity has therefore provided the main motivation for the construction of Gauge Theories of Gravity. Earlier, however, in 1958-76 several such attempts were initiated, for aesthetic or heuristic reasons, to provide a better understanding of the algebraic structure of GR. A third motivation has come from the interest in Unification, making it necessary to bring GR into a form compatible with an enlargement of the Standard Model. Models can be classified according to the relevant structure group in the fibre. Within the Poincaré group, this has been either the \(R^{4}\) translations, or the Lorentz group \({\rm SL}(2,C)\) — or the entire Poincaré \({\rm SL}(2,C)\times R^{4}\). Enlarging the group has involved the use of the Conformal \({\rm SU}(2,2)\), the special Affine \(\overline {{\rm SA}}(4,R)=\overline {{\rm SL}}(4,R)\times R^{4}\) or Affine \(\overline {A}(4,R)\) groups. Supergroups have included supersymmetry, i.e. the graded-Poincaré group \((n=1\ldots 8\) in its extensions) or the superconformal \({\rm SU}(2,2/n)\). These supergravity theories have exploited the lessons of the aesthetic-heuristic models — Einstein–Cartan etc. — and also achieved the Unification target. Although perturbative renormalizability has been achieved in some models, whether they satisfy unitarity is not known. The nonperturbative Ashtekar program has exploited the understanding of instantons and self-dual solutions in QCD, in the complexification and in the selection of new variables. Note that supergravity involves Lie Derivatives as supertranlations, and several models have treated local spacetime translations similarly. The reduction of the larger groups, down to Poincaré, has involved spontaneous fibration and spontaneous symmetry breakdown. In this context, noncommutative geometry may allow for further geometrization.

abstract

The group structure of quasicrystallographic space groups is described in an alternative way, using the language of group extensions. The rôle of the lattice of translations in classical crystallography is pointed out to be taken by the abstract dual of the generalized kind of lattice in Fourier space that one has in the quasi case.

abstract

The canonical formalism for general relativity on a null surface is compared with that on a space-like surface using Ashtekar variables, the self- dual connection and a densitized triad. The principal difference lies in the appearance of second class constraints. These arise in part because the metric on a null surface is singular, in part because on a null surface there is a preferred direction, and in part because a compact mapping will not map a null surface into a null surface. Second class constraints are eliminated by the use of Dirac brackets. It is shown that, in principle, this is particularly straightforward in this case.

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We study the question to what extend classical Hodge–deRham theory for harmonic differential forms carries over to harmonic spinors. Despite some special phenomena in very low dimensions and despite the Atiyah–Singer index theorem which provides a link between harmonic spinors and the topology of the underlying manifold it turns out that in many dimensions harmonic spinors are not topologically obstructed. In this respect harmonic spinors behave very differently from harmonic differential forms. We also discuss parallel spinors and Killing spinors.

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The teleparallelism equivalent GR\(_\parallel \) of gravity is considered. After its complexification via a canonical transformation, it becomes a true Yang–Mills theory of translations. It is shown that states constructed from the translational Chern–Simons term \(\underline {\cal C}_{\rm TT}\) fully solve the corresponding Hamiltonian and diffeomorphism constraints.

abstract

Basic features of the conservation laws in the Hamiltonian approach to the Poincaré gauge theory are presented. It is shown that the Hamiltonian is given as a linear combination of ten first class constraints. The Poisson bracket algebra of these constraints is used to construct the gauge generators. By assuming that the asymptotic symmetry is the global Poincaré symmetry, we derived the improved form of the asymptotic generators, and discussed the related conservation laws of energy, momentum, etc.

abstract

In Kaluza–Klein theory one usually computes the scalar curvature of the principal bundle manifold using the Levi–Civita connection. Here we consider a natural family of invariant connections on a soldered principal bundle which is then parallelizable and hence spinable. This 3-parameter family includes the Levi–Civita connection and the flat connection. By varying the connection instead of merely scaling the metric on the fibers, there is greater independence among the coupling constants in the scalar curvature. In particular, a large cosmological constant can be avoided in spite of tiny fibers.

abstract

It is conjectured that space-time and momentum space may be both conformally compactified and correlated by conformal inversion, rendering a priori impossible the empirical realization of the concept of both infinity and infinitesimal. It appears that in such a world momentum space is appropriate for the description of quantum mechanics in spinorial form. An exactly soluble, two-dimensional model is presented and discussed.

abstract

New variables related to spin-structures are introduced with the aim of replacing the metric in the description of gravity. These new variables provide a general framework which allows one to deal with interactions between spinors and a dynamical gravitational field, thus generalizing the notion of spinors on curved spaces. In this framework there is no action of space-time diffeomorphisms on the configuration bundle, but there is covariance with respect to automorphisms of a suitable principal bundle, as it is standard in gauge theories. A concrete example of spinors interacting with gravity is considered as an application.

abstract

The Coulomb field of the proton in a hydrogen atom is a completely classical object. We know it from the success of the Dirac equation in which the classical Coulomb field is put in. However, the proton’s charge, which gives the scale of the Coulomb field, is quantized. Thus the Coulomb field behaves like Bohr’s orbits in the old quantum theory: its spatio-temporal shape is classical but its magnitude is quantized. The Author explains this curious state of affairs. There are two distinct regimes of the electromagnetic field: the regime described by the standard quantum electrodynamics and zero-frequency regime, which is translationally invariant and has only the Lorentz group as its symmetry group. The electric charge is a part of the translationally invariant zero-frequency regime and as such can indeed be quantized.

abstract

Propagation of scalar field waves interacting with a strong gravitational field exhibits effects of backscattering. Due to this the total flux of radiation can diminish if sources are placed in a region adjoining a compact body. Backscattering can be neglected in the case when the emitter is located at a distance much larger than the Schwarzschild radius. The effect should be detectable in some astrophysical sources of electromagnetic or gravitational radiation.

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The problem of cosmological singularity of the general relativity theory is discussed in the frame of gauge approach to gravitation. Equations of isotropic cosmology obtained in the frame of gauge theories of gravitation are given. Some regular cosmological solutions of these equations are discussed.

abstract

The effect of constraints on the initial value and acausal propagation problems in the Poincaré Gauge Theory is considered with the aid of the linearized theory and the Hamiltonian analysis. To linear order there are no difficulties, however non-linearities in any extra “if” constraints can cause serious problems involving a change in the number and type of constraints as well as acausal propagation modes. Specific examples are given. Only very special parameter choices are expected to avoid these problems. A similar story is predicted for most other gauge theories of gravity. This type of analysis holds promise as a strong test for alternate gravity theories.

abstract

Regarding Pauli’s matrices as proper Higgs fields one can deduce an effective approximation for gravity in flat space. In this work we extend this approximation up to the second order, reaching complete agreement in the special case of gravitational waves. Unification in view, we introduce isospinorial degrees of freedom. In this way the mass spectrum and chiral asymmetry can be generated with the help of an additional scalar Higgs field. The Higgs modes corresponding to gravity are discussed.

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In this contribution we discuss a recent result which shows that a gravitational collapse cannot in generic situations lead to the formation of a final state resembling the Kerr solution with a naked singularity. This result supports the validity of the cosmic censorship hypothesis.

abstract

Motivated by the invariance of actions under gauge symmetries the definitions of standard clocks in theories of gravitation are discussed. We argue that standard Einsteinian clocks can be defined in non-Riemannian theories of gravitation and that atomic clocks may be adopted to measure proper time in the presence of non-Riemannian gravitational fields. These ideas are illustrated in terms of a recently developed model of gravitation based on a non-Riemannian space-time geometry.

abstract

A dual description of 3-dimensional topological Seiberg–Witten theory in terms of the Alexander invariant on manifolds obtained via surgery on a knot is proposed. The description directly follows from a low-energy analysis of the corresponding SUSY theory, in full analogy to the 4-dimensional case.

abstract

A hamiltonian description of the physical system composed of a dust shell interacting with the gravitational field is considered. In the spherically symmetric case, the phase space of the system is effectively reduced with respect to the Gauss–Codazzi constraints. The Hamiltonian of the system (numerically equal to the value of the A. D. M. mass) is explicitly calculated in terms of the “true degrees of freedom”, i.e. as a function on the reduced phase space.

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We draw attention to a novel type of geometric gauge invariance relating the autoparallel equations of motion in different Riemann-Cartan spacetimes with each other. The novelty lies in the fact that the equations of motion are invariant even though the actions are not. As an application we use this gauge transformation to map the action of a spinless point particle in a Riemann-Cartan spacetime with a gradient torsion to a purely Riemann spacetime, in which the initial torsion appears as a nongeometric external field. By extremizing the transformed action in the usual way, we obtain the same autoparallel equations of motion as those derived in the initial spacetime with torsion via a recently-discovered variational principle.

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We discuss a construction of a wormhole with the following properties: the wormhole connects to the same asymptotic region and is one-way traversable i.e. there exist timelike curves that start and end in the same asymptotic region and go through the wormhole. Moreover it is possible to satisfy the energy conditions. From any point in the asymptotic region there exist closed timelike curves.

abstract

The lecture explains the geometric basis for the recently-discovered nonholonomic mapping principle which specifies certain laws of nature in spacetimes with curvature and torsion from those in flat spacetime, thus replacing and extending Einstein’s equivalence principle. An important consequence is a new action principle for determining the equation of motion of a free spinless point particle in such spacetimes. Surprisingly, this equation contains a torsion force, although the action involves only the metric. This force changes geodesic into autoparallel trajectories, which are a direct manifestation of inertia. The geometric origin of the torsion force is a closure failure of parallelograms. The torsion force changes the covariant conservation law of the energy-momentum tensor whose new form is derived.

abstract

As is well-known, Newton’s gravitational theory can be formulated as a four-dimensional space-time theory and follows as singular limit from Einstein’s theory, if the velocity of light tends to the infinity. Here ’singular’ stands for the fact, that the limiting geometrical structure differs from a regular Riemannian space-time. Geometrically, the transition Einstein \(\rightarrow \) Newton can be viewed as an ’opening’ of the light cones. This picture suggests that there might be other singular limits of Einstein’s theory: Let all light cones shrink and ultimately become part of a congruence of singular world lines. The limiting structure may be considered as a nullhypersurface embedded in a five-dimensional spacetime. While the velocity of light tends to zero here, all other velocities tend to the velocity of light. Thus one may speak of an ultrarelativistic limit of General Relativity. The resulting theory is as simple as Newton’s gravitational theory, with the basic difference, that Newton’s elliptic differential equation is replaced by essentially ordinary differential equations, with derivatives tangent to the generators of the singular congruence. The Galilei group is replaced by the Carroll group introduced by Lévy-Leblond. We suggest to study near ultrarelativistic situations with a perturbational approach starting from the singular structure, similar to post-Newtonian expansions in the \(c \rightarrow \infty \) case.

abstract

The role of the Equivalence Principle (EP) in classical and quantum mechanics is reviewed. It is shown that the weak EP has a counterpart in quantum theory, a Quantum Equivalence Principle (QEP). This implies that also in the quantum domain the geometrization of the gravitational interaction is an operational procedure similar to the procedure in classical physics. This QEP can be used for showing that it is only the usual Schrödinger equation coupled to gravito–inertial fields which obeys our equivalence principle. In addition, the QEP applied to a generalized Pauli equation including spin results in a characterization of the gravitational fields which can be identified with the Newtonian potential and with torsion. Also, in the classical limit it is possible to state beside the usual EP for the path an EP for the spin which again may be used for introducing torsion as a gravitational field.

abstract

In recent numerical simulations of spherically symmetric gravitational collapse a new type of critical behaviour, dominated by a sphaleron solution, has been found. In contrast to the previously studied models, in this case there is a finite gap in the spectrum of black-hole masses which is reminiscent of a first order phase transition. We briefly summarize the essential features of this phase transition and describe the basic heuristic picture underlying the numerical phenomenology.

abstract

It is well known that fundamental linear higher-spin (\(\geq 2\)) fields are unphysical: they cannot be a a source of gravity, i.e. their dynamics is inconsistent unless they exist as test fields in an empty space. A certain kind of nonlinear spin-2 field arises from vacuum nonlinear metric gravity theories (Lagrangian being any smooth scalar function of Ricci tensor) as a component of a multiplet of tensor fields describing gravity. These theories can be reformulated as Einstein gravity theory with gravity described by the metric field alone and the other fields contained in the multiplet acting as a “matter” source in Einstein field equations. This framework provides a consistent gravitational interaction for the spin-2 field. A number of open problems still remains. This paper is a progress report on a joint work done by Guido Magnano and myself.

abstract

World spinors are objects that transform w.r.t. double covering group \(\overline {\rm Diff}(4,R)\) of the Group of General Coordinate Transformations. The basic mathematical results and the corresponding physical interpretation concerning these, infinite-dimensional, spinorial representations are reviewed. The role of groups Diff\((4,R)\), GA\((4,R)\), GL\((4,R)\), SL\((4,R)\), SO(3,1) and the corresponding covering groups is pointed out. New results on the infinite dimensionality of spinorial representations, explicit construction of the \(\overline {\rm SL}(4,R)\) representations in the basis of finite-dimensional non-unitary SL\((2,C)\) representations, SL\((4,R)\) representation regrouping of tensorial and spinorial fields of an arbitrary spin lagrangian field theory, as well as its SL\((5,R)\) generalization in the case of infinite-component world spinor and tensor field theories are presented.

abstract

The uncertainty relations associated to the covariant \(\kappa \)-deformation of \(D=4\) relativistic symmetries, with quantum “time” coordinate and modified Heisenberg algebra, are shown to be consistent with independent heuristic estimates of limitations on the measurability of space-time distances. Our analysis generalizes the one previously reported by one of us, which considered only the space-time coordinate sector.

abstract

We find a class of electrically charged exact solutions for a toy model of metric-affine gravity. Their metric is of the Plebański-Demiański type and their nonmetricity and torsion are represented by a triplet of covectors with dilation, shear, and spin charges.

abstract

We point out that in the gauge approach to gravity it is not always possible to reduce translation invariance to diffeomorphism invariance. It is argued that a proper generator of translations on the spacetime manifold is given by a gauge covariant Lie derivative. A reduction to diffeomorphism invariance is obtained if the gauging of the translation group does not involve homogeneous frame transformations. Possible consequences are shortly discussed.

abstract

We consider the classical theory of the Dirac massive particle in the Riemann-Cartan spacetime. We demonstrate that the translational and the Lorentz gravitational moments, obtained by means of the Gordon type decompositions of the canonical energy-momentum and spin currents, are consistently coupled to torsion and curvature, as expected.