Principle of event symmetry

To accommodate topology change, the symmetry of space-time must be extended from the diffeomorphism group of a manifold to the symmetric group acting on the discrete set of space-time events. This is the principle ofevent-symmetric space-time. I investigate a number of physical toy models with this symmetry to gain some insight into the likely nature of event-symmetric space-time. In the more advanced models the symmetric group is embedded into larger structures such as matrix groups which provide scope to unify space-time symmetry with the internal gauge symmetries of particle physics. I also suggest that the symmetric group of space-time could be related to the symmetric group acting to exchange identical particles, implying a unification of space-time and matter. I end with a definition of a new type of loop symmetry which is important in event-symmetric superstring theory.     

Operator-valued connections,Lie connections,and Gauge field theory

Noether's first theorem tells us that the global symmetry groupG _r of an action integral is a Lie group of point transformations that acts on the Cartesian product of the space-time manifold with the space of states and their derivatives. Gauge theory constructs are thus required for symmetry groups that act indiscriminately on the independent and dependent variables where the group structure can not necessarily be realized as a subgroup of the general linear group. Noting that the Lie algebra of a general symmetry groupG _r can be realized as a Lie algebrag _r of Lie derivatives on an appropriately structured manifold,G _r -covariant derivatives are introduced through study of connection 1-forms that take their values in the Lie algebrag _r of Lie derivatives (operator-valued connections). This leads to a general theory of operator-valued curvature 2-forms and to the important special class of Lie connections. The latter are naturally associated with the minimal replacement and minimal coupling constructs of gauge theory when the symmetry groupG _r is allowed to act locally. Lie connections give rise to the gauge fields that compensate for the local action ofG _r in a natural way. All governing field equations and their integrability conditions are derived for an arbitrary finite dimensional Lie group of symmetries. The case whereG _r contains the ten-parameter Poincaré group on a flat space-timeM ₄ is considered. The Lorentz structure ofM ₄ is shown to give a pseudo-Riemannian structure of signature 2 under the minimal replacement associated with the Lie connection of the local action of the Poincaré group. Field equations for the matter fields and the gauge fields are given for any system of matter fields whose action integral is invariant under the global action of the Poincaré group.     

Towards a unified gauge theory of gravitational and strong interactions

We study the space-time properties of leptons and hadrons and find it necessary to extend general relativity to the gauge theory based on the four-dimensional affine group. This group translates and deforms the tetrads of the locally Minkowskian space-time. Its conserved currents, momentum, and hypermomentum, act as sources in the two field equations of gravity. A Lagrangian quadratic in torsion and curvature allows for the propagation of two independent gauge fields: translationale-gravity mediated by the tetrad coefficients, and deformational -gravity mediated by the connection coefficients. For macroscopic mattere-gravity coincides with general relativity up to the post-Newtonian approximation of fourth order. For microscopic matter -gravity represents a strong Yang-Mills type interaction. In the linear approximation, for a static source, a confinement potential is found.This essay received an honorable mention (1979) from the Gravity Research Foundation.-Ed.     

Differentiable probabilities: A new viewpoint on spin,gauge invariance,gauge fields,and relativistic quantum mechanics

A new approach to developing formulisms of physics based solely on laws of mathematics is presented. From simple, classical statistical definitions for the observed space-time position and proper velocity of a particle having a discrete spectrum of internal states we derive u generalized Schrödinger equation on the space-time manifold. This governs the evolution of an N component wave function with each component square integrable over this manifold and is structured like that for a charged particle in an electromagnetic field but also includes SU(N) gauge field couplings. This construction reveals a new hasis for gauge invariance and new insight into the appearance of spin and other such properties in relativistic quantum mechanics and suggests a new charged particle model.     

Towards a quantization of gauge fields on de Sitter group by functional integral method

A formulation of the de Sitter symmetry as a purely inner symmetry defined on a fixed Minkowski space-time is presented. We define the generators of the de Sitter group and write the structure equations using a constant deformation parameter λ. The conserved gauge currents are calculated, and their physical meaning is given. Local gauge transformations and the corresponding covariant derivative depending on the gauge fields are also obtained. We study the behavior of gauge fields, the torsion and curvature tensors and give a regularization technique in terms of the ζ function.     

Riemannian geometry in spaces with Grassman coordinates

The use of spaces containing Grassman (anticommuting) coordinates (in addition to the usual space-time coordinates) as a framework for unified gauge theories is described. The theory developed represents a local gauge-invariant extension of conventional (global) supersymmetry. Aside from containing the usual general coordinate invariance group of gravitational theory, the gauge supersymmetry group is seen to also encompass other symmetries of particle physics, e.g., electromagnetic (or Yang-Mills) invariance. The role of spontaneous symmetry breaking and the field equations unifying the Einstein, Maxwell, and Dirac interactions are discussed.Research supported in part by the National Science Foundation.Invited talk at the conference, The Riddle of Gravitation, on the Occasion of the 60th Birthday of Peter G. Bergmann, Syracuse, New York, March 1975.     

Quark confinement and the short-range component of general affine gauge gravity

Within the framework of a gauge field theory based on the general affine space-time symmetry, we propose a certain purely quadratic gauge field lagrangian. In the large-scale region it yields an Einstein-Cartan-like gravity with Newton's constant generated spontaneously, while in the particle domain it yields a renormalizable theory with a confining potential applying to quarks and not to leptons.     

Generalized coupling constants for broken gauge symmetries in geometrical terms

A geometrical treatment of the gauge coupling constant is proposed in terms of a generalized connection form using fibre-bundle language. This extends the notion of the coupling constant to a notion of a field. The reduction of a curvature form for the generalized connection form is described in the case of a reduction of a structure group G to a subgroup H (broken gauge symmetry), and a coupling constant for the gauge group H is constructed from the corresponding one for the gauge group G.     

Effective lagrangian for spontaneously broken N = 2 superconformal symmetry

In a general framework of non-linear realizations for space-time symmetries, we investigate the effective lagrangian for N = 2 superconformal symmetry which is spontaneously broken down to N = 2 super-Poincaré symmetry. For the case in which the dilation multiplet is a massless N = 2 gauge multiplet, we derive a low-energy effective lagrangian which describes the interaction of Nambu-Goldstone particles.     

Stability of proton and maximally symmetric minimal unification model for basic forces and building blocks of matter

With the hypothesis that all independent degrees of freedom of basic building blocks should be treated equally on the same footing and correlated by a possible maximal symmetry, we arrive at a 4-dimensional space-time unification model. In this model the basic building blocks are Majorana fermions in the spinor repre- sentation of 14-dimensional quantum space-time with a gauge symmetry GM4D = SO(1,3)×SU(32)×U(1)A×SU(3)F. The model leads to new physics including mirror particles of the standard model. It enables us to issue some fundamental questions that include: why our living space-time is 4-dimensional, why parity is not con- served in our world, how the stability of proton is, what the origin of CP violation is and what the dark matter can be.     

Gravity and gauge: A new perspective

Realization of the Poincaré groupP ₁₀ as a subgroup ofGL(5,R) that maps a 4-dimensional affine set into itself has been shown to lead to a direct Yang-Mills gauging process. This paper discusses the differences between direct gauge theory forP ₁₀ and previously published works. These differences are fundamental, both physically and mathematically, and lead to marked departures from previous concepts and interpretations. The translation subgroup is correctly gauged; the metric structure and metric compatibility are derived from the gauging process rather than assumed; spin structures are automatically incorporated in a consistent manner; the local holonomy group is shown to be the component of the Lorentz group connected to the identity; the geometric analog of Yang-Mills minimal coupling precludes dependence of the free gauge field Lagranian on torsion; and the theory reduces exactly to general relativity when the momentumenergy complex is symmetric and all matter fields are spin-free. Gravitational effects on neutral test particles are shown to arise from the compensating 1-forms for local action of Lorentz boosts. The compensating 1-forms for local action of the translation subgroup may be interpreted as space-time dislocations, while the compensating 1-forms for the rotation subgroup can be viewed as space-time disclinations. Unfortunately, there are no clear physical meanings that can be ascribed to space-time dislocations or disclinations.     

Gauge fields,space-time geometry and gravity

A general framework for the gauge theory of the affine group and its various subgroups in terms of connections on the bundle of affine frames and its subbundles is given, with emphasis on the correct gauging of groups including space-time translations. For consistency of interpretation, the appropriate objects to be identified with gravitational vierbeins in such theories are not the translational gauge fields themselves, but their pull backs,via appropriate bundle homomorphisms, to the bundle of frames. This automatically solves the problems usually encountered in constructing a gauge theory of the conventional sort for groups containing translations. We give a consistent formulation of the Poincare gauge theory and also of the theory based on translational gauge invariance which, in the absence of matter fields with intrinsic spin, gives a local Lorentz invariant theory equivalent to Einstein gravity.     

The generalized radiation gauge in curved space-time

In the following we give a necessary and sufficient criterion for the existence of theU(1) radiation gauge in a curved space-time with isometry. This criterion is purely geometric and leads to a (local) 3+1-split of space-time with vanishing extrinsic curvature. If the symmetry is timelike and in absence of charges the generalization of the Coulomb gauge leads to a time evolution of the Maxwell field which is analogous to that in flat space-time.     

Rotational symmetry of classical orbits, arbitrary quantization of angular momentum and the role of the gauge field in two-dimensional space

We study quantum–classical correspondence in terms of the coherent wave functions of a charged particle in two- dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 2π-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.     

T4 gauge theory of gravity and new general relativity

We formulate a space-time translationT ₄ gauge theory of gravity on the Minkowski space-time with appropriate choice of the Lagrangian. By comparing the energy-momentum law of this theory with that of new general relativity constructed on the Weitzenböck space-time we find that in the classical limit the gauge potentials correspond to the parallel vector fields in the Weitzenböck space-time and the gauge field equation coincides with the field equation of gravity in new general relativity in the linearized version. Thus we conclude that in the classical limit theT ₄ gauge theory of gravity leads to the new general relativity.     

Critical behavior at finite-temperature confinement transitions

Confinement in a pure gauge theory at non-zero temperature may be discussed in terms of an order parameter which transforms under a global symmetry group, the center of the gauge group. Integrating out all degrees of freedom except this order parameter generates an effective scalar field theory for the order parameter, globally invariant under the center symmetry. We argue that the effective theory possesses only short-range couplings, and hence that the finite-temperature confinement phase transition (when continuous) is accompanied by long-range fluctuations only in the order parameter. Universality ideas then lead to predictions for the critical properties of U(1), Z(N), and SU(N) gauge theories for all dimensionalities of space-time. An explicit renormalization-group calculation is presented for the U(1) gauge theory in (2 + 1) dimensions, the results of which fit the general picture.