Gravitational and electroweak unification by replacing diffeomorphisms with larger group

The covariance group for general relativity, the diffeomorphisms, is replaced by a group of coordinate transformations which contains the diffeomorphisms as a proper subgroup. The larger group is defined by the assumption that all observers will agree whether any given quantity is conserved. Alternatively, and equivalently, it is defined by the assumption that all observers will agree that the general relativistic wave equation describes the propagation of light. Thus, the group replacement is analogous to the replacement of the Lorentz group by the diffeomorphisms that led Einstein from special relativity to general relativity, and is also consistent with the assumption of constant light velocity that led him to special relativity. The enlarged covariance group leads to a non-commutative geometry based not on a manifold, but on a nonlocal space in which paths, rather than points, are the most primitive invariant entities. This yields a theory which unifies the gravitational and electroweak interactions. The theory contains no adjustable parameters, such as those that are chosen arbitrarily in the standard model.     

Consequences of the inertial equivalence of energy

The usual macroscopic theory of relativistic mechanics and electromagnetism is formulated so that all assumptions but one are consistent with both special relativity and Newtonian mechanics, the distinguishing assumption being that to any energyE, whatever its form, there corresponds an inertial massE/c ² . The speed of light enters this formulation only as a consequence of the inertial equivalent of energy1/c ² . While, for1/c ² >0 the resulting theory has symmetry under the Poincaré group, including Lorentz transformations, all its physical consequences can be derived and tested in any one inertial frame. In particular, an account is given in one inertial frame for the dynamic causes of relativistic effects for simple accelerated clocks and roads.     

The foundations of the poincaré group and the validity of general relativity

After discussions about accepted ideas concerning the nonlocalisability of the photon, the interpretation of the Minkowski space-time, the wave-corpuscle duality ideas of Niels Bohr and the concept of elementary particle by Eugene Wigner, the validity of the Poincaré group is brought into question and some other ideas are developed. Lukierski, Nowicki and Ruegg showed that the successes of the Poincaré group are preserved if we deform the group by introducing a constant κ. Such deformation replaces the Poincaré Hopf algebra by another one. We call such a deformation a mathematical deformation. The main inconvenience of this mathematical deformation is that the coproduct is not commutative. The consequence is that a two-particle state is defined in an ambiguous way because we must say which is the first particle and which is the second one. The only mathematical deformation of the Poincaré group which preserves the commutativity of the coproduct is the trivial one, that is the Poincaré Hopf algebra itself. That is why we reject the mathematical deformation of Lukierski, Nowicki and Ruegg. That is also why we propose what we call a physical deformation of the Poincaré group, which means that we reinterpret the Poincaré Hopf algebra, with the same constant κ. Our proposal has four advantages:

1.

1. The constant x has the dimensions of a mass. When this constant becomes infinite, we are left with the Poincaré group with its main successes.

2.

2. The two-particle states are unambiguously defined.

3.

3. The constant κ may be chosen in such a way that the search for a missing mass in the universe is useless.

4.

4. It consists in the disappearing of unphysical irreducible representations of the Poincaré group.

With the constant κ, we arrive at a reformulation of special relativity where the energy is no longer additive. This would imply a change in general relativity where the density of matter must be different from the density of energy. Unfortunately, we are not able to propose a substitute for the general relativity theory. Obviously, when the constant κ goes to infinity, the new general relativity would become the standard general relativity.     

Physical symmetries in a theory of several scalar real fields

Let us consider a theory ofn scalar, real, local, Poincaré covariant quantum fields forming an irreducible set and giving rise to one particle states belonging to the same mass different from zero. The vacuum is unique. It is shown under fairly weak assumptions that every Poincaré and TCP invariant symmetry of the theory, implemented unitarily, which mapps localized elements of the field algebra into operators almost local with respect to the former (such a symmetry we call a physical one) can be defined uniquely in terms of the incoming or outgoing fields and ann-dimensional (real) orthogonal matrix. The symmetry commutes with the scattering matrix. Incidentally we show also that the symmetry groups are compact. A special case of these symmetries are the internal symmetries and symmetries induced by locally conserved currents local with respect to the basic fields and transforming under the same representation of the Poincaré group. We may make linear combinations out the original fields resulting in complex fields and its complex conjugate in a suitable way. The inspection of the representations of the groupsSO(n) and their subgroups sheds some light on the s.c. generalized Carruthers Theorem concerning the self- and pair-conjugate multiplets.     

Relativistic Nonlocal Quantum Field Theory

A theory is defined to be relativistic if its Hamiltonian, total momenta, and boost's generators satisfy commutation relations of the Poincaré group. Field theories with usual local interactions are known to be relativistic. A simple example of a relativistic nonlocal theory is found. However, it has divergences. Some conditions are obtained which are necessary in order that a nonlocal theory be relativistic and divergenceless.     

Representations of the Poincaré Semigroup and Relativistic Causality

This paper provides the mathematical tools for addressing issues of two kinds of causality in relativistic scattering theory: general causality, i.e., an effect can only be measured after its cause, and Einstein causality, i.e., no propagation of probability outside of the forward light cone. Starting from Wigner's unitary irreducible representations of the Poincaré group for noninteracting, one particle states, we describe the mathematical tools necessary to describe scattering states, the Lippmann-Schwinger Dirace kets, and to describe resonances and decaying states, the relativistic Gamow ket. An important step for their derivations is the Hardy space hypothesis. Investigating the transformation properties of scattering and resonance states under the dynamical Poincaré semigroup reveals that both kinds of causality result from this hypothesis about nature of the spaces of states and observables.     

Relativistic internal time operator

We construct a self-adjoint time operator for massless relativistic systems in terms of the generators of the Poincaré group. The Lie algebra generated by the time operator and the generators of the Poincaré group turns out to be an infinitedimensional extension of the Poincaré algebra. The internal time operator generates two new entities, namely the velocity operator and the internal position operator. The transformation properties of the internal time and position operator under Lorentz boosts are different from what one would expect from relativity theory. This difference reflects the fact that the time concept associated with the internal time operator is radically different from the time coordinate of Minkowski space, due to the nonlocality of the time operator. The spectral projections of the time operator allow us to construct incoming subspaces for the wave equation without invoking Huygens' principle, as in two and one spatial dimensions where Huygens' principle does not hold.     

Weak equivalence principle and gauge theory of the tetrad gravitational field

The tetrad theory of gravitation corresponding to the Treder formulation of the weak equivalence principle is incompatible with the customary method for constructing a gauge theory for a tetrad gravitational field. In this formulation, the Lagrangian of the nongravitating mass is a direct covariant generalization of the partially relativistic expression to a Riemannian space-time V₄. This incompatibility is at odds with the resutt found in the tetrad formulation of the general theory of relativity derived from the requirement of localization of the Poincaré group.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 18–21, April, 1978.     

The symmetry aspects of quantum field theory in general relativity 1

Some proposed models for a quantum field theory in general relativity are briefly analyzed. Their main difficulties are a consequence of the initial choice of the group of symmetries of the (quantum) field equations. The necessity of selecting space-time isometries in a general covariant theory and the unphysical character of the Poincaré translations in a tangent plane theory are discussed. Starting from some basic requirements, a model is proposed in which the groups of symmetries are derived from the proper homogeneous groups of isometries of the minimal isometric local embedding spaces of space-times.This essay received an honorable mention (1977) from the Gravity Research Foundation-Ed.     

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.     

Particle dynamics on Snyder space

We examine Hamiltonian formalism on Euclidean Snyder space. The latter corresponds to a lattice in the quantum theory. For any given dynamical system, it may not be possible to identify time with a real number parametrizing the evolution in the quantum theory. The alternative requires the introduction of a dynamical time operator. We obtain the dynamical time operator for the relativistic (nonrelativistic) particle, and use it to construct the generators of Poincaré (Galilei) group on Snyder space.     

Magnetohydrodynamics and cosmology

It is well known that the Maxwell equations are connected to Minkowski space-time and to the Poincaré group. If we pass to the De Sitter universe with constant curvature, i.e., to the projective relativity, we must generalize the Maxwell equations in such a way to make them invariants for the Fantappié group. We thus obtain more general equations which can be interpreted as equations of magnetohydrodynamics and which reunite in a single theory electromagnetism and relativistic hydrodynamics.     

Constraint algebra from local Poincaré symmetry

A rigorous derivation of the constraint algebra between lapse, shift and Lorentz Hamiltonians is presented assuming that only local Poincaré symmetry constraints are present in the theory. It is also shown that the Dirac-Arnowitt-Deser-Misner form of the Hamiltonian is merely a consequence of the local Poincaré symmetry identities.     

Some aspects of the geometry of first-quantized theories

The ECSK and Yang-Mills theories are constructed with emphasis on their fiber bundle structure. In particular, the momentum tensor is derived as the Noether current of translational symmetry. The structure of the ECSK theory as a gauge theory of the Poincaré group is discussed. A theory of a Dirac field exhibiting internal affine symmetry, i.e., full internal Poincaré symmetry, is described. Aspects of the topological-geometric foundations of these theories are discussed, and some intuitive interpretations are presented.     

Algebraic interpretation of the Yang-Mills field

Einstein's principle of general relativity is a dynamical-group approach in that all dynamics is implied by the invariance and no force is introduced (as an external, symmetry-breaking factor). In this spirit we take a Poincaré-invariant free wave equation and, deforming the Poincaré group to the de Sitter group, obtain interaction. This illustrates our algebraic approach to gauge invariance, whereby the (generalized) Maxwell tensor of the Yang-Mills field appears as structure constants of the homogeneous algebra obtained as a deformation of an inhomogeneous one, with interaction appearing via the same tensor, which plays a role corresponding to the curvature tensor in Einstein's general relativity.     

Relativistic action-at-a-distance interactions: Lagrangian and Hamiltonian to terms of second order

Relativistic systems of particles interacting pairwise at a distance (interactions not mediated by fields) in flat spacetime are studied. It is assumed that the interactions propagate at the speed of light in vacuum and that all masses are scalars under Poincaré transformations. The action functional of the theory depends on multiple times (the proper times of the particles). In the static limit, the theory has three components: a linearly rising potential, a Coulomb-like interaction and a dynamical component to the Poincaré invariant mass. In this Letter we obtain explicitly, to terms of second order, the Lagrangian and the Hamiltonian with all the dynamical variables depending on a single time. Approximate solutions of the relativistic two-body problem are presented.     

Lorentz violation, two-time physics, and strings

The aim of this paper is to study the generalization of the relativistic particle recently proposed by Kostelecký. An alternative action for this system is presented, and it is shown that this action can be interpreted as a particle in curved space. Furthermore, the following results are established for the model: (i) there exists a limit where the system has more local symmetries than the usual relativistic particle; (ii) in this limit when Lorentz symmetry is restored, a direct relationship with the two-time physics is determined; (iii) if also Poincaré symmetry is recovered, the action of a relativistic bosonic string is obtained.     

Gravitational and Electroweak Unification

Einstein suggested that a unified field theorybe constructed by replacing the diffeomorphisms (thecoordinate transformations of general relativity) withsome larger group. We have constructed a theory that unifies the gravitational and electroweakfields by replacing the diffeomorphisms with the largestgroup of coordinate transformations under whichconservation laws are covariant statements. Thisreplacement leads to a theory with field equations whichimply the validity of the Einstein equations of generalrelativity, with a stress-energy tensor that is justwhat one expects for the electroweak field andassociated currents. The electroweak field appears as aconsequence of the field equations (rather than as a"compensating field" introduced to secure gaugeinvariance). There is no need for symmetry breaking toaccommodate mass, because the U(1) × SU(2) gaugesymmetry is approximate from the outset. Thegravitational field is described by the space-timemetric, as in general relativity. The electroweak fieldis described by the "mixed symmetry" part of the Riccirotation coefficients. The gauge symmetry-breakingquantity is a vector formed by contracting theLevi-Civita symbol with the totally antisymmetric partof the Ricci rotation coefficients.     

Direct gauging of the Poincaré group V. Group scaling,classical gauge theory,and gravitational corrections

Homogeneous scaling of the group space of the Poincaré group,P ₁₀, is shown to induce scalings of all geometric quantities associated with the local action ofP ₁₀. The field equations for both the translation and the Lorentz rotation compensating fields reduce toO(1) equations if the scaling parameter is set equal to the general relativistic gravitational coupling constant 8Gc ^–4. Standard expansions of all field variables in power series in the scaling parameter give the following results. The zeroth-order field equations are exactly the classical field equations for matter fields on Minkowski space subject to local action of an internal symmetry group (classical gauge theory). The expansion process is shown to breakP ₁₀-gauge covariance of the theory, and hence solving the zeroth-order field equations imposes an implicit system ofP ₁₀-gauge conditions. Explicit systems of field equations are obtained for the first- and higher-order approximations. The first-order translation field equations are driven by the momentum-energy tensor of the matter and internal compensating fields in the zeroth order (classical gauge theory), while the first-order Lorentz rotation field equations are driven by the spin currents of the same classical gauge theory. Field equations for the first-order gravitational corrections to the matter fields and the gauge fields for the internal symmetry group are obtained. Direct Poincaré gauge theory is thus shown to satisfy the first two of the three-part acid test of any unified field theory. Satisfaction of the third part of the test, at least for finite neighborhoods, seems probable.