Spatial Gauge Research Articles

Odd dimensional gauge theories and current algebra

We quantize gauge field theories in odd dimensional space-time with actions including both a Chern-Simons term and a Yang-Mills term. Such theories will be referred to as CSYM theories. We show that there are deep connections between these theories and chiral anomalies and current algebra in even dimensional space-time. The classical canonical structure of the CSYM theories is intimately related to the algebraic properties of the consistent and covariant chiral anomalies. The quantization of the CSYM theories involves a one-cocycle which is the Wess-Zumino functional and, depending on the dimension of space-time and the gauge group, the consistent realization of gauge invariance at the quantum level imposes a quantization condition on the Chern-Simons coupling parameter. The associated cocycle behavior of physical states may be understood in terms of an Abelian functional curvature on the space of all spatial gauge fields. By considering the CSYM theories on a space-time with a spatial boundary we show that the algebra of Gauss law generators acquires a boundary-valued anomaly which is cohomologous to the Faddeev-Shatashvili proposal for the anomaly in the equal-time commutator of Gauss law operators in the theory of massless chiral fermions interacting with a gauge field in even dimensional space-time.

Stochastic quantisation of a gauge field in the spatial axial gauge

The stochastic quantisation method of Nelson can be applied to quantise an Abelian gauge field in the spatial axial gauge.

Material and spatial gauge theories of solids—III. dynamics of disclination free states

The dynamics of disclination free states of material bodies are analyzed by restricting the spatial and material gauge groups so that only the translation subgroups act locally. This eliminates the spatial and material curvature tensors for the rotation sectors and simplifies the problem. Field equations for the matter fields, x i , the spatial translation compensating fields, φ a i and the material translation compensating fields, ψ b a , are obtained. All of the field equations are shown to be gauge covariant. Effects of local material translations are pervasive because they arise from local transformations of the base manifold. In the absence of material defects, the field equations for the x i and φ a i fields reduce to those previously reported, and the integrability conditions of the field equations for the φ's are the field equations for the x's in all cases. The new field equations for the material translation compensating fields are shown to have sources that are linear in the components of the gauge field momentum-energy complex, and hence they are intrinsically nonlinear. The integrability conditions for the field equations for the ψ's are shown to be equations of balance for gauge momentum-energy of the matter fields with explicitly calculated source terms that vanish in the absence of defects.

Material and spatial gauge theories of solids — I. Gauge constructs, geometry, and kinematics

A gauge-theoretic approach for modeling continuous distributions of defects in solids is undertaken, based on local internal and external symmetries of an appropriate Lagrangian for those materials. The invariance of the Lagrangian under global action of internal (spatial) and external (material) symmetry groups is extended to invariance under local action of both groups of symmetries through an application of the minimal replacement construct of gauge theory. This is achieved by introduction of nontrivial spatial and material gauge connections. Both connections are used for the construction of two sets of Cartan equations of structure with two different types of torsion and curvature (spatial and material). The spatial torsion and curvature are shown to model “microcracks” and “microrotations”, while the material torsion and curvature model dislocations, disclinations, and energy dissipation mechanisms.

Material and spatial gauge theories of solids—II. Problems with given material dislocation densities

Effects of material defects that arise from local action of the material translation symmetry group are analyzed. These problems are uncoupled by the assumptions that only the material translation group acts locally and that the material dislocation density 2-forms (material Cartan torsion) are both specified and 2-plane supported (elementary). All essential geometric quantities are evaluated. Since the material dislocation densities are specified, the only dynamical variables are the three displacement functions of the deformed body. Explicit forms of the dynamic equilibrium equations are obtained. These give new definitions of effective linear momentum and effective stress. Exact solutions of the field equations reproduce the stress field of straight screw dislocations, but the stress field for straight edge dislocations only obtain in the weak defect field limit. Both of these classes of problems arise through the local action of spatial material translations. The governing equations for problems with local action of the material time translation group are derived. They are shown to yield stress distributions that depend on the velocity components. Solutions of the field equations indicate that such models can represent relaxation phenomena.

A possibility of a lattice form in the broken-phase symmetry state

A possibility of the existence of a lattice form in the broken-phase symmetry state is investigated from the viewpoint of spontaneous breakdown of the spatial translational and gauge invariance. It is proved that there can exist at least four gapless phonons; three correspond to lattice phonons and the fourth to a phase phonon.

Spatial rotations, electromagnetic gauge transformations and the fine-structure constant

Since it expresses a connection between the basic units of angular momentum /~ and electric charge e, the experimentally observed value of the fine-structure constant = eS/4~t~c may signify an underlying relationship between the associated groups for spatial rotations SOs and electromagnetic gauge transformations SOs. In the present letter, at tention is given to Haar-induced volumes of the fourand ten-dimensional envelopments #[SOs • SOs] and #[[S03] s • SOs]. I t is observed that if the lat ter volumes, denoted as V and V', are related to the radius a of SOs and the half-period b of SO s by the simple conditions in (12), then the fine-structure constant is required to have the theoretical value shown in (17). I t is customary to parametrize the SOs.group of proper rotations in physical Euclidean space in terms of r a real three-component vector defined through a sphere of radius z with antipodal surface points identified topologically: r ~-r for I~al = ~. For any SOs element the direction and magnitude (radians) of the rotation is given by the direction and magnitude of ~ . An elementary calculation yields the associated normalized Haar measure (1)

Perfect fluid sources for spatially homogeneous spacetimes

Spatially homogeneous perfect fluid spacetimes are studied from a point of view which emphasizes the spatial geometry and the action of that subgroup of the spatial gauge group of the three-plus-one formulation of general relativity which is compatible with the spatial homogeneity. The specializations of the dynamics which correspond to the existence of additional spacetime symmetries are classified. An unconstrained set of gravitational and fluid variables is obtained by elimination of the gravitational constraints using an approach which obtains the gravitational evolution equations from a suitably modified Lagrangian/Hamiltonian formalism. A slightly different choice of variables is then described which allows one to take full advantage of the spatial gauge group and of the 1-parameter group of scale transformations of the unit of length.