L. K. Norris

Covariant field theory on frame bundles of fibered manifolds

We show that covariant field theory for sections of π : E→M lifts in a natural way to the bundle of vertically adapted linear frames LπE. Our analysis is based on the fact that LπE is a principal fiber bundle over the bundle of 1-jets J1π. On LπE the canonical soldering 1-forms play the role of the contact structure of J1π. A lifted Lagrangian L: LπE→R is used to construct modified soldering 1-forms, which we refer to as the Cartan–Hamilton–Poincare 1-forms. These 1-forms on LπE pass to the quotient to define the standard Cartan–Hamilton–Poincare m-form on J1π. We derive generalized Hamilton–Jacobi and Hamilton equations on LπE, and show that the Hamilton–Jacobi and canonical equations of Caratheodory–Rund and de Donder–Weyl are obtained as special cases.

The Robertson-Walker metric and the symmetries belong to the family of contracted Ricci collineations

It is shown that the general form of the Robertson-Walker cosmological metric admits symmetry properties that are members of the symmetry family of contracted Ricci collineations. A particular form for the conservation law generator given by ▽j[(−g)1/2(Tij−1/2δijT)ηi] = 0 following in consequence of these symmetries is obtained and interpreted.

n-symplectic algebra of observables in covariant Lagrangian field theory

n-symplectic geometry on the adapted frame bundle λ:LπE→E of an n=(m+k)-dimensional fiber bundle π:E→M is used to set up an algebra of observables for covariant Lagrangian field theories. Using the principle bundle ρ:LπE→J1π we lift a Lagrangian L:J1π→R to a Lagrangian L≔ρ*(L):LπE→R, and then use L to define a “modified n-symplectic potential” θL on LπE, the Cartan–Hamilton–Poincare (CHP) Rn-valued 1-form. If the lifted Lagrangian is nonzero, then (LπE,dθL) is an n-symplectic manifold. To characterize the observables we define a lifted Legendre transformation φL from LπE into LE. The image QL≔φL(LπE) is a submanifold of LE, and (QL,d(θ|QL)) is shown to be an n-symplectic manifold. We prove the theorem that θL=φL*(θ|QL), and pull back the reduced canonical n-symplectic geometry on QL to LπE to define the algebras of observables on the n-symplectic manifold (LπE,dθL). To find the reduced n-symplectic algebra on QL we set up the equations of n-symplectic reduction, and apply the general theory to the mo...

Underlying fibre bundle structure of A(4) gauge theories

Abstract The underlying fibre bundle structure for gauge theories of gravitation and their extensions possessing the affine structure group A(4) is considered. We point out that the identification of the torsion as the R4-curvature leads to restricted A(4)-theories. Our results suggest immediate extensions for metric-affine type theories.

Splitting of the connection in gauge theories with broken symmetry

We obtain a fully geometric analog of the Higgs mechanism whereby a symmetry‐breaking Higgs field is used to impart mass to gauge fields. We do this by showing that under fairly general hypotheses a symmetry‐breaking Higgs field φ on a ‘‘principal bundle with connection’’ (P,ω) allows the decomposition of the connection ω into a pair (ω′,τ) where ω′ is a connection on P that reduces to a φ‐subbundle of P and where τ is a tensorial field on P. The gauge fields that remain massless are identified with the components of ω′ while the gauge fields that acquire mass are identified with the components of τ. This decomposition of the connection is exploited in the case where the group of the bundle is the conformal group which scales some fixed metric of arbitrary signature. The geometry of such a bundle with connection generalizes Weyl geometry and provides a bundle setting for conformal gauge theories. We then show that the Weinberg–Salam electroweak theory can be recast as a conformal gauge theory. A primary f...

Fluid space–times including electromagnetic fields admitting symmetry mappings belonging to the family of contracted Ricci collineations

This paper investigates certain symmetry mappings belonging to the family of contracted Ricci collineations (FCRC) (satisfying gijLRij=0) admitted by the general fluid space–times, including electromagnetic fields, that were classified and studied earlier by Stewart and Ellis (1967). Many of the results obtained are applicable to the perfect fluid models treated by Wainwright (1970) and Krasinski (1974,1975). A major part of this paper represents an extension of previous investigations (1976) of the Robertson–Walker metrics and more general perfect fluid space–times that admit FCRC symmetry mappings and concomitant conservation expressions. More specifically, these results provide a number of theorems relating to the more general fluid space–times that admit FCRC symmetry mappings (including both timelike and spacelike symmetry vectors) that lead to conservation expressions and specific conditions on the metric tensors for the given particular cases of these space–times. Also the form of the symmetry mapp...

The affine geometry of the Lanczos H-tensor formalism

We identify the fiber-bundle-with-connection structure that underlies the Lanczos H-tensor formulation of Riemannian geometrical structure. We consider linear connections to be type (1,2) affine tensor fields, and we sketch the structure of the appropriate fiber bundle that is needed to describe the differential geometry of such affine tensors, namely the affine frame bundleA12M with structure groupA12 (4) =GL(4) ⓈT12ℝ4 over spacetimeM. Generalized affine connections on this bundle are in 1-1 correspondence with pairs(Γ, K) onM, where thegl(4)-componentΓ denotes a linear connection and the T12ℝ4-componentK is a type (1,3) tensor field onM. We show that the Lanczos H-tensor arises from a gauge fixing condition on this geometrical structure. The resulting translation gauge, theLanczos gauge, is invariant under the transformations found earlier by Lanczos. The other Lanczos variablesQμmandq are constructed in terms of the translational component of the generalized affine connection in the Lanczos gauge. To complete the geometric reformulation we reconstruct the Lanczos Lagrangian completely in terms of affine invariant quantities. The essential field equations derived from ourA12 (4)-invariant Lagrangian are the Bianchi and Bach-Lanczos identities for four-dimensional Riemannian geometry.

Projective and volume-preserving bundle structures involved in the formulation ofA(4) gauge theories

The bundle structures required by volume-preserving and related projective properties are developed and discussed in the context ofA(4) gauge theories which may be taken as the proper framework for Poincaré gauge theories. The results of this paper include methods for extending both tensors and connections to a principal fiber bundle havingG1(4,R)xG1(4,R) as its structure group. This bundle structure is shown to be a natural arena for the generalized (±) covariant differentiation utilized by Einstein for his extended gravitational theories involving nonsymmetric connections. In particular, it is shown that this generalized (±) covariant differentiation is actually a special case of ordinary covariant differentiation with respect to a connection on theG1(4,R) xG1(4,R) bundle. These results are discussed in relation to certain properties of generalized gravitational theories based on a nonsymmetric connection which include the metric affine theories of Hehl et al. and the general requirement that it should be possible to formulate well-defined local conservation laws. In terms of the extended bundle structure considered in this paper, it is found that physically distinct particle number type conservation expressions could exist for certain given types of matter currents.

Algebraically special Yang–Mills solutions without sources

We consider source‐free Yang–Mills solutions for which the curvature is decomposable in the sense that the curvature 2‐form is the product of a single Lie‐algebra‐valued function and a real 2‐form. If the curvature is everywhere nonnull (or null with twisting rays), then the solution is a connection in a principal fiber bundle, which is reducible to a source‐free Maxwell principal bundle. All such solutions are therefore readily obtained, locally or globally, from Maxwell solutions. Our analysis uses the Ambrose–Singer theorem to show that the holonomy group is one‐dimensional. A principal bundle‐with‐connection is reducible to the holonomy subbundle of any point, and, in this case, since the holonomy group is one‐dimensional, the reduced bundle has the structure of a Maxwell bundle. On the other hand, if the curvature is null and twist‐free on a full neighborhood of some point, then the bundle need not be reducible. The holonomy group is generally the entire gauge Lie group. The solutions can still be co...

Gromov’s theorem in n-symplectic geometry on LRn

If M is an n-dimensional manifold, then the associated bundle of linear frames LM of M supports the canonically defined Rn-valued soldering 1-form θ^. The pair (LM,dθ^) is an n-symplectic manifold, where dθ^ is the n-symplectic 2-form. We adapt the proofs of de Gosson and McDuff and Salamon of Gromov’s non-squeezing theorem on R2n to give a proof of Gromov’s theorem for affine n-symplectomorphisms on LRn.