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Given an Ehresmann connection γ on a fibered manifold p:E→M, a covariant Hamiltonian density Hγ is then associated to each Lagrangian density L on J1E. Assume E is the bundle of connections of a principal bundle and that L is gauge invariant. Our goal in this paper is to determine conditions on γ under which Hγ is also gauge invariant. The general conclusion is that there is no gauge-invariant Ehresmann connection but there is plenty of such connections providing gauge-invariant covariant Hamiltonians. The relevant cases of U(1) bundles and SU(2) bundles are discussed in detail.

A gauge theoretical treatment proves to have been successful in the study of systems of point particles; the center-of-mass system is made into a principal fiber bundle, on which is defined a natural connection. The gauge theoretical approach may be generalized to be applicable to a system of rigid bodies. The present article deals with a system of two identical axially symmetric cylinders jointed together by a special type of joint. This system is the model made by Kane and Scher and reformulated later by Montgomery, in order to study the falling cats who can land on their legs when released upside down. With the no-twist condition, the system turns out to have the configuration space diffeomorphic with SO(3), which is made into a principal O(2) bundle over RP2, the real projective space of dimension two, and endowed with a natural connection. An optimal control problem for this system with the vanishing total angular momentum is satisfactorily treated in this bundle picture. Along with a certain perform...

A classic result in the foundations of Yang-Mills theory, due to J. W. Barrett ["Holonomy and Path Structures in General Relativity and Yang-Mills Theory." Int. J. Th. Phys. 30(9), (1991)], establishes that given a "generalized" holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such that the holonomy map corresponds to the holonomies of that connection. Barrett also provided one sense in which this "recovery theorem" yields a unique bundle, up to isomorphism. Here we show that something stronger is true: with an appropriate definition of isomorphism between generalized holonomy maps, there is an equivalence of categories between the category whose objects are generalized holonomy maps on a smooth, connected manifold and whose arrows are holonomy isomorphisms, and the category whose objects are principal connections on principal bundles over a smooth, connected manifold. This result clarifies, and somewhat improves upon, the sense of "unique recovery" in Barrett's theorems; it also makes precise a sense in which there is no loss of structure involved in moving from a principal bundle formulation of Yang-Mills theory to a holonomy, or "loop", formulation., Comment: 20 pages

In this paper the velocity field of a planar fluid with point vortices is recognized as induced by a connection on a principal bundle. Various geometric/topological invariants, including braiding of the trajectories of the vortices and the Berry’s phase, are discussed in this bundle context. These invariants give rise to quantization schemes for the system.

It is shown that any Lagrangian distribution on a 2n‐dimensional symplectic manifold Γ is equivalent to a principal O(n) bundle obtained as a reduction of the frame bundle. Bisymplectic manifolds with Nijenhuis recursion operators are studied and it is shown that the set of all bi‐Lagrangian distributions is a trivial bundle the structure group being the homogeneous space U(1)n/O(1)n. Various formulas for Maslov indices of closed curves are given including one using only data from the recursion operator.

In this, the first of a sequence of papers applying infinite dimensional natural bundle techniques to problems in general relativity and other field theories, we construct a minimal resolution of the singularities in the space of geometries or superspace of a closed manifold M. The basic properties of two natural bundles associated with M are also discussed. Subsequent articles will develop the theory in the context of spin and the diffeomorphism group, the Yang–Mills moduli space, and nonlinear sigma models.

A natural map between group‐cohomology of the structure group of a principal fiber bundle with coefficients in the space of functions from the total space into an Abelian group and Cech‐cohomology of the base space is defined. A differential complex of local group‐cochains is constructed and an analog of the Poincare lemma for group‐cohomology is proven. By using the machinery of spectral sequences the cohomology of this complex is calculated and the connection between group‐cohomology and Cech‐cohomology of the given principal fiber bundle is elucidated. Finally, the non‐Abelian and Witten anomaly in this context is reviewed and the relevance of our results for lifting principal group actions is discussed.

The Higgs mechanism employed in the standard model of electroweak interactions (Glashow–Weinberg–Salam) is formulated in terms of equivariant mappings on a principal fiber bundle with values in the space of sesquilinear forms defined on a complex vector space. A classification of symmetry‐breaking equivariant maps of this nature is given and compared to the phenomenological results of standard model.

Given a principal bundle P→X with a structure group G and an associated Higgs bundle Σ with a standard fiber G/H, the case of a matter bundle E whose standard fiber admits action only of an exact symmetry subgroup H of G is examined. In the presence of a fixed Higgs field σ: X→Σ, matter fields are represented by sections of a matter bundle Eh associated with the corresponding reduced subbundle Ph of P. The totality of matter fields and Higgs fields is described by sections of the bundle E which is the composite bundle EH→Σ→X where EH→Σ is the bundle associated with the principal H‐bundle P→Σ. The bundle E fails to be associated with a principal bundle. To construct a connection Γ: E→J1E on E, the canonical jet bundle morphism J1EH×J1Σ→J1E is used.

In this paper the concept of bundle realization of a Lie group on an Abelian principal bundle is defined. This definition is based on the theory of locally operating realizations of Lie groups. Afterward the bundle realizations are studied and characterized into pseudoequivalence classes. This theory is applied to a systematic study and classification of invariant connections under a Lie group. In particular, some examples of gauge invariant potentials under some subgroups of the Poincare group are worked out.

The relationship between the group Γ of pure gauge transformations of Atiyah, Hitchin, and Singer [Proc. R. Soc. London Ser. A 362, 425 (1978)] of a principal fiber bundle and the group G of gauge transformations consisting of the direct product of the local gauge groups on the base space is studied. Γ is an invariant subgroup of G and the quotient G/Γ is identified with the group of inequivalent gauge transformations. In the framework of the category of principal fiber bundles with connections, a natural explanation for the relevance of the group Γ in the classical and quantum theories of gauge fields is presented. The paper is made self‐contained by an introductory discussion of the concepts of principal coordinate fiber bundle (gauge fixed principal fiber bundle) and principal fiber bundle, and of the equivalence between the three different versions of the group of vertical automorphisms of the bundle.

Representations of the semidirect product group D■Diff(M) in the context of fiber bundle theory are studied. Here, D is the group of compactly supported functions and Diff(M) is the group of compactly supported diffeomorphisms of a manifold M. The carrier space is taken as the space of equivariant functions on a flat principal bundle over M where M is multiply connected. Two principal bundles are taken as equivalent if they are related by a gauge transformation. For the U(1) case it is found that the representations are irreducible, and the equivalence classes of representations are in one to one correspondence with the equivalence classes of bundles. Simple examples are discussed.

A correspondence between (not necessarily self‐dual nor anti‐self‐dual) Yang–Mills fields on Minkowski space, and pairs of cohomology classes γ, φ, on null twistor space PN■ is established; γ∈H1CR (PN■,g ) defines a deformed Cauchy–Riemman (C–R)(γ) structure on a principal bundle over PN, and φ∈H1CR(γ) (PN■,O(−4)⊗g) . The correspondence depends on the choice of a spacelike hyperplane in Minkowski space. Here, γ and φ provide initial values in the spin bundle over this hyperplane for a system of evolution equations, along the null geodesic spray congruence in the spin bundle over Minkowski space.

The motivation of this paper stems from the need to construct explicit isomorphisms of (possibly nontrivial) principal G-bundles on the space of loops or, more generally, of paths in some manifold M, over which we consider a fixed principal bundle P; the aforementioned bundles are then pull backs of P with respect to evaluation maps at different points. The explicit construction of these isomorphisms between pulled-back bundles relies on the notion of parallel transport. We introduce and discuss extensively at this point the notion of generalized gauge transformation between (a priori) different principal G-bundles over the same base M, and we see immediately that the parallel transport is a generalized gauge transformation for two special kinds of bundles on the space of loops or paths; at this point, we may try and generalize the previous arguments for more general pulled-back bundles. Finally, we discuss how flatness of the reference connection, with respect to which we consider parallel transport, is related to horizontality of the associated generalized gauge transformation.

We provide a coordinate-free version of the local classification, due to A. G. Walker [Quart. J. Math. Oxford (2) 1, 69 (1950)], of null parallel distributions on pseudo-Riemannian manifolds. The underlying manifold is realized, locally, as the total space of a fibre bundle, each fibre of which is an affine principal bundle over a pseudo-Riemannian manifold. All structures just named are naturally determined by the distribution and the metric, in contrast with the non-canonical choice of coordinates in the usual formulation of Walker's theorem., Comment: 12 pages

We analyze in this paper the group theoretical meaning of the covariant derivatives, and show that they are horizontal left‐invariant vector fields on superspace obtained from a (super)Lie group which at the same time exhibits the structure of a principal bundle with a canonical connection. The geometrical construction is general and not restricted to the super‐Poincare group.

A new fiber bundle approach to the gauge theory of a group G that involves space‐time symmetries as well as internal symmetries is presented. The ungauged group G is regarded as the group of left translations on a fiber bundle G(G/H,H), where H is a closed subgroup and G/H is space‐time. The Yang–Mills potential is the pullback of the Maurer–Cartan form and the Yang–Mills fields are zero. More general diffeomorphisms on the bundle space are then identified as the appropriate gauged generalizations of the left translations, and the Yang–Mills potential is identified as the pullback of the dual of a certain kind of vielbein on the group manifold. The Yang–Mills fields include a torsion on space‐time.

We exhibit the complexified spin and conformally weighted functions as sections of holomorphic line bundles over P1(C) ×P1(C). As an example of a nontrivial bundle, we discuss the complex null cone in some detail.

Take a smooth manifold M and a Lie algebra action (g action) ϑ on M as the geometrical arena of a physical system moving on M with momenta given by ϑ. We propose to quantize the system with a Mackey‐like method via the associated vector bundle ξρ of a principal bundle ξ= (P,π,M,H) with model dependent structure group H and with G‐action φ on P lifted from ϑ on M. This (quantization) bundle ξρ gives the Hilbert space H=L2(ξρ,ω) of the system as the linear space of sections in ξρ being square integrable with respect to a volume form ω on M; the usual position operators are obtained; φ leads to a vector field representation D (φρ,ϑ) of g in H and hence to momentum operators. So H carries the quantum kinematics. In this quantization the physically important connection between geometrical properties of the system, e.g., quasicomqleteness of ρ and G maximality of φρ, and global properties of its quantized kinematics, e.g., skew‐adjointness of the momenta and integrability of D (φρ,ϑ) can easily be studied. The ...

This paper deals with representations of connected Lie groups by bundle maps of fiber bundles. It is pointed out that a large class of such representations can be obtained from the bundle structure theroem, and explicit constructions are given, first on principal bundles and then on associated bundles. Examples are provided to show that, for bundle representations, the theorem of full reducibility breaks down even for compact Lie groups. Finally, a general construction is given for obtaining representations of a Lie group on an arbitrary principal bundle. However, it is not known whether this exhausts all possibilities.

It is shown that the transition functions that give the global structure of the fiber bundle play an important role in the construction of the metric. The invariance properties of this metric under general gauge transformations are discussed and it is found that the usual requirement of a gauge‐invariant metric leads to severe constraints on the gauge fields. To avoid them, it is shown that the metric should instead be covariant with respect to these transformations. Moreover the existence of global actions that are essential in the context of the consistency problem is also discussed. The presence of such actions is studied in both the principal and their associated bundles. In the case of a homogeneous bundle with G/H as the typical fiber, it is shown that a ‘‘spliced’’ bundle with G×N(H)/H as the structure group has to be used. The unified space is then taken as the bundle space of its associated bundle.

It is shown that for any principal bundle over a Riemannian symmetric space G/G0 which admits G as automorphism group, the canonical G‐invariant connection satisfies the source free gauge field equations. Extending this to product manifolds V×G/G0 and assuming the metric and gauge fields decompose in a natural way, this result is still valid and the Einstein equations with gauge fields as source may also be satisfied. For G/G0, this is so automatically, but with a cosmological term present. For dimV=1 or 2, solutions are found, yielding metrics of the Robertson–Walker and Reissner–Nordstrom type.

We examine a class of two‐dimensional Lorentz manifolds which are ’’singular’’ in a certain sense. It is shown that, for such a manifold (M, g), the bundle boundary is a single point whose only neighborhood is all of M [the bundle completion of M; see B. G. Schmidt, Gen. Rel. Gravitation 1, 269–80 (1971)]. The four‐dimensional Schwarzschild and Friedmann–Robertson–Walker solutions are then investigated. We show that the bundle completions of these spaces are not Hausdorff.

A Zeeman topology is defined in the general framework of any set W of events which has been equipped with an acyclic signal relation ∼ →. The Wssumption that the ∼ → structure of W is locally that of Minkowski space and that the ’’piecing together’’ maps are smooth in an appropriate sense, allows a tangent bundle p:E→W to be defined. This bundle has, as structure group, the group G of linear causal automorphisms of Minkowski space.

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...

Let F be the curvature of some connection on some principal bundle over R4. I show that if F decays as fast as (r2lnr)−1 as r tends to infinity, then (1/8π2) ∫ trF2 is an integer. If F decays like r−2, any real value is possible. There is an analogous statement for R2n(n≳2), although it fails for R2.

A contraction of the structural group with respect to the stability subgroup is performed in a fiber bundle with Cartan connection. The relation of the connections in the original and in the contracted bundle is examined. As an example interesting for physics the contraction of the SO(4,1) de Sitter bundle over space–time to the affine tangent bundle over space–time is discussed with the latter bundle possessing the Poincare group as structural group.

A gauge theory of supergravity which allows one to obtain first‐order Lagrangians that are locally gauge invariant by construction is presented. The formalism makes use of supertwistors as a representation space for the construction of a typical fiber of a vector bundle associated with a principal bundle, where the structural group is the super‐Poincare group. The approach proposed provides a means of resolving one of the central problems of gauge field theories of external symmetry groups, that is the satisfactory treatment of translations.

Let Pn be the trivial principal bundle with structural group G and base space Pn−1, P1 being the usual fiber bundle of gauge theories. In order to give a geometrical interpretation to the Faddeev–Popov fields, as well as to the Becchi, Rouet, and Stora transformations, we need to use the fiber bundle P3. The gauge fields and the Faddeev–Popov ghost and antighost fields appear as part of certain 1‐forms defined on the base space P2. The anticommuting character of the ghost and antighost fields is essentially due to their identification with 1‐forms. The Becchi, Rouet, and Stora transformations are identified with generalized infinitesimal gauge transformations on P3 of parameters related to the ghost fields. We obtain a further invariance of the action given by a similar generalized infinitesimal gauge transformation on P3 related to the antighost fields.

The trivial bundle of orthonormal frames over flat space‐time is decomposed into two subbundles with structure groups SO(1,1) and SO(2), respectively. The curvature in the SO(2) bundle is identified with the electromagnetic field. It is shown that on certain conditions imposed upon the bundle decomposition the exterior derivative of the mean curvature 1‐form in the SO(2) bundle is equal to the curvature 2‐form in the SO(1,1) bundle. These conditions are (i) the Frobenius integrability of certain distributions generated by the splitting of the associated tangent bundle, and (ii) the vanishing of the mean curvature in the SO(1,1) bundle. For a single point charge the curvature in the SO(1,1) subbundle is identical to the radiation part of the Lienard–Wiechert field.

A hyper‐relativistic system is defined as one whose equation of motion is form invariant under coordinate transformations induced by a semisimple group whose algebra is contractible to the algebra of the Poincare group. Such a system lies, categorically, in the domain between the special theory of relativity and the general theory, for whereas the former requires covariance under transformations between inertial systems, the latter imposes covariance with respect to arbitrary continuous transformations. In this paper, a new interpretation of a particular fiber‐bundle structure constructed on the timelike homogeneous space M=SO(4,2)/SO(4,1) is presented, and Minkowski space‐time is realized as a subspace of the standard fiber of the tangent bundle over this hyperquadric. Through the process of group contraction, coupled with the commutation of the momentum vector fields with the principal bundle of linear frames with which the tangent bundle is associated, a hierarchy of ‘‘Heisenberg commutation relations,...

A notion of Hilbert bundle is proposed which leads to the construction of a ’’big’’ Hilbert space H starting from a family of Hilbert spaces. For this, such a family is equipped with a suitable structure, called Borel field structure. A meaningful relationship is established between the Borel structures which can be defined on the union of the Hilbert spaces of the family and the Borel field structures with which the family can be equipped. For a topological group G, the structure of G‐Hilbert bundle is defined linking in a suitable way a Hilbert bundle with actions of G. In the framework of a G‐Hilbert bundle, a continuous unitary representation of G in H can be constructed. The transitive G‐Hilbert bundles which are often used in the theory of induced representations of groups are shown to be a subclass of the class of the G‐Hilbert bundles which are proposed in this paper.

Using the natural connection form on a principal bundle P(M,G) and the bundle A (A /G,G) a systematic derivation of the double‐cohomological series constituted by the exterior differential d on space‐time M and arbitrary, horizontal, and vertical variations in connection space is given. The relationship between these cohomologies and the family index theorem is clarified. The formalism is then used to analyze Abelian gauge structure inside non‐Abelian gauge theory. The pertinent functional U(1) connection form, curvature form, and three‐form ‘‘curvature’’ are identified and computed, and are related to the θ vacuum, anomalous commutation relation, and Jacobi identity, respectively. Some of the results differ from those obtained by Wu and Zee [Nucl. Phys. B 258, 157 (1985)] and Niemi and Semenoff [Phys. Rev. Lett. 55, 227 (1985)] and the results of this paperrecover theirs under certain conditions. Finally the generalization of the formalism to a nontrivial principal bundle by introduction of a fixed backg...

It is proved that the equivalence class of bundle representations of a group G on the product bundle B0 with total space B0=X×Y includes all representations of G on bundles B which are homeomorphic (but not necessarily naturally homeomorphic) to the product X×Y, provided the G has the same action on the fibres of B0 and B. The group A of the bundle B is immaterial.

The problem of solving the combined gravitational and Yang–Mills field systems is regarded as a purely geometrical problem of determining a linear connection on the principal frame bundle L(M) from a connection on a SU(2) principal bundle over a space‐time M. It is suggested that mapping theorems of connections on bundles may provide a means of actually solving ‘‘field equations.’’

The principal bundle of orthogonal frames over M4 is explicitly constructed from certain pairs of Ō(3,3)bar)= spinors that transform as (associated) twistors under the action of the covering group of the Poincare group. In particular, flat space–time is constructed from these associated twistors, and is thus shown to be an object derivable from geometric structures more basic than the vectors of M4. Associated twistors describe massive elementary particles. The position in M4 of such a particle can be explicitly defined in terms of the components of these twistors. The usual momentum and angular momentum variables which coordinatize the classical phase space of this elementary relativistic system of nonzero mass and arbitrary spin may also be realized in terms of this pair of associated twistors. This realization is not equivalent to descriptions of massive particles using twistors which have previously appeared in the literature.