Flux Homomorphisms and Principal Bundles over Infinite Dimensional Manifolds

Neeb, Karl-Hermann; Vizman, Cornelia

doi:10.1007/s00605-002-0001-6

Cited by 20 publications

(41 citation statements)

References 8 publications

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“…This is consistent with the results of [40] and also with the general relation between holonomies on homogeneous spaces and symplectic fluxes (see e.g. [43]), although we are not aware of any reference that describes analogous observations for semi-direct products. In the context of Thomas precession, the phase (4.19) is generally neglected, precisely because it is blind to spin; for the same reason, from now on we let the translational path α(t) be constant (α = 0) so that the only non-zero contribution to the Berry holonomy (4.17) comes from its rotational piece.…”

Section: Energy Eigenstatessupporting

confidence: 91%

Probing Wigner rotations for any group

Oblak

2018

Journal of Geometry and Physics

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Wigner rotations are transformations that affect spinning particles and cause the observable phenomenon of Thomas precession. Here we study these rotations for arbitrary symmetry groups with a semi-direct product structure. In particular we establish a general link between Wigner rotations and Thomas precession by relating the latter to the holonomies of a certain Berry connection on a momentum orbit. Along the way we derive a formula for infinitesimal, Liealgebraic transformations of one-particle states. *

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Section: Energy Eigenstatessupporting

confidence: 91%

Probing Wigner rotations for any group

Oblak

2018

Journal of Geometry and Physics

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“…For a discussion of the relation between quantomorphisms and Hamiltonian diffeomorphisms, extending some of these structures, such as Kostant's Theorem ( [Kos70]) to infinite dimensional manifolds, we refer to [NV03].…”

Section: Theorem V28 If G Is a Connected Lie Group And S : G → Autmentioning

confidence: 99%

Towards a Lie theory of locally convex groups

2006

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In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We further describe how regularity or local exponentiality of a Lie group can be used to obtain quite satisfactory answers to some of the fundamental problems. These results are illustrated by specialization to some specific classes of Lie groups, such as direct limit groups, linear Lie groups, groups of smooth maps and groups of diffeomorphisms.

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“…φm : G → M , g → g · m, for any m ∈ M [17]. 26 Principal bundle automorphisms are diffeomorphisms which commute with the right action of the structure group on P .…”

Section: An Equivalent Action With Manifest Symmetry and Localitymentioning

confidence: 99%

Quantum mechanics in magnetic backgrounds with manifest symmetry and locality

Davighi

Gripaios

Tooby-Smith

2020

J. Phys. A: Math. Theor.

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The usual methods for formulating and solving the quantum mechanics of a particle moving in a magnetic field respect neither locality nor any global symmetries which happen to be present. For example, Landau's solution for a particle moving in a uniform magnetic field in the plane involves choosing a gauge in which neither translation nor rotation invariance are manifest. We show that locality can be made manifest by passing to a redundant description in which the particle moves on a U (1)-principal bundle over the original configuration space and that symmetry can be made manifest by passing to a corresponding central extension of the original symmetry group by U (1). With the symmetry manifest, one can attempt to solve the problem by using harmonic analysis and we provide a number of examples where this succeeds. One is a solution of the Landau problem in an arbitrary gauge (with either translation invariance or the full Euclidean group manifest). Another example is the motion of a fermionic rigid body, which can be formulated and solved in a manifestly local and symmetric way via a flat connection on the non-trivial U (1)-central extension of the configuration space SO(3) given by U (2). Contents1 For the sake of simplicity, we will consider here only examples where G and M are connected.in the lagrangian representing the magnetic field that is valid globally on M . Instead, the best that one can do is to cover M by overlapping patches and to use multiple lagrangians, each of which is valid only locally on some patch. The most famous example, due to Dirac [1], 2 is given by the motion of an electrically-charged particle in the presence of a magnetic monopole, but we will see that there exists an example that is arguably even simpler (and certainly more prevalent in everyday life!), given by the motion of a rigid body which happens to be a fermion. 3The second complication is that the corresponding lagrangian (or lagrangians) will not be invariant under the action of G, but rather will shift by a total derivative. Perhaps the simplest example, made famous by Landau [5], is given by the motion of a particle in a plane in the presence of a uniform magnetic field, where there is no choice of gauge such that the lagrangian is invariant under translations in more than one direction. 4At the classical level, neither of these complications causes any problems, since they disappear once we pass from the lagrangian to the classical equations of motion. Indeed, the equations of motion are both globally valid and invariant (or rather covariant) under G. Thus, we can attempt to solve for the classical dynamics using our usual arsenal of techniques. But this is not the case at the quantum level. There, our usual technique is to convert the hamiltonian into an operator on L 2 (M ) and to exploit the conserved charges corresponding to G to solve, at least partially, the resulting Schrödinger equation. Here though, we do not have a unique hamiltonian, but rather several; even if we did have a unique hamiltonian, we would, in g...

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Flux Homomorphisms and Principal Bundles over Infinite Dimensional Manifolds

Cited by 20 publications

References 8 publications

Probing Wigner rotations for any group

Probing Wigner rotations for any group

Towards a Lie theory of locally convex groups

Quantum mechanics in magnetic backgrounds with manifest symmetry and locality

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