Rieffel induction as generalized quantum Marsden-Weinstein reduction

Landsman, N.P.

doi:10.1016/0393-0440(94)00034-2

Cited by 99 publications

(102 citation statements)

References 41 publications

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“…A simple version for the product of two groups was given in Marsden and Weinstein [1974]. Other versions are due to Sjamaar and Lerman [1991] and Landsman [1995Landsman [ , 1998.…”

Section: The Setting For Reduction By Stagesmentioning

confidence: 99%

“…Even though special symplectic reduction generalizes Marsden-Weinstein reduction, the special reduction by stages theorem in Landsman [1995] studies a setup that, in general, is different to the ones in the reduction by stages theorems of this book.…”

Section: This Takes Us Up To About 1972mentioning

confidence: 99%

“…The notion of symplectic induction that we just reviewed should not be confused with the symplectic induction procedure introduced in Landsman [1995] based on previous work of Xu [1991]. Due to this terminological conflict, Landsman's construction was renamed "special symplectic induction" in Landsman [1998].…”

Section: The Setting For Reduction By Stagesmentioning

confidence: 99%

“…The semidirect product reduction theorem has been proved in Landsman [1995], Landsman [1998, Chapter 4] as an application of a stages theorem for his special symplectic reduction method. Even though special symplectic reduction generalizes Marsden-Weinstein reduction, the special reduction by stages theorem in Landsman [1995] studies a setup that, in general, is different to the ones in the reduction by stages theorems of this book.…”

Section: This Takes Us Up To About 1972mentioning

confidence: 99%

See 3 more Smart Citations

Hamiltonian Reduction by Stages

Marsden¹,

Misiołek²,

Ortega³

et al. 2007

Lecture Notes in Mathematics

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PrefaceThis book is about Symplectic Reduction by Stages for Hamiltonian systems with symmetry. Reduction by stages means, roughly speaking, that we have two symmetry groups and we want to carry out symplectic reduction by both of these groups, either sequentially or all at once. More precisely, we shall start with a "large group" M that acts on a phase space P and assume that M has a normal subgroup N . The goal is to carry out reduction of the phase space P by the action of M in two stages; first by N and then by the quotient group M/N . For example, M might be the Euclidean group of R 3 , with N the translation subgroup so that M/N is the rotation group. In the Poisson context such a reduction by stages is easily carried out and we shall show exactly how this goes in the text. However, in the context of symplectic reduction, things are not nearly as simple because one must also introduce momentum maps and keep track of the level set of the momentum map at which one is reducing. But this gives an initial flavor of the type of problem with which the book is concerned.As we shall see in this book, carrying out reduction by stages, first by N and then by M/N , rather than all in "one-shot" by the "large group" M is often not only a much simpler procedure, but it also can give nontrivial additional information about the reduced space. Thus, reduction by stages can provide an essential and useful tool for computing reduced spaces, including coadjoint orbits, which is useful to researchers in symplectic geometry and geometric mechanics.Reduction theory is an old and time-honored subject, going back to the early roots of mechanics through the works of Euler, Lagrange, Poisson, vi Preface Liouville, Jacobi, Hamilton, Riemann, Routh, Noether, Poincaré, and others. These founding masters regarded reduction theory as a useful tool for simplifying and studying concrete mechanical systems, such as the use of Jacobi's elimination of the node in the study of the n-body problem to deal with the overall rotational symmetry of the problem. Likewise, Liouville and Routh used the elimination of cyclic variables (what we would call today an Abelian symmetry group) to simplify problems and it was in this setting that the Routh stability method was developed.The modern form of symplectic reduction theory begins with the works of Arnold [1966a], Smale [1970], Meyer [1973], and Marsden and Weinstein [1974]. A more detailed survey of the history of reduction theory can be found in the first Chapter of the present book. As was the case with Routh, this theory has close connections with the stability theory of relative equilibria, as in Arnold [1969] and Simo, Lewis and Marsden [1991]. The symplectic reduction method is, in fact, by now so well known that it is used as a standard tool, often without much mention. It has also entered many textbooks on geometric mechanics and symplectic geometry, such as Abraham and Marsden [1978], Arnold [1989], Guillemin and Sternberg [1984], Libermann and Marle [1987], and McDuff and Salamon [1995]. De...

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“…A simple version for the product of two groups was given in Marsden and Weinstein [1974]. Other versions are due to Sjamaar and Lerman [1991] and Landsman [1995Landsman [ , 1998.…”

Section: The Setting For Reduction By Stagesmentioning

confidence: 99%

Section: This Takes Us Up To About 1972mentioning

confidence: 99%

Section: The Setting For Reduction By Stagesmentioning

confidence: 99%

Section: This Takes Us Up To About 1972mentioning

confidence: 99%

See 2 more Smart Citations

Hamiltonian Reduction by Stages

Marsden¹,

Misiołek²,

Ortega³

et al. 2007

Lecture Notes in Mathematics

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show abstract

“…Much work [11,36,9,10,5] has been done on the quantization of the reduced phase spaces of constrained systems and of a particle in a gauge field, most notably by Landsman for the case of homogeneous configuration spaces [25] using induced representations and for the general case [27,26] using Rieffel's notion of "strict deformation quantization" and Rieffel induction respectively. However, these approaches suffer from the inability, in general, to quantize classical observables which are unbounded.…”

Section: Introductionmentioning

confidence: 99%

Geometric quantization of reduced contangent bundles

Robson¹

1996

Journal of Geometry and Physics

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The method of geometric quantization is applied to a particle moving on an arbitrary Riemannian manifold Q in an external gauge field, that is a connection on a principal H-bundle N over Q. The phase space of the particle is a Marsden-Weinstein reduction of T * N , hence this space can also be considered to be the reduced phase space of a particular type of constrained mechanical system. An explicit map is found from a subalgebra of the classical observables to the corresponding quantum operators. These operators are found to be the generators of a representation of the semi-direct product group, Aut N ⋉ C ∞ c (Q). A generalised Aharanov-Bohm effect is shown to be a natural consequence of the quantization procedure. In particular the rôle of the connection in the quantum mechanical system is made clear. The quantization of the Hamiltonian is also considered.Additionally, our approach allows the related quantization procedures proposed by Mackey and by Isham to be fully understood.

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Coherent State Quantization of Constraint Systems

Klauder

1997

Annals of Physics

225

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A careful reexamination of the quantization of systems with first- and second-class constraints from the point of view of coherent-state phase-space path integration reveals several significant distinctions from more conventional treatments. Most significantly, we emphasize the importance of using path-integral measures for Lagrange multipliers which ensure that the quantum system satisfies the quantum constraint conditions. Our procedures involve no delta-functionals of the classical constraints, no need for gauge fixing of first-class constraints, no need to eliminate second-class constraints, no potentially ambiguous determinants, and have the virtue of resolving differences between canonical and path-integral approaches. Several examples are considered in detail.Comment: Latex, 38 pages, no figure

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Rieffel induction as generalized quantum Marsden-Weinstein reduction

Cited by 99 publications

References 41 publications

Hamiltonian Reduction by Stages

Hamiltonian Reduction by Stages

Geometric quantization of reduced contangent bundles

Coherent State Quantization of Constraint Systems

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