Coadjoint Orbits and the Beginnings of a Geometric Representation Theory

Raţiu, Tudor S.

doi:10.1007/978-0-8176-4741-4_13

Cited by 5 publications

(9 citation statements)

References 72 publications

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“…There is an extensive literature on the subject (see e.g. [40] and references therein), but we will concentrate on physically important examples and will not develop a general theory. This will allow for relatively easy proofs of the facts we will need in the sequel.…”

Section: The Unitary Group and Topologymentioning

confidence: 99%

“…Now the mapping Φ : ρ ′ → U gives us a local embedding of O ρ into U(H). It follows from (40) and the construction of U that this mapping is continuous with respect to the norm topology. Since ρ has finite rank, on O ρ the norm topology is equivalent to the topology of the space of trace-class operators and thus we have a continuous local embedding of O ρ into U(H).…”

Section: Local Embedding Of Pure States Into the Unitary Groupmentioning

confidence: 99%

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Geometry of quantum dynamics in infinite-dimensional Hilbert space

Grabowski

Kuś

Marmo

et al. 2018

J. Phys. A: Math. Theor.

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We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional and including a Lagrangian formalism in which self-adjoint (Schrödinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we obtain also results concerning coadjoint orbits of the unitary group in infinite dimension, embedding of the Hilbert projective space of pure states in the unitary group, and an approach to self-adjoint extensions of symmetric relations. * Thus, the dynamics is integrable. Actually, it is the properly understood phase dynamics of vertical rolling disc on a plane in a Dirac algebroid setting studied in [22].All this can be generalized to ordinary implicit differential equations of arbitrary order. In this case we consider as a subset of higher jet bundles, the n-th tangent bundle Ì n N in case of an equation of order n, and consider γ as a solution if its n-th jet prolongation takes values in . If we call the n-th jet prolongations admissible in Ì n N , then solutions of are exactly projections γ N to N of admissible curves (or paths) γ in Ì n N lying in .Remark 2.1. The implicit differential equations described above are called by some authors differential relations. Let us explain that we use the most general definition, not requiring from any differentiability properties, since in real life the dynamics we encounter are often not submanifolds.

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Section: The Unitary Group and Topologymentioning

confidence: 99%

Section: Local Embedding Of Pure States Into the Unitary Groupmentioning

confidence: 99%

Geometry of quantum dynamics in infinite-dimensional Hilbert space

Grabowski

Kuś

Marmo

et al. 2018

J. Phys. A: Math. Theor.

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“…is a smooth section of T 𝑀 , but may not be a vector field on 𝑀 , because T𝑀 Ď T 𝑀 and T𝑀 -T 𝑀 in general. In order to solve this problem, Odzijewicz and Ratiu [253] proposed (see also further discussion and results in [254,27,28,255,351,287,252]) to define a real Banach Poisson manifold as a pair p𝑀, t¨, ¨uq of a real Banach smooth manifold 𝑀 and a Poisson structure t¨, ¨u on it such that the function 6 : T 𝑀 Ñ T 𝑀 defined above satisfies 6pT 𝑀 q Ď T𝑀 . If p𝑀, t¨, ¨uq is a real Banach Poisson manifold, then every 𝑘 P C 8 F p𝑀 ; Rq determines a unique vector field X 𝑘 P T𝑀 defined by (38), and called a hamiltonian vector field.…”

Section: )mentioning

confidence: 99%

“…Because suppp𝜔q " I for each 𝜔 P 𝒩 ‹0 , the bundle projection e reduces in this case to a cartesian product projection. Orbits of any Poisson flow leave 𝒩 ‹0 Ď 𝒩 sa ‹ invariant [38,27,287], while 𝛼 𝑡 ‹ is norm preserving, so the restrictions of Poisson compatible isometries 𝛼 𝑡 ‹ to 𝒩 ‹0 are automorphisms of this space. As a result, we obtain a remarkable geometric correspondence: every weakly-‹ continuous representation 𝛼 : R Q 𝑡 Þ Ñ 𝛼 𝑡 P Autp𝒩 q satisfying the Poisson compatibility condition (97) determines a unique globally integrable hamiltonian vector field X ℎ 𝛼 P T𝒩 sa ‹ and a family of standard liouvillean operators 𝒩 ‹0 Q 𝜔 Þ Ñ 𝐾 𝛼 𝜔 P pLinpℋ 𝜔 qq sa acting pointwise on the GNS bundle of Hilbert spaces.…”

Section: Case Study: Algebraic Hamiltonian Vector Fieldsmentioning

confidence: 99%

“…which gives the Schrödinger equation. This leads to a question whether some modification of the variational principle (287) on Pℋ could result in an interesting form of the temporal behaviour of quantum theoretic models? In particular, it seems that for not vanishing metric term, provided by the finite values of 𝜆 corresponding to finite values of 𝜐 in ( 281) and ( 282), the resulting temporal behaviour would be nonunitary.…”

Section: Hilbert Space Geometry and Coherent State Path Integralsmentioning

confidence: 99%

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Local quantum information dynamics

Kostecki

2016

Preprint

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This paper is intended to: 1) show how the local smooth geometry of spaces of normal quantum states over W ˚-algebras (generalised spaces of density matrices) may be used to substantially enrich the description of quantum dynamics in the algebraic and path integral approaches; 2) provide a framework for construction of quantum information theories beyond quantum mechanics, such that quantum mechanical linearity holds only locally, while the nonlocal multi-user dynamics exhibits some similarity with general relativity. In the algebraic setting, we propose a method of incorporating nonlinear Poisson and relative entropic local dynamics, as well as local gauge and local source structures, into an effective description of local temporal evolution of quantum states by using fibrewise perturbations of liouvilleans in the fibre bundle of Hilbert spaces over the quantum state manifold. We apply this method to construct an algebraic generalisation of Savvidou's action operator. In the path integral setting, motivated by the Savvidou-Anastopoulous analysis of the role of Kähler space geometry in the Isham-Linden quantum histories, we propose to incorporate local geometry by means of a generalisation of the Daubechies-Klauder coherent state phase space propagator formula. Finally, we discuss the role of Brègman relative entropy in the Jaynes-Mitchell-Favretti renormalisation scheme. Using these tools we show that: 1) the propagation of quantum particles (in Wigner's sense) can be naturally explained as a free fall along the trajectories locally minimising the quantum relative entropy; 2) the contribution of particular trajectories to the global path integral is weighted by the local quantum entropic prior, measuring user's lack of information; 3) the presence of nonlinear quantum control variables results in the change of the curvature of the global quantum state space; 4) the behaviour of zero-point energy under renormalisation of local entropic dynamics is maintained by local redefinition of information mass (prior), which encodes the curvature change. We conclude this work with a proposal of a new framework for nonequilibrium quantum statistical mechanics based on quantum Orlicz spaces, quantum Brègman distances and Banach Lie algebras.

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Bäcklund Transformations in Discrete Variational Principles for Lie-Poisson Equations

Barbero-Lin͂án

Diego

2018

Discrete Mechanics, Geometric Integration and Lie–Butcher Series

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Coadjoint Orbits and the Beginnings of a Geometric Representation Theory

Cited by 5 publications

References 72 publications

Geometry of quantum dynamics in infinite-dimensional Hilbert space

Geometry of quantum dynamics in infinite-dimensional Hilbert space

Local quantum information dynamics

Bäcklund Transformations in Discrete Variational Principles for Lie-Poisson Equations

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