The Duflo non-commutative Fourier transform for the Lorentz group

2020 ◽  
Vol 17 (supp01) ◽  
pp. 2040011
Author(s):  
Giacomo Rosati
Keyword(s):  
Fourier Transform ◽  
Phase Space ◽  
Quantum System ◽  
Lie Group ◽  
Lorentz Group ◽  
Commutative Algebra ◽  
Cotangent Bundle ◽  
Irreducible Representations ◽  
Algebra Representation

For a quantum system whose phase space is the cotangent bundle of a Lie group, like for systems endowed with particular cases of curved geometry, one usually resorts to a description in terms of the irreducible representations of the Lie group, where the role of (non-commutative) phase space variables remains obscure. However, a non-commutative Fourier transform can be defined, intertwining the group and (non-commutative) algebra representation, depending on the specific quantization map. We discuss the construction of the non-commutative Fourier transform and the non-commutative algebra representation, via the Duflo quantization map, for a system whose phase space is the cotangent bundle of the Lorentz group.

scholarly journals THE SCHWINGER REPRESENTATION OF A GROUP: CONCEPT AND APPLICATIONS

2006 ◽  
Vol 18 (08) ◽  
pp. 887-912 ◽  
Author(s):  
S. CHATURVEDI ◽  
G. MARMO ◽  
N. MUKUNDA ◽  
R. SIMON ◽  
A. ZAMPINI
Keyword(s):  
Quantum Mechanics ◽  
Lie Group ◽  
Irreducible Representations ◽  
Quantum Mechanical ◽  
Direct Sum ◽  
Pure States ◽  
Group Concept ◽  
Set Up ◽  
Multiplicity Free

The concept of the Schwinger Representation of a finite or compact simple Lie group is set up as a multiplicity-free direct sum of all the unitary irreducible representations of the group. This is abstracted from the properties of the Schwinger oscillator construction for SU(2), and its relevance in several quantum mechanical contexts is highlighted. The Schwinger representations for SU(2), SO(3) and SU(n) for all n are constructed via specific carrier spaces and group actions. In the SU(2) case, connections to the oscillator construction and to Majorana's theorem on pure states for any spin are worked out. The role of the Schwinger Representation in setting up the Wigner–Weyl isomorphism for quantum mechanics on a compact simple Lie group is brought out.

scholarly journals On the Geometry of Relativistic Anyon

1997 ◽  
Vol 12 (24) ◽  
pp. 1783-1789 ◽  
Author(s):  
A. Nersessian
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Irreducible Representations ◽  
Hamiltonian Reduction ◽  
Poincare Group ◽  
Covariant Model ◽  
Lobachevsky Plane ◽  
Spin Generator

A twistor model is proposed for the free relativistic anyon. The Hamiltonian reduction of this model by the action of the spin generator leads to the minimal covariant model; whereas that by the action of spin and mass generators leads to the anyon model with free phase space which is a cotangent bundle of the Lobachevsky plane with twisted symplectic structure. Quantum mechanics of that model is described by irreducible representations of the (2+1)-dimensional Poincaré group.

scholarly journals RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

2021 ◽  
pp. 1-29
Author(s):  
DREW HEARD
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Algebraic Model ◽  
Compact Lie Group ◽  
Loop Spaces ◽  
Local Systems ◽  
Special Case ◽  
Finite Loop ◽  
Finite Loop Space

Abstract Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group W G K is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

scholarly journals Generalized Hamiltonian biodynamics and topology invariants of humanoid robots

2002 ◽  
Vol 31 (9) ◽  
pp. 555-565 ◽  
Author(s):  
Vladimir Ivancevic
Keyword(s):  
Phase Space ◽  
Lie Group ◽  
Humanoid Robots ◽  
And Topology ◽  
Hamiltonian Model

Humanoid robots are anthropomorphic mechanisms with biodynamics that resembles human musculo-skeletal dynamics. This paper proposes a new generalized (dissipative, muscle-driven, stochastic) Hamiltonian model of humanoid biodynamics. Also, (co)homological analysis is performed on its Lie-group based configuration and momentum phase-space manifolds.

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