Quantum mechanics and the geometry of the Weyl character formula

Journal of the Australian Mathematical Society, 2005

We develop a notion of a *-product on a general abelian group, establish a Weyl calculus for operators on the group and connect these with the representation theory of an associated Heisenberg group. This can all be viewed as a generalization of the familiar theory for R. A symplectic group is introduced and a connection with the classical Cayley transform is established. Our main application is to finite groups, where consideration of the symbol calculus for the cyclic groups provides an interesting alternative to the usual matrix form for linear transformations. This leads to a new basis for sl(n) and a decomposition of this Lie algebra into a sum of C*artan subalgebras.

Journal of Mathematical Physics

In this work, we consider fixed 1/2 spin particles interacting with the quantized radiation field in the context of quantum electrodynamics (QED). We investigate the time evolution operator in studying the reduced propagator (interaction picture). We first prove that this propagator belongs to the class of infinite dimensional Weyl pseudodifferential operators recently introduced in [3] on Wiener spaces. We give a semiclassical expansion of the symbol of the reduced propagator up to any order with estimates on the remainder terms. Next, taking into account analyticity properties for the Weyl symbol of the reduced propagator, we derive estimates concerning transition probabilities between coherent states.

Quantum Studies: Mathematics and Foundations, 2015

In this paper, we introduce the generalized Weyl operators canonically associated with the one-mode oscillator Lie algebra as unitary operators acting on the bosonic Fock space (C). Next, we establish the generalized Weyl relations and deduce a group structure on the manifold R 2 × [−π, π[×R generalizing the well-known Heisenberg one in a natural way.

Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 1974

The construction of field theory which exhibits invariance under the Weyl group with parameters dependent on space–time is discussed. The method is that used by Utiyama for the Lorentz group and by Kibble for the Poincaré group. The need to construct world-covariant derivatives necessitates the introduction of three sets of gauge fields which provide a local affine connexion and a vierbein system. The geometrical implications are explored; the world geometry has an affine connexion which is not symmetric but is semi-metric. A possible choice of Lagrangian for the gauge fields is presented, and the resulting field equations and conservation laws discussed.

Letters in Mathematical Physics, 1977

The formal expansion defining the twisted exponential of an element of the Lie algebra ~n [] ( en S~P( 2, IR)) can be summed and this result is used to explicitly obtain the classical function u t corresponding to an evolution operator associated to a quantum Hamiltonian belonging to the above mentioned Lie algebra.

Groups and Analysis

Expositiones Mathematicae, 2016

We produce a connection between the Weil 2-cocycles defining the local and adèlic metaplectic groups defined over a global field, i.e. the double covers of the attendant local and adèlic symplectic groups, and local and adèlic Maslov indices of the type considered by Souriau and Leray. With the latter tied to phase integrals occurring in quantum mechanics, we provide a formulation of quadratic reciprocity for the underlying field, first in terms of an adèlic phase integral, and then in terms of generalized time evolution unitary operators. c

Journal of Lie theory

The aim of this paper is to present a new character formula for finite-dimensional representations of finite-dimensional complex semisimple Lie Algebras and compact semisimple Lie Groups. Some applications of the new formula include the exact determination of the number of weights in a representation, new recursion formulas for multiplicities and, in some cases, closed formulas for the multiplicities themselves.

Noncommutative Geometry and Physics 3, 2012

2006

A covariant Wigner-Weyl quantization formalism on the manifold that uses pseudo-differential operators is proposed. The asymptotic product formula that leads to the symbol calculus in the presence of gauge and gravitational fields is presented. The new definition is used to get covariant differential operators from momentum polynomial symbols. A covariant Wigner function is defined and shown to give gauge-invariant results for the Landau problem. An example of the covariant Wigner function on the 2-sphere is also included. iv To my family v ACKNOWLEDGMENTS First, I would like to thank my advisor Stephen A. Fulling for his endless support during my entire study at Texas A&M. He is undoubtedly the best academic mentor that I have met with his willingness to interact in the most productive way. I thank the members of my Ph.D. committee for their support. I also thank Tom Osborn, Robert Littlejohn, Ivan Avramidi, Lance Drager and Frank Molzahn for valuable conversations and correspondence, direct or indirect. Throughout my study, I was supported by the Physics Department as a graduate teaching assistant. I particularly thank Prof. Kenefick and Prof. Ford and the rest of the physics faculty at Texas A&M. I also thank Prof. Kattawar for his financial and academic support during my last semester. Finally, I thank my friends EsadÖzmetin, Serkan Erdin, Ali Kaya,