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GEOMETRIC PHASES IN THE QUANTISATION OF BOSONS AND FERMIONS

Part of: Symplectic geometry, contact geometry General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  29 June 2011

SIYE WU
SIYE WU*
Affiliation:
Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong, China (email: swu@maths.hku.hk)
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Abstract

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After reviewing geometric quantisation of linear bosonic and fermionic systems, we study the holonomy of the projectively flat connection on the bundle of Hilbert spaces over the space of compatible complex structures and relate it to the Maslov index and its various generalisations. We also consider bosonic and fermionic harmonic oscillators parametrised by compatible complex structures and compare Berry’s phase with the above holonomy.

Keywords

geometric quantisationHermitian symmetric spacesMaslov indexBerry’s phase

MSC classification

Secondary: 53D50: Geometric quantization 53D12: Lagrangian submanifolds; Maslov index 81Q70: Differential-geometric methods, including holonomy, Berry and Hannay phases, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 90 , Issue 2 , April 2011 , pp. 221 - 235
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

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