2010/08/31 by Siye Wu, Wu, Siye
Mathematics · Physics and Astronomy · #15A66 #32M15 (Secondary) #53D50 (Primary) #58C50 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Quantum Physics (quant-ph) #Symplectic Geometry (math.SG) #math.SG #msc:15A66 #msc:32M15 #msc:53D50 #msc:58C50 #quant-ph
paper · pdf · doi:10.48550/arxiv.1008.5333
20 pages, typo and other minor corrections made
openalex publication_date 2010/08/31 · arxiv created 2010/10/06 · arxiv updated 2010/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We compare the quantisation of linear systems of bosons and fermions. We recall the appearance of projectively flat connection and results on parallel transport in the quantisation of bosons. We then discuss pre-quantisation and quantisation of fermions using the calculus of fermionic variables. We then define a natural connection on the bundle of Hilbert spaces and show that it is projectively flat. This identifies, up to a phase, equivalent spinor representations constructed by various polarisations. We introduce the concept of metaplectic correction for fermions and show that the bundle of corrected Hilbert spaces is naturally flat. We then show that the parallel transport in the bundle of Hilbert spaces along a geodesic is the rescaled projection or the Bogoliubov transformation provided that the geodesic lies within the complement of a cut locus. Finally, we study the bundle of Hilbert spaces when there is a symmetry.