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Projective flatness in the quantisation of bosons and fermions. (English) Zbl 1330.81136

This paper makes a comparison of the quantization of linear systems of bosons and fermions. Firstly a bosonic system is considered on a real symplectic vector space \((V,\omega)\) of even dimension. The pre-quantum Hilbert space is \(\mathcal{H}_0:=L^2(V,\ell)\), where the pre-quantum line bundle \(\ell\) has a connection \(\nabla_x\), \(x\in V\), whose curvature is \(\omega/{\sqrt{-1}}\). To \(\alpha\in V^*\) there are attached pre-quantum operators \(\hat{\alpha}\) acting on \(\mathcal{H}_0\), which verify Heisenberg canonical commutation relations. If \(\mathcal{J}_{\omega}\) denotes the space of compatible complex structures on \((V,\omega)\), let \(\mathcal{H}\rightarrow \mathcal{J}_{\omega}\) be the bundle of quantum Hilbert spaces whose fibre over \(J\in \mathcal{J}_{\omega}\) is the quantum Hilbert space \(\mathcal{H}_J\) associated to \(J\). Let \(\mathcal{V}\rightarrow \mathcal{J}_{\omega}\) be the vector bundle whose fibre over \(J\in \mathcal{J}_{\omega}\) is \(V^{1,0}_J\) and let us denote \(\mathcal{K}:=(\text{det}\mathcal{V})^*\). Then the bundle of quantum Hilbert spaces with metaplectic corrections is \(\hat{\mathcal{H}}=\mathcal{H}\otimes \sqrt{\mathcal{K}}\) and has fibres \(\hat{\mathcal{H}}_J=\mathcal{H}_J\otimes \sqrt{\mathcal{K}_J}\). Based on the references [S. Axelrod et al., J. Differ. Geom. 33, No. 3, 787–902 (1991; Zbl 0697.53061)], [W. D. Kirwin and S. Wu, Commun. Math. Phys. 266, No. 3, 577–594 (2006; Zbl 1116.53063)], and [N. M. J. Woodhouse, Geometric quantization. 2nd ed. Oxford Mathematical Monographs. Oxford: Clarendon Press (1992; Zbl 0747.58004)], it is proved that the bundle of quantum Hilbert spaces is projectively flat and that the bundle of quantum Hilbert spaces with metaplectic correction is flat. Results from the mentioned paper of W. D. Kirwin of S. Wu on parallel transport in the bundle of Hilbert spaces along geodesics in the base space are reviewed. The existence of a symmetry of the bosonic system is supposed. If there is a a symplectic group action on \((V,\omega)\), then the bundle of Hilbert spaces is written down as a direct sum of projectively flat sub-bundles. Next a fermionic system is considered. The phase space is a real finite dimensional vector space \(V\) equipped with a Euclidean inner product \(q\). The pre-quantum Hilbert space of fermions \(\mathcal{H}_0\) is taken \(\bigwedge^*(V^{\mathbb{C}})^*\), equipped with a Hermitian form given by Berezin integral. The equivalent of the curvature is a symmetric bilinear form. The classical observables \(\alpha\in V^*\) are pre-quantised, giving rise to self-adjoint operators which satisfies the Clifford algebra relations. \(V\) it is supposed to have even dimension. The space of complex structures on \(V\) compatible with the metric \(q\) and the orientation is denoted \(\mathcal{J}_q\). Each \(J\in \mathcal{J}_q\) defines a polarisation and the quantum Hilbert space \(\mathcal{H}_J\). Next we have the bundle of quantum Hilbert spaces \(\mathcal{H}\rightarrow \mathcal{J}_q\) whose fibre over \(J\in\mathcal{J}_q\) is \(\mathcal{H}_J\). A natural connection is defined on \(\mathcal{H}\) and it is proved that it is projectively flat. Also the quantum Hilbert space with metaplectic corrections is flat. It is proved that parallel transport in the bundle of Hilbert spaces is a rescaled projection if the geodesic lies within the complement of the cut locus relative to the starting point in \(\mathcal{J}_q\). As in the case of bosons, a decomposition of the bundle of fermionic Hilbert spaces with symmetry is studied. The paper has an appendix dedicated to the geometry of the space of complex structures. Another appendix summarises some facts used in the paper concerning the Berezin calculus. The last appendix refers to the invariant real, complex and quaternionic structures.

MSC:

81S10 Geometry and quantization, symplectic methods
53D50 Geometric quantization
81S05 Commutation relations and statistics as related to quantum mechanics (general)
81T70 Quantization in field theory; cohomological methods
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
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References:

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