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geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
Given a symplectic vector space one may consider the restriction of its quantomorphism group to the affine symplectic group (Robbin-Salamon 93, corollary 9.3)
Sometimes (e.g. Robbin-Salamon 93, p. 30) this is called the extended symplectic group, but maybe to be more specific one should at the very least say “extended affine symplectic group” or “extended inhomogeneous symplectic group” (ARZ 06, prop. V.1).
Notice that the further restriction to regarded as the translation group over itself is the Heisenberg group
The group is that of those quantomorphisms which come from Hamiltonians that are quadratic Hamiltonians. Those elements covering elements in the symplectic group instead of the affine symplectic group come from homogeneously quadratic Hamiltonians (e.g. Robbin-Salamon 93, prop. 10.1). In fact is the semidirect product of the metaplectic group with the Heisenberg group (ARZ 06, prop. V.1, see also Low 12)
Let be the 2-dimensional symplectic vector space.
Write
for its two canonical coordinate functions (the “canonical coordinates and canonical momenta”).
Write
for the constant function with value 1.
The Poisson bracket is
Any smooth function we may call a Hamiltonian. Given a Hamiltonian , its Hamiltonian flow is the flow given by the vector field (the Hamiltonian vector field) corresponding to the derivation on .
Those Hamiltonians whose Hamiltonian flows are linear functions on are precisely the homogeneously quadratic Hamiltonians:
The general element of the metaplectic group is hence
By differentiating this by at we obtain a basis for the Lie algebra of, both, the symplectic group as well as its metaplectic group
The Hamiltonians that generate translations are precisely the homogeneously linear Hamiltonians:
This is the key point where the extension appears: While these two linear translation operations themselves (i.e. the underlying symplectomorphisms) of course commute with each other, their generating Hamiltonians do not Poisson commute but instead form the Heisenberg algebra extension of the translation group.
The general element of the extended affine symplectic group is
Joel Robbin, Dietmar Salamon, Feynman path integrals on phase space and the metaplectic representation, Math. Z. 221 (1996), no. 2, 307–-335, (MR98f:58051, doi, pdf), also in Dietmar Salamon (ed.), Symplectic Geometry, LMS Lecture Note series 192 (1993)
Sergio Albeverio, J. Rezende and J.-C. Zambrini, Probability and Quantum Symmetries. II. The Theorem of Noether in quantum mechanics, Journal of Mathematical Physics 47, 062107 (2006) (pdf)
Stephen G. Low, Maximal quantum mechanical symmetry: Projective representations of the inhomogenous symplectic group, J. Math. Phys. 55, 022105 (2014) (arXiv:1207.6787)
Last revised on July 18, 2020 at 19:22:27. See the history of this page for a list of all contributions to it.