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In this note we gather together some diverse facts about the moment map for symplectic group actions and the corresponding reductions of the symplectic manifold on which the group acts. We also relate moment maps with the “character Lagrangian” so as to obtain “intertwining Lagrangians” for Hamiltonian group actions.
We show that both in classical and quantum theory of the relativistic electron there are three sets of independent dynamical variables: position, velocity and momentum. The independence of velocity and momentum is interpreted by internal degrees of freedom. The geometry of the internal phase-space is discussed. The close analogy between the classical and quantum equations and their algebraic and symplectic...
The quantum mechanically admissible definitions of the factor exp [i/ℏ S(y)] in Feynman's integral—are put in bijection with the prequantisations of Kostant and Souriau. The different allowed expressions of this factor— the inequivalent prequantisations—are classified in terms of algebraic topology.
A Choquet type of an integral representation is found for a class of normalized positive operator valued (POV) measures on a Hilbert space. An arbitrary POV-measure within this class is thereby represented uniquely as an integral over projection valued (PV) measures. As an application, the case of a commutative system of covariance (representing a generalization of the imprimitivity theorem of Mackey)...
Path integrals are considered for the cases where the underlying manifold is multiply connected or non-flat. In case of multiple connectivity, the contributions of different homotopy classes of paths are analyzed with the help of covering spaces. In case of a non-flat manifold, it is pointed out that a judicious choice of the free Hamiltonian operator and of normalizing factors can eliminate the explicit...
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