The formulation of covariant brackets on the space of solutions to a variational problem is analyzed in the framework of contact geometry. It is argued that the Poisson algebra on the space of functionals on fields should be read as a Poisson subalgebra within an algebra of functions equipped with a Jacobi bracket on a suitable contact manifold.
Using the recently developed groupoidal description of Schwinger’s picture of Quantum Mechanics, a new approach to Dirac’s fundamental question on the role of the Lagrangian in Quantum Mechanics is provided. It is shown that a function [Formula: see text] on the groupoid of configurations (or kinematical groupoid) of a quantum system determines a state on the von Neumann algebra of the histories of the system. This function, which we call q-Lagrangian, can be described in terms of a new function [Formula: see text] on the Lie algebroid of the theory. When the kinematical groupoid is the pair groupoid of a smooth manifold M, the quadratic expansion of [Formula: see text] will reproduce the standard Lagrangians on TM used to describe the classical dynamics of particles.
Linear dynamics restricted to invariant submanifolds generally gives rise to nonlinear dynamics. Submanifolds in the quantum framework may emerge for several reasons: one could be interested in specific properties possessed by a given family of states, either as a consequence of experimental constraints or inside an approximation scheme. In this work we investigate such issues in connection with a one parameter group φ t of transformations on a Hilbert space, H, defining the unitary evolutions of a chosen quantum system. Two procedures will be presented: the first one consists in the restriction of the vector field associated with the Schrödinger equation to a submanifold invariant under the flow φ t . The second one makes use of the arXiv:1908.03699v1 [quant-ph]
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