Geometry of quantum dynamics in infinite-dimensional Hilbert space

Abstract

We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional and including a Lagrangian formalism in which self-adjoint (Schrödinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we obtain also results concerning coadjoint orbits of the unitary group in infinite dimension, embedding of the Hilbert projective space of pure states in the unitary group, and an approach to self-adjoint extensions of symmetric relations. * Thus, the dynamics is integrable. Actually, it is the properly understood phase dynamics of vertical rolling disc on a plane in a Dirac algebroid setting studied in [22].All this can be generalized to ordinary implicit differential equations of arbitrary order. In this case we consider as a subset of higher jet bundles, the n-th tangent bundle Ì n N in case of an equation of order n, and consider γ as a solution if its n-th jet prolongation takes values in . If we call the n-th jet prolongations admissible in Ì n N , then solutions of are exactly projections γ N to N of admissible curves (or paths) γ in Ì n N lying in .Remark 2.1. The implicit differential equations described above are called by some authors differential relations. Let us explain that we use the most general definition, not requiring from any differentiability properties, since in real life the dynamics we encounter are often not submanifolds.

Unitary group orbits versus groupoid orbits of normal operators

“…If Ker a = {0}, some modifications of the previous argument as in [GKMS18,Th. 5.1] also show that the orbit is not closed.…”

Manifolds of classical probability distributions and quantum density operators in infinite dimensions

“…The last step we want to take is to write down a tangent vector V at ∈ O + , where O + may be any of the orbits in equation (67). At this purpose, we consider the canonical immersion i + : O + −→ A * sa , and we recall equation (32), from which it follows that…”

“…In the context of quantum information theory, the geometrization of some of the relevant structures, for instance the Kähler-Hilbert manifold structure on the space of pure quantum states given by the complex projective space of an infinite-dimensional, complex, separable Hilbert space H, together with a Hamiltonian formulation of the unitary evolutions of quantum mechanics, as given for instance in [7,23,24,25,43], allows a simpler treatment of the differentiable structures of the corresponding infinitedimensional groups present in the theory as it is shown in [4,5,6,13,16,32,44], where the action of Banach-Lie groups of unitary operators on an infinite-dimensional, complex, separable Hilbert space H is used to give a Banach manifold structure to appropriate subsets of quantum density operators, positive semidefinite linear operators and elements of Banach Lie-Poisson spaces or, as it will be shown in this paper, to certain orbits of the group of invertible elements on a C * -algebra.…”

From Classical Trajectories to Quantum Commutation Relations

“…Indeed, given the Lagrangian function L, we have dL : TQ → T * TQ, and thus dL(TQ) is a submanifold of T * TQ. By means of the inverse of the canonical Tulczyjew isomorphism τ : TT * Q → T * TQ ( [23]), we obtain a submanifold of TT * Q, i.e., an implicit differential equation on T * Q. Explicitly, we have…”