Quantum mechanics as an infinite-dimensional Hamiltonian system with uncertainty structure: Part II

Abstract

Making reference to the formalism developed in Part I to formulate Schrödinger quantum mechanics, the properties of Kählerian functions in general, almost Kählerian manifolds, are studied.

The structure of classical extensions of quantum probability theory

“…Moreover, the projective Hilbert space is a differentiable manifold carrying a natural symplectic structure which allows one to reformulate quantum dynamics in terms of Hamiltonian mechanics (cf. [4,5,12,13,19,23]). Hence, quantum mechanics can be interpreted to be a reduced classical statistical mechanics on the phase space P(H).…”

Reduction Procedures in Classical and Quantum Mechanics

“…This picture of Quantum Mechanics is not widely known but it arises in connection with the so called Ehrenfest theorem which may be seen from the point of view of ⋆-products on phase space (see [26]). Some aspects of this picture have been considered by Weinberg [54] and more generally appear in the geometrical formulation of Quantum Mechanics [16,17,18,19].…”

Poisson Spaces with a Transition Probability

“…This shows that the symplectic structure on each leaf is ω F S , where ω F S is the Fubini-Study structure [53,38,18,19,20,39]. (A closely related fact is that the Kähler metric associated to ω F S is determined, up to a multiplicative constant, by its invariance under the induced action of all unitary operators on H, cf.…”

“…Inspired by [1,19], we define a (locally non-trivial) fiber bundle B(P), whose base space B is the space of sectors, equipped with the discrete topology, and whose fiber above a given base point α is B(H α ) sa ; here H α is such that the sector α is PH α . Moreover, P itself may be seen as a fiber bundle over the same base space; now the fiber above α is PH α .…”

“…This Poisson bracket is defined by letting the sectors PH α of P(A) coincide with its symplectic leaves, and making each PH α into a symplectic manifold by endowing it with the (suitably normalized) Fubini-Study symplectic form [53,38,18,19,20,39]. The reason that this structure is uniquely determined by (1.21) is that in an irreducible representation π(A C ) on a Hilbert space H the collection of differentials {d π(A), A ∈ A} is dense in the cotangent space at each point of PH.…”