In this paper we propose a geometrization of the non-relativistic quantum mechanics for mixed states. Our geometric approach makes use of the Uhlmann's principal fibre bundle to describe the space of mixed states and as a novelty tool, to define a dynamic-dependent metric tensor on the principal manifold, such that the projection of the geodesic flow to the base manifold gives the temporal evolution predicted by the von Neumann equation. Using that approach we can describe every conserved quantum observable as a Killing vector field, and provide a geometric proof for the Poincaré quantum recurrence in a physical system with finite energy levels.
Received 15 February 2013Accepted 03 May 2013Published online 23 May 2013
This work was supported by the Spanish Ministry of Science and Innovation under the project DGI grant MTM2010-21206-C02-02.
Article outline: I. INTRODUCTION II. DENSITY MATRICES SPACE AS A BASE OF A PRINCIPAL FIBRE BUNDLE III. HAMILTONIAN VECTOR FIELD, DYNAMIC RIEMANNIAN METRIC, SHG-QUANTUM FIBRE BUNDLE AND MAIN THEOREM A. Hamiltonian vector field, dynamic metric, and its relation with other metrics B. Geometric structure of the SHg-quantum fibre bundle IV. GEOMETRIC APPROACH TO QUANTUM POINCARÉ RECURRENCE A. Physical systems with discrete energy eigenvalues
17.J. M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics Vol. 218 (Springer-Verlag, New York, 2003), pp. xviii+628.
18.M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000), pp. xxvi+676.
19.Observe that in Eq. (1) and throughout this paper we use the natural units system (h = 1).
20.The map τ is well defined because a positive operator admits a unique positive square root. It is a section because .
21.B. O'Neill, Semi-Riemannian Geometry. With Applications to Relativity, Pure and Applied Mathematics Vol. 103 (Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983), pp. xiii+468.
22.I. C. Percival, “Almost periodicity and the quantal H theorem,” J. Math. Phys.2, 235–239 (1961).