WAVE PACKETS AND THE QUADRATIC MONGE-KANTOROVICH DISTANCE IN QUANTUM MECHANICS

Abstract In this paper, we extend the upper and lower bounds for the “pseudo-distance” on quantum densities analogous to the quadratic Monge–Kantorovich(–Vasershtein) distance introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016) 165–205] to positive quantizations defined in terms of the family of phase space translates of a density operator, not necessarily of rank 1 as in the case of the Toplitz quantization. As a corollary, we prove that the uniform as ħ → 0 convergence rate for the mean-field limit of the N-particle Heisenberg equation holds for a much wider class of initial data than in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016) 165–205]. We also discuss the relevance of the pseudo-distance compared to the Schatten norms for the purpose of metrizing the set of quantum density operators in the semiclassical regime.

[1]  François Golse,et al.  Derivation of the Schrödinger–Poisson equation from the quantum N-body problem , 2002 .

[2]  D. Haar,et al.  Quantum Mechanics Vol. 1 , 1965 .

[3]  H. Spohn.  Kinetic equations from Hamiltonian dynamics: Markovian limits , 1980 .

[4]  Benjamin Schlein,et al.  Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics , 2007, 0711.3087.

[5]  E. Schrödinger.  Der stetige Übergang von der Mikro- zur Makromechanik , 1926, Naturwissenschaften.

[6]  T. Paul,et al.  Sur les mesures de Wigner , 1993 .

[7]  M. Reed.  Methods of Modern Mathematical Physics. I: Functional Analysis , 1972 .

[8]  T. Paul,et al.  On the Mean Field and Classical Limits of Quantum Mechanics , 2015, Communications in Mathematical Physics.

[9]  L. Landau.  Quantum Mechanics-Nonrelativistic Theory , 1958 .

[10]  Horng-Tzer Yau,et al.  Derivation of the nonlinear Schr\"odinger equation from a many body Coulomb system , 2001 .

[11]  M. Hauray,et al.  Rigorous derivation of Lindblad equations from quantum jumps processes in 1D , 2016, 1603.07969.

[12]  A. Grossmann,et al.  Transforms associated to square integrable group representations. I. General results , 1985 .

[13]  François Golse,et al.  Weak Copling Limit of the N-Particle Schrödinger Equation , 2000 .

[14]  E. M. Lifshitz,et al.  Quantum mechanics: Non-relativistic theory, , 1959 .

[15]  M. Reed,et al.  Methods of Modern Mathematical Physics. 2. Fourier Analysis, Self-adjointness , 1975 .

[16]  C. Villani.  Topics in Optimal Transportation , 2003 .

[17]  P. Pickl.  A Simple Derivation of Mean Field Limits for Quantum Systems , 2009, 0907.4464.