The modal-Hamiltonian interpretation and the Galilean covariance of quantum mechanics

Abstract The aim of this paper is to analyze the modal-Hamiltonian interpretation of quantum mechanics in the light of the Galilean group. In particular, it is shown that the rule of definite-value assignment proposed by that interpretation has the same properties of Galilean covariance and invariance as the Schrodinger equation. Moreover, it is argued that, when the Schrodinger equation is invariant, the rule can be reformulated in an explicitly invariant form in terms of the Casimir operators of the Galilean group. Finally, the possibility of extrapolating the rule to quantum field theory is considered.

[1]  Sunny Y. Auyang,et al.  How Is Quantum Field Theory Possible , 1995 .

[2]  Shiro Ishikawa,et al.  A New Interpretation of Quantum Mechanics , 2011, J. Quantum Inf. Sci..

[3]  井上 研三.  H. Georgi: Lie Algebras in Particle Physics; From Isospin to Unified Theories, Benjamin/Cummings, Advanced Book Program, Massachusetts, 1982, xxiii+256ページ, 24.5×17cm, 5,650円(Frontiers in Physics; A Lecture Note and Reprint Series, No. 54). , 1982 .

[4]  Rob Clifton,et al.  Independently Motivating the Kochen—Dieks Modal Interpretation of Quantum Mechanics , 1995, The British Journal for the Philosophy of Science.

[5]  Pieter E. Vermaas,et al.  The Modal Interpretation of Quantum Mechanics , 1998 .

[6]  C. cohen-tannoudji,et al.  Quantum Mechanics: , 2020, Fundamentals of Physics II.

[7]  J. Earman.  Curie's Principle and spontaneous symmetry breaking , 2004 .

[8]  Dennis Dieks,et al.  The Formalism of Quantum Theory: An Objective Description of Reality? , 1988 .

[9]  Arthur Fine.  Probability and the Interpretation of Quantum Mechanics* , 1973, The British Journal for the Philosophy of Science.

[10]  On Symplectic Reduction in Classical Mechanics , 2005, physics/0507194.

[11]  O. Lombardi,et al.  The cosmological origin of time asymmetry , 2002, quant-ph/0211162.

[12]  Harvey R. Brown,et al.  The Galilean covariance of quantum mechanics in the case of external fields , 1999 .

[13]  Olimpia Lombardi,et al.  Quantum Mechanics: Modal Interpretation and Galilean Transformations , 2009 .

[14]  S. Bose.  The galilean group in 2+1 space-times and its central extension , 1995 .

[15]  J. Earman.  Laws, Symmetry, and Symmetry Breaking: Invariance, Conservation Principles, and Objectivity , 2004, Philosophy of Science.

[16]  Katherine Brading,et al.  Symmetries and invariances in classical physics , 2005 .

[17]  K. Brading,et al.  Symmetries in physics: philosophical reflections , 2003, quant-ph/0301097.

[18]  A. Raczyński,et al.  Symposium on the foundations of modern physics : edited by P. Lahti and P. Mittelstaedt, World Scientific Publishing Co. Pte. Ltd., Singapore 1985 , 1986 .

[19]  A Bicycle Built for Two: The Galilean and U(1) Gauge Invariance of the Schr\"odinger Field , 2008, 0802.2401.

[20]  Albert Einstein,et al.  The Principle of Relativity , 2014 .

[21]  L. Ballentine.  Quantum mechanics : a modern development , 1998 .

[22]  Mario Bunge,et al.  Philosophy of Physics , 1972 .

[23]  H. Georgi.  Lie Algebras in Particle Physics: From Isospin to Unified Theories , 1994 .

[24]  H. Minkowski,et al.  Space and time , 1952 .

[25]  N L Harshman,et al.  Galilean and dynamical invariance of entanglement in particle scattering. , 2007, Physical review letters.

[26]  Robert Nozick.  Invariances: The Structure of the Objective World , 2001 .

[27]  E. Specker,et al.  The Problem of Hidden Variables in Quantum Mechanics , 1967 .

[28]  M. Suárez,et al.  Are Sharp Values of Observables always Objective Elements of Reality , 1998 .

[29]  Pieter E. Vermaas,et al.  The modal interpretation of quantum mechanics and its generalization to density operators , 1995 .

[30]  Invariance, Symmetry and Meaning , 2000 .

[31]  V. Colussi,et al.  Galilean and U(1)-gauge symmetry of the Schrödinger field , 2008 .

[32]  J. M. Lévy-Leblond,et al.  The pedagogical role and epistemological significance of group theory in quantum mechanics , 1974 .

[33]  Olimpia Lombardi,et al.  A modal-Hamiltonian interpretation of quantum mechanics , 2006, quant-ph/0610121.