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Intimate Connections: Symmetries and Conservation Laws in Quantum vs. Classical Mechanics

Throughout his career, Eugene Wigner repeatedly claimed that symmetries and conservation laws play different roles in quantum and in classical mechanics. In this paper, I show that Wigner's claims are correct in the sense that the nature of the connection between symmetries and conservation laws is different in the two theories. I provide, in particular, a list of three specific differences that obtain for the Lagrangian formulation of classical mechanics and for the Hilbert space formulation of quantum mechanics. I also show that these differences are due to two main facts: the fact that Noether's theorems are not needed in quantum mechanics, in order to derive a conservation law from a symmetry; and the fact that the Schrodinger equation is linear, and its solutions can be superimposed to yield new solutions.Author Information:

Pablo Ruiz de Olano

University of Notre Dame