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Progress of Theoretical Physics

Conservation Laws in Classical and Quantum Physics

Eugene Paul Wigner

Published Apr 1, 1954• 42 citations
Publisher

Abstract

The relation between the conservation laws and the symmetry properties of space and time is examined. If the simplest possible equation of motion takes the place of Newton’s equation, the usual conservation laws are not obtained. In the quantum physics, on the contrary, there is a superbundance of conservation laws and their physical significance is not obvious.

DOI
10.1143/ptp.11.437

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The structures are physical systems and/or their mathematical descriptions consisting of elements such as particles, fields, observables, operators etc and of relations between these elements: the equations of motion and the rules of quantum mechanics. Examples of symmetries are reflections, permutations of particles, time reversal. The physical system has a symmetry group means, then, that there is a group of transformations leaving the equations of motion and the rules of quantum mechanics invariant. The discrete symmetries lead to conservation laws frequently expressible in terms of quantum numbers, but sometimes requiring a more elaborate description. The observables corresponding to the super-selection quantum numbers can be measured without disturbing any other measurement; their values are always sharp. The chapter shows that the invariant subspace may be labelled by continuous quantum numbers (for instance momentum) when the direct sum becomes a direct integral.
In this article, I use a number of remarks made by Eugene Wigner to defend the claim that the nature of the connection between symmetries and conservation laws is different in quantum and in classical mechanics. In particular, I provide a list of three differences that obtain between the Hilbert space formulation of quantum mechanics and the Lagrangian formulation of classical mechanics. I also show that these differences are due to the fact that conservation laws are not the only consequence that symmetries have in quantum mechanics and to the fact that, in classical mechanics, the connection between symmetries and conservation laws does not always obtain.