On conservation laws in quantum mechanics

We raise fundamental questions about the very meaning of conservation laws in quantum mechanics, and we argue that the standard way of defining conservation laws, while perfectly valid as far as it goes, misses essential features of nature and has to be revisited and extended.

Conservation laws, such as those for energy, momentum, and angular momentum, are among the most fundamental laws of nature. As such, they have been intensively studied and extensively applied. First discovered in classical Newtonian mechanics, they are at the core of all subsequent physical theories, nonrelativistic and relativistic, classical and quantum. Here, we present a paradoxical situation in which such quantities are seemingly not conserved. Our results raise fundamental questions about the very meaning of conservation laws in quantum mechanics, and we argue that the standard way of defining conservation laws, while perfectly valid as far as it goes, misses essential features of nature and has to be revisited and extended.That paradoxical processes must arise in quantum mechanics in connection with conservation laws is to be expected. Indeed, on the one hand, physics is local: Causes and observable effects must be locally related, in the sense that no observations in a given space–time region can yield any information about events that take place outside its past light cone.^* On the other hand, measurable dynamical quantities are identified with eigenvalues of operators, and their corresponding eigenfunctions are not, in general, localized. Energy, for example, is a property of an entire wave function. However, the law of conservation of energy is often applied to processes in which a system with an extended wave function interacts with a local probe. How can the local probe “see” an extended wave function? What determines the change in energy of the local probe? These questions lead us to uncover quantum processes that seem, paradoxically, not to conserve energy.The present paper (which is based on a series of unpublished results, first described in refs. 3 and 4), presents the paradox and discusses various ways to think of conservation laws, but does not offer a resolution of the paradox.