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Journal of Mathematical Physics

Note on Wigner's Theorem on Symmetry Operations

V. Bargmann

Published Jul 1, 1964• 346 citations
Publisher

Abstract

Wigner's theorem states that a symmetry operation of a quantum system is induced by a unitary or an anti-unitary transformation. This note presents a detailed proof which closely follows Wigner's original exposition.
DOI
10.1063/1.1704188

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A cornerstone of quantum mechanics is the characterisation of symmetries provided by Wigner's theorem. Wigner's theorem shows that every symmetry of the quantum state space must be either a unitary transformation, or a antiunitary transformation. Here we extend Wigner's theorem from quantum states to quantum evolutions, including both the deterministic evolution associated to the dynamics of closed systems, and the stochastic evolution associated to quantum measurements. We prove that every symmetry of the space of quantum evolutions can be decomposed into two state space symmetries that are either both unitary or both antiunitary. Building on this result, we show that it is impossible to extend the time reversal symmetry of unitary quantum dynamics to a symmetry of the full set of quantum evolutions. Our no-go theorem implies that any time symmetric formulation of quantum theory must either restrict the set of the allowed evolutions, or modify the operational interpretation of quantum states and processes. We propose one such modification, wherein the evolutions are restricted to a suitable set of time symmetric operations, including both unitary evolution and the state reductions associated to projective measurements.
Wigner's theorem is a fundamental part of the mathematical formulation of quantum mechanics. The theorem characterizes unitary and anti-unitary operators as symmetries of quantum mechanical systems, and is a key result when relating preserver problems to quantum mechanics. At the heart of this book is a geometric approach to Wigner-type theorems, unifying both classical and more recent results. Readers are initiated in a wide range of topics from geometric transformations of Grassmannians to lattices of closed subspaces, before moving on to a discussion of applications. An introduction to all the key aspects of the basic theory is included as are plenty of examples, making this book a useful resource for beginning graduate students and non-experts, as well as a helpful reference for specialist researchers.
A cornerstone of quantum mechanics is the characterization of symmetries provided by Wigner's theorem. Wigner's theorem establishes that every symmetry of the quantum state space must be either a unitary transformation or an antiunitary transformation. Here we extend Wigner's theorem from quantum states to quantum evolutions, including both the deterministic evolution associated with the dynamics of closed systems and the stochastic evolutions associated with the outcomes of quantum measurements. We prove that every symmetry of the space of quantum evolutions can be decomposed into two state space symmetries that are either both unitary or both antiunitary. Building on this result, we show that it is impossible to extend the time-reversal symmetry of unitary quantum dynamics to a symmetry of the full set of quantum evolutions. Our no-go theorem implies that any time-symmetric formulation of quantum theory must either restrict the set of the allowed evolutions or modify the operational interpretation of quantum states and processes. Here we propose a time-symmetric formulation of quantum theory where the allowed quantum evolutions are restricted to a suitable set, which includes both unitary evolution and projective measurements but excludes the deterministic preparation of pure states. The standard operational formulation of quantum theory can be retrieved from this time-symmetric version by introducing an operation of conditioning on the outcomes of past experiments.