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Journal of Mathematical Physics

Phenomenological Distinction between Unitary and Antiunitary Symmetry Operators

Eugene Paul Wigner

Published Sep 1, 1960• 23 citations

Abstract

It is well known that one always can find as many orthogonal states (i.e., states between which the transition probability is zero) as the Hilbert space has dimensions which are invariant under a given unitary transformation. The corresponding vectors are characteristic vectors of the unitary operator. In contrast, most antiunitary operators leave not more than one state invariant. However, if there are two orthogonal invariant states, a consideration of the states for which the transition probability is ½ into both invariant states surely provides a distinction. In the antiunitary case, one of these states is also invariant, another one is transformed into an orthogonal state, the rest are in between. In the unitary case, the transition probability between original state and transformed state is the same for all states for which the transition probability is ½ into two orthogonal states. This provides a ``directly observable'' distinction between unitary and antiunitary transformations.

DOI: 10.1063/1.1703673

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It is well known that one always can find as many orthogonal states (i.e., states between which the transition probability is zero) as the Hilbert space has dimensions which are invariant under a given unitary transformation. The corresponding vectors are characteristic vectors of the unitary operator. In contrast, most antiunitary operators leave not more than one state invariant. However, if there are two orthogonal invariant states, a consideration of the states for which the transition probability is ½ into both invariant states surely provides a distinction. In the antiunitary case, one of these states is also invariant, another one is transformed into an orthogonal state, the rest are in between. In the unitary case, the transition probability between original state and transformed state is the same for all states for which the transition probability is ½ into two orthogonal states. This provides a ``directly observable'' distinction between unitary and antiunitary transformations.

A fundamental problem which arises in determining whether two quantum mechanical systems are essentially identical is whether a unitary or antiunitary transformation exists which maps one set of dynamical variables into another. Since an elementary dynamical system is specified by giving an irreducible set of dynamical variables, we are led to investigate the following problem: Given two irreducible sets of operators with a one-to-one correspondence between them, find the algebraic properties of the two sets which make it possible to infer the existence of a unitary or antiunitary operator relating them. A series of theorems is obtained from such considerations for finite dimensional spaces. It is shown that if the second set of operators contains some of the algebraic properties of the first set, the two sets are related by a similarity transformation. By altering the requirements, this transformation is a unitary transformation. Indications are also given to show how the theorems can be extended to Hilbert spaces. The rigorous statements of the theorems and the proofs will be given in a second paper. Finally, in the Appendix there is given a definition of invariance of elementary quantum-mechanical systems based on the above theorems, giving the same results as Wigner's definition in terms of transition probabilities.

Transformations on the set of all

n

-dimensional subspaces of a Hilbert space preserving orthogonality

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.

We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation

AHA^\dagger

that leaves the Hamiltonian

H

unchanged represents a symmetry of the Hamiltonian, which implies the commutativity

[H,A]=0

, and a conservation law, namely the invariance of expectation values of

A

. For non-Hermitian Hamiltonians,

H^\dagger

comes into play as a distinct operator that complements

H

in generalized unitarity relations. The above description of symmetries has to be extended to include also

A

-pseudohermiticity relations of the form

AH=H^\dagger A

. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of

A

operators forming Klein's 4-group: parity, time-reversal, parity\&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be more subtle for non-Hermitian than for Hermitian Hamiltonians.