Wigner's theorem


Wigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT are represented on the Hilbert space of states.
According to the theorem, any symmetry transformation of ray space is represented by a linear and unitary or antilinear and antiunitary transformation of Hilbert space. The representation of a symmetry group on Hilbert space is either an ordinary representation or a projective representation.

Rays and ray space

It is a postulate of quantum mechanics that vectors in Hilbert space that are scalar nonzero multiples of each other represent the same pure state. A ray is a set
and a ray whose vectors have unit norm is called a unit ray. If, then is a representative of. There is a one-to-one correspondence between physical pure states and unit rays. The space of all rays is called ray space.
Formally, if is a complex Hilbert space, then let be the subset
of vectors with unit norm. If is finite-dimensional with complex dimension, then has real dimension. Define a relation ≅ on by
The relation ≅ is an equivalence relation on the set. Unit ray space,, is defined as the set of equivalence classes
If is finite, has real dimension hence complex dimension. Equivalently for these purposes, one may define ≈ on by
where is the set of nonzero complex numbers, and set
This definition makes it clear that unit ray space is a projective Hilbert space. It is also possible to skip the normalization and take ray space as
where ≅ is now defined on all of by the same formula. The real dimension of is if is finite. This approach is used in the sequel. The difference between and is rather trivial, and passage between the two is effected by multiplication of the rays by a nonzero real number, defined as the ray generated by any representative of the ray multiplied by the real number.
Ray space is sometimes awkward to work with. It is, for instance, not a vector space with well-defined linear combinations of rays. But a transformation of a physical system is a transformation of states, hence mathematically a transformation of ray space. In quantum mechanics, a transformation of a physical system gives rise to a bijective unit ray transformation of unit ray space,
The set of all unit ray transformations is thus the permutation group on. Not all of these transformations are permissible as symmetry transformations to be described next. A unit ray transformation may be extended to by means of the multiplication with reals described above according to
To keep the notation uniform, call this a ray transformation. This terminological distinction is not made in the literature, but is necessary here since both possibilities are covered while in the literature one possibility is chosen.

Symmetry transformations

Loosely speaking, a symmetry transformation is a change in which "nothing happens" or a "change of our view" that does not change the outcomes of possible experiments. For example, translating a system in a homogeneous environment should have no qualitative effect on the outcomes of experiments made on the system. Likewise for rotating a system in an isotropic environment. This becomes even clearer when one considers the mathematically equivalent passive transformations, i.e. simply changes of coordinates and let the system be. Usually, the domain and range Hilbert spaces are the same. An exception would be the Hilbert space of electron states that is subjected to a charge conjugation transformation. In this case the electron states are mapped to the Hilbert space of positron states and vice versa. To make this precise, introduce the ray product,
where is the Hilbert space inner product, and are normalized elements of this space. A surjective ray transformation is called a symmetry transformation if
It can also be defined in terms of unit ray space; i.e., with no other changes. In this case it is sometimes called a Wigner automorphism. It can then be extended to by means of multiplication by reals as described earlier. In particular, unit rays are taken to unit rays. The significance of this definition is that transition probabilities are preserved. In particular the Born rule, another postulate of quantum mechanics, will predict the same probabilities in the transformed and untransformed systems,
It is clear from the definitions that this is independent of the representatives of the rays chosen.

Symmetry groups

Some facts about symmetry transformations that can be verified using the definition:
The set of symmetry transformations thus forms a group, the symmetry group of the system. Some important frequently occurring subgroups in the symmetry group of a system are realizations of
These groups are also referred to as symmetry groups of the system.

Statement of Wigner's theorem

Preliminaries

Some preliminary definitions are needed to state the theorem. A transformation of Hilbert space is unitary if
and a transformation is antiunitary if
A unitary operator is automatically linear. Likewise an antiunitary transformation is necessarily antilinear. Both variants are real linear and additive.
Given a unitary transformation of Hilbert space, define
This is a symmetry transformation since
In the same way an antiunitary transformation of Hilbert space induces a symmetry transformation. One says that a transformation of Hilbert space is compatible with the transformation of ray space if for all,
or equivalently
Transformations of Hilbert space by either a unitary linear transformation or an antiunitary antilinear operator are obviously then compatible with the transformations or ray space they induce as described.

Statement

Wigner's theorem states a converse of the above:
Proofs can be found in, and.
Antiunitary and antilinear transformations are less prominent in physics. They are all related to a reversal of the direction of the flow of time.

Representations and projective representations

A transformation compatible with a symmetry transformation is not unique. One has the following.
The significance of this theorem is that it specifies the degree of uniqueness of the representation on. On the face of it, one might believe that
would be admissible, with for, but this is not the case according to the theorem. If is a symmetry group, and if with, then
where the are ray transformations. From the last theorem, one has for the compatible representatives,
where is a phase factor.
The function is called a -cocycle or Schur multiplier. A map satisfying the above relation for some vector space is called a projective representation or a ray representation. If, then it is called a representation.
One should note that the terminology differs between mathematics and physics. In the linked article, term projective representation has a slightly different meaning, but the term as presented here enters as an ingredient and the mathematics per se is of course the same. If the realization of the symmetry group,, is given in terms of action on the space of unit rays, then it is a projective representation in the mathematical sense, while its representative on Hilbert space is a projective representation in the physical sense.
Applying the last relation to the product and appealing to the known associativity of multiplication of operators on, one finds
They also satisfy
Upon redefinition of the phases,
which is allowed by last theorem, one finds
where the hatted quantities are defined by

Utility of phase freedom

The following rather technical theorems and many more can be found, with accessible proofs, in.
The freedom of choice of phases can be used to simplify the phase factors. For some groups the phase can be eliminated altogether.
In the case of the Lorentz group and its subgroup the rotation group SO, phases can, for projective representations, be chosen such that. For their respective universal covering groups, SL and Spin, it is according to the theorem possible to have, i.e. they are proper representations.
The study of redefinition of phases involves group cohomology. Two functions related as the hatted and non-hatted versions of above are said to be cohomologous. They belong to the same second cohomology class, i.e. they are represented by the same element in, the second cohomology group of. If an element of contains the trivial function, then it is said to be trivial. The topic can be studied at the level of Lie algebras and Lie algebra cohomology as well.
Assuming the projective representation is weakly continuous, two relevant theorems can be stated. An immediate consequence of continuity is that the identity component is represented by unitary operators.