Wigner's theorem

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

Ψ _ = { e i α Ψ | α ∈ R } , ∅ ≠ Ψ ∈ H ,

and a ray whose vectors have unit norm is called a unit ray. If Φ ∈ Ψ, then Φ is a representative of Ψ. 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 H is a complex Hilbert space, then let B be the subset

B = { Ψ ∈ H : | | Ψ | | = 1 }

of vectors with unit norm. If H is finite-dimensional with complex dimension N, then B has real dimension 2N − 1. Define a relation ≅ on B by

Ψ ≅ Φ ⇔ Ψ = e i α Φ , α ∈ R .

The relation ≅ is an equivalence relation on the set B. Unit ray space, S, is defined as the set of equivalence classes

S = B / ≅ .

If N is finite, S has real dimension 2N − 2 hence complex dimension N − 1. Equivalently for these purposes, one may define ≈ on H by

Ψ ≈ Φ ⇔ Ψ = z Φ , z ∈ C ∖ { 0 } ,

where ℂ {0} is the set of nonzero complex numbers, and set

S ′ = H ∖ ∅ / ≈ .

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

R = H ∖ ∅ / ≅ ,

where ≅ is now defined on all of H by the same formula. The real dimension of R is 2N − 1 if N is finite. This approach is used in the sequel. The difference between R and S 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 T of unit ray space,

T : S ∋ Ψ _ ⊂ H ↦ S ′ ∋ Ψ ′ _ = T Ψ _ ⊂ H ′ .

The set of all unit ray transformations is thus the permutation group on S. Not all of these transformations are permissible as symmetry transformations to be described next. A unit ray transformation may be extended to R by means of the multiplication with reals described above according to

T : R → R ′ ; T ( λ Ψ _ ) ≡ λ T Ψ _ , Ψ _ ∈ S , λ ∈ R .

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 (in a non-relativistic theory) 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,

Ψ _ ⋅ Φ _ = | ( Ψ , Φ ) | ,

where (Ψ,Φ) is the Hilbert space inner product. A surjective ray transformation T: R → R' is called a symmetry transformation if

T Ψ _ ⋅ T Φ _ = Ψ _ ⋅ Φ _ , ∀ Ψ , Φ ∈ H .

It can also be defined in terms of unit ray space, i. e. T: S → S' with no other changes. In this case it is sometimes called a Wigner automorphism. It can then be extended to R 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,

P ( Ψ → Φ ) = | ( Ψ , Φ ) | 2 = [ Ψ _ ⋅ Φ _ ] 2 = [ T Ψ _ ⋅ T Φ _ ] 2 = | ( Ψ ′ , Φ ′ ) | 2 = P ( Ψ ′ → Φ ′ ) , Ψ ′ ∈ T Ψ _ , Φ ′ ∈ T Φ _ .

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 product of two symmetry transformations, i.e. two symmetry transformations applied in concession, is a symmetry transformation.

Any symmetry transformation has an inverse.

The identity transformation is a symmetry transformation.

Multiplication of symmetry transformations is associative.

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

The symmetric group with its subgroups. This is important on the exchange of particle labels.

The Poincaré group. It encodes the fundamental symmetries of spacetime.

Internal symmetry groups like SU(2) and SU(3). They describe so called internal symmetries, like isospin and color charge peculiar to quantum mechanical systems.

These groups are also referred to as symmetry groups of the system.

Preliminaries

Some preliminary definitions are needed to state the theorem. A transformation U of Hilbert space is unitary if

( U Ψ , U Φ ) = ( Ψ , Φ ) ,

and a transformation is antiunitary if

( A Ψ , A Φ ) = ( Ψ , Φ ) ∗ = ( Φ , Ψ ) .

A unitary operator is automatically linear. Likewise an antiunitary transformation is necessarily antilinear. Both variants are real linear and additive.

Given a unitary transformation U of Hilbert space, define

T : Ψ _ = { e i α Ψ | α ∈ R } ↦ Ψ ′ _ = { e i β U Ψ | β ∈ R } .

This is a symmetry transformation since

T Ψ _ ⋅ T Φ _ = Ψ ′ _ ⋅ Φ ′ _ = | ( e i α U Ψ , e i β U Φ ) | = | ( Ψ , Φ ) | = Ψ _ ⋅ Φ _ .

In the same way an antiunitary transformation of Hilbert space induces a symmetry transformation. One says that a transformation U of Hilbert space is compatible with the transformation T of ray space if for all Ψ,

T Ψ _ = { e i α U Ψ | α ∈ R } ,

or equivalently

U Ψ ∈ T Ψ _ .

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:

Wigner's theorem (1931): If H and K are Hilbert spaces and if T : Ψ _ ⊂ H ↦ Ψ ′ _ ⊂ K is a symmetry transformation, then there exists a transformation V:H → K which is compatible with T and such that V is either unitary or antiunitary if dim H ≥ 2. If dim H = 1 there exists a unitary transformation U:H → K and an antiunitary transformation A:H → K, both compatible with T.

Proofs can be found in Wigner (1931, 1959), Bargmann (1964) and Weinberg (2002).

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 (additive transformations include both linear and antilinear transformations).

Theorem: If U and V are two additive transformations of H onto K, both compatible with the ray transformation T with dim H ≥ 2, then V = U e i α , α ∈ R .

The significance of this theorem is that it specifies the degree of uniqueness of the representation on H. On the face of it, one might believe that

V h = U e i α ( h ) h , α ∈ R , h ∈ H ( wrong unless  α ( h ) = const )

would be admissible, with α(h) ≠ α(k) for ⟨h|k⟩ = 0, but this is not the case according to the theorem. If G is a symmetry group (in this latter sense of being embedded as a subgroup of the symmetry group of the system acting on ray space), and if f, g, h ∈ G with fg = h, then

T ( f ) T ( g ) = T ( h ) ,

where the T are ray transformations. From the last theorem, one has for the compatible representatives U,

U ( f ) U ( g ) = ω ( f , g ) U ( f g ) = e i ξ ( f , g ) U ( f g ) ,

where ω(f, g) is a phase factor.

The function ω is called a 2-cocycle or Schur multiplier. A map U:G → GL(V) satisfying the above relation for some vector space V is called a projective representation or a ray representation. If ω(f, g) = 1, 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, g → T(g), is given in terms of action on the space of unit rays S = PH, then it is a projective representation G → PGL(H) in the mathematical sense, while its representative on Hilbert space is a projective representation G → GL(H) in the physical sense.

Applying the last relation (several times) to the product fgh and appealing to the known associativity of multiplication of operators on H, one finds

ω ( f , g ) ω ( f g , h ) = ω ( g , h ) ω ( f , g h ) , ξ ( f , g ) + ξ ( f g , h ) = ξ ( g , h ) + ξ ( f , g h ) ( mod ⁡ 2 π ) .

Thy also satisfy

ω ( g , e ) = ω ( e , g ) = 1 , ξ ( g , e ) = ξ ( e , g ) = 0 ( mod ⁡ 2 π ) , ω ( g , g − 1 ) = ω ( g − 1 , g ) , ξ ( g , g − 1 ) = ξ ( g − 1 , g ) = 0 ( mod ⁡ 2 π ) .

Upon redefinition of the phases,

U ( g ) ↦ U ^ ( g ) = η ( g ) U ( g ) = e i ζ ( g ) U ( g ) ,

which is allowed by last theorem, one finds

ω ^ ( g , h ) = ω ( g , h ) η ( g ) η ( h ) η ( g h ) − 1 , ξ ^ ( g , h ) = ξ ( g , h ) + ζ ( g ) + ζ ( h ) − ζ ( g h ) ( mod ⁡ 2 π ) ,

where the hatted quantities are defined by

U ^ ( f ) U ^ ( g ) = ω ^ ( f , g ) U ^ ( f g ) = e i ξ ^ ( f , g ) U ^ ( f g ) .

Utility of phase freedom

The following rather technical theorems and many more can be found, with accessible proofs, in Bargmann (1954).

The freedom of choice of phases can be used to simplify the phase factors. For some groups the phase can be eliminated altogether.

Theorem:If G is semisimple and simply connected, then ω(g, h) = 1 is possible.

In the case of the Lorentz group and its subgroup the rotation group SO(3), phases can, for projective representations, be chosen such that ω(g, h) = ± 1. For their respective universal covering groups, SL(2,C) and Spin(3), it is according to the theorem possible to have ω(g, h) = 1, 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 ω above are said to be cohomologous. They belong to the same second cohomology class, i.e. they are represented by the same element in H²(G), the second cohomology group of G. If an element of H²(G) contains the trivial function ω = 0, 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 g → T(g) is weakly continuous, two relevant theorems can be stated. An immediate consequence of (weak) continuity is that the identity component is represented by unitary operators.

Theorem: (Wigner 1939). The phase freedom can be used such that in a some neighborhood of the identity the map g → U(g) is strongly continuous.

Theorem (Bargmann). In a sufficiently small neighborhood of e, the choice ω(g₁, g₂) ≡ 1 is possible for semisimple Lie groups (such as SO(n), SO(3,1) and affine linear groups, (in particular the Poincaré group). More precisely, this is exactly the case when the second cohomology group H²(G, ℝ) of the Lie algebra g of G is trivial.