Little Groups and Stabiliser Groups

Consider the group action of a group, G acting on a set X. This is defined by elements g ∈ G acting on elements x ∈ X, such that the image of g acting on x

is in X: g·x∈ X. Multiple actions of elements ofG acting on x are associative:

g1·(g2·x) = (g1g2)·x, and the identity of G acting on x yields x: eg ·x=x.

The orbit of x is defined as the set of elements in X that can be possibly ob- tained through elements ofGacting onx, and is denoted byG ·x. In mathematical notation, this is:

G ·x={g·x∈X :g ∈ G}.

If there are elements other than the identity of G, that yield the input xas an image of the group action, then the set of these elements of G forms a subgroup of G, called an isotropy group. This is denoted by Gx_{. In mathematical notation,}

Gx _{={g ∈ G :g·x=x, x∈X}.}

In the case of a semidirect product group, consider the group action of the homogeneous group,K, acting on the unitary dual3 Nb of the normal subgroup,N.

This unitary dual space consists of the set of equivalence classes [ξ] of continuous unitary irreducible representations ξ of elements a ∈ N (cf. ch. 16 §1 of [4]). These representations act on states |ψiin a Hilbert space Hξ in the following way:

ξ(a) :Hξ→Hξ :|ψi 7→ |ψ˜i=ξ(a)|ψi.

The homogeneous group has representations σ(k) which act in a similar manner on the Hilbert space Hσ. Elementsk ∈ K act on the representationsξ:

k :N →b N˜ :ξ 7→ξ˜=k·ξ.

This action of elements k ∈ K acting on representations ξ (and so by extension, on the equivalence classes of representations, [ξ]) is defined to be:

k·ξ =ξ(kak−1). (2.25)

3_{Though it uses the same notation, this should not be confused with central extensions of}

Two representations ξ and ˜ξ are equivalent if: ˜

ξ =U ξU−1, (2.26)

U being some unitary operator. Therefore, the equivalence class of ξ is defined to be:

[ξ] ={ξ˜=U ξU−1}; (2.27) it is the set of all representations ξ that can be reached under conjugation of operators U. Nb is partitioned by its equivalence classes [ξ], but the action of k

can take one equivalence class into another. Therefore each equivalence class has orbit: K ·[ξ] ={k·[ξ]∈Nb, k∈ K} ={[k·ξ]∈Nb, k∈ K}, since for ξ,ξ˜∈[ξ]: k·ξ˜(a) = ˜ξ(k−1ak) =U ξ(k−1ak)U−1 =U(k·ξ(a))U−1.

The orbit is written as [K ·ξ]. On each orbit, there is defined a little group, Kξ

such that:

k·[ξ] = [ξ], (2.28)

for any k∈ Kξ_{. This little group is a specific case of an isotropy group. Equation}

2.28 in turn defines a fixed-point condition for ξ: since [ξ] is defined by Equation 2.27, it follows that the fixed-point condition for the little group Kξ _is:

k·ξ= ˜ξ=ρkξρk−1, (2.29)

since Nb consists of the set of equivalence classes of ξ, and where ρ is a unitary

operator that depends on k in some way in the group action. The fixed-point condition means the little group consists of the elements of K that act on rep- resentations ξ by moving them to another representation within its equivalence class.

For k1, k2 ∈ Kξ, two successive group actions on ξ yields: (k1k2)·ξ=ρk1k2ξρ−1k1k2,

from Equation 2.29. But if the actions are taken one at a time, the two successive group actions yield:

(k1)·(k2·ξ) =ρk1ρk2ξρ−1k2 ρ

−1

Equating the right hand sides of these two equations gives us the equation:

ρk1k2ξρ−1k1k2 =ρk1ρk2ξρ−1k2ρ

−1

k1 .

Defining a new operator Ω in the following manner: Ωk1k2 :=ρ−1k2 ρ

−1

k1 ρk1k2

allows the following condition to be set:

Ωk1k2ξ=ξΩk1k2.

ξis irreducible, so Schur’s lemma demands Ω to be a scalar multiple of the identity operator:

Ωk1k2 =φ(k1, k2)×I. Therefore,

ρk1k2 =φ(k1, k2)ρk1ρk2. (2.30)

This is a homomorphism condition for a projective representation, and so ρ(k) is a projective representation of elements k in the little group Kξ _{(therefore the}

subscript notation will be dropped in favour of a bracket notation appropriate for representations).

The semidirect product of a little group with the normal subgroup defines a new semidirect product group, known as the stabiliser group, Gξ_:

Gξ_=Kξ

nN. (2.31)

Sometimes in the literature the little groupKξ_{is referred to as a “stabiliser group”}

or an “isotropy group”. It is referred to here as a little group so as to avoid confusion with the stabiliser group defined in Equation 2.31 and to indicate that it is a specific kind of isotropy group.

Now, ρ is a projective extension of the representationξ toGξ_{, such that:}

ρ(g) :Hξ →Hξ, ∀g ∈ Gξ_;

the projective extension representations act on the Hilbert space of the normal subgroup. Clearly, the representations ρ of elements of the normal subgroup are equivalent to the representations ξ of those elements:

ρ|N ∼=ξ. In other words, ρ(a)∼=ξ(a) for a∈ N.

Equation 2.25 can be modified to incorporate ρ:

k·ξ(a) = ρ(kak−1),

and so the fixed-point condition becomes:

This condition must be true at every point of the unitary dual spaceNb. Therefore,

the little groups of semidirect product groups are comprised of elements:

Kξ ={k ∈ K:k·ξ =ρ(k)ξρ−1(k)}. (2.32) The representations of elements of these little groups are just the ordinary repre- sentations of the homogeneous subgroup:

σ(k) :Hσξ →Hσξ, k ∈ Kξ_,

where the Hilbert space Hσξ is the Hilbert space of the homogeneous group Hσ

restricted to the little group. The representations τξ of the stabiliser groups are (from [55]):

τξ =σ⊗ρ:Hτξ →Hτξ, (2.33) where Hτξ =Hσξ ⊗Hξ.