Projective unitary representation - Imported Knowlpedia Content

Definition

Let $G$ be a topological group and $H$ a complex Hilbert space. Write

PU(H)=U(H)/\{\lambda I:|\lambda|=1\},

with the quotient of the strong operator topology on $U(H)$ . A projective unitary representation is a continuous group homomorphism

\overline\pi:G\longrightarrow PU(H).

Thus each $g$ acts by a unitary operator determined only up to a scalar phase, while multiplication is exact after passing to projective classes. Equivalently, it is an action of $G$ on the rays of $H$ induced by unitaries; it need not admit a globally continuous choice of unitary representatives.

Multipliers

If representatives $U_g\in U(H)$ can be chosen, then

U_gU_h=\sigma(g,h)U_{gh}

for phases $\sigma(g,h)\in\mathbb T$ . Associativity gives the cocycle identity

\sigma(g,h)\sigma(gh,k)=\sigma(h,k)\sigma(g,hk).

Changing representatives by $U_g\mapsto b(g)U_g$ changes $\sigma$ by a coboundary. The resulting cohomology class records the obstruction to replacing the projective representation by a genuine one.

Lifts and central extensions

In a category where the chosen multiplier is continuous or Borel as required, the projective representation lifts to a genuine unitary representation of $G$ exactly when its multiplier class is trivial. Even when it does not lift on $G$ , it gives a genuine representation of the associated central extension by $\mathbb T$ . Global lifts and continuous representatives require hypotheses beyond the bare quotient-valued definition Varadarajan, Chapter VIII.

Example

The spin- $\tfrac12$ representation of $SU(2)$ does not descend to an ordinary representation of $SO(3)=SU(2)/\{\pm I\}$ , because the nontrivial central element acts as $-I$ . After quotienting operators by phases, it does descend to a projective unitary representation of $SO(3)$ .

References

V. S. Varadarajan, Geometry of Quantum Theory, 2nd ed., Springer, 1985. DOI record. Relevant: Chapter VIII, “Multipliers.”
V. Bargmann, “On Unitary Ray Representations of Continuous Groups,” Annals of Mathematics 59 (1954), 1–46. DOI record. Relevant: multipliers, ray representations, and lifting.