From Classical Mechanics to Quantum Field Theory

164 From Classical Mechanics to Quantum Field Theory. A Tutorial

unitary projective representation ofGin a complex Hilbert space is equivalent to a
strongly continuous unitary representation ofGif, for every bilinear antisymmetric
mapΘ:g×g→Rsuch that

Θ([u, v],w)+Θ([v,w],u)+Θ([w,u],v)=0, ∀u, v, w∈g

there is a linear mapα:g→Rsuch thatΘ(u, v)=α([u, v]), for allu, v∈g.

Remark 2.3.56.The condition is equivalent to require that the second cohomology
groupH^2 (G,R)is trivial.SU(2)for instance satisfies the requirement.

However, non-unitarizable unitary projective representations do exist and one has
to deal with them. There is nevertheless a way to circumvent the technical prob-
lem. Given a unitary projective representationGg→Ugwith multiplicatorsω,
let us put onU(1)×Gthe group structure arising by the product◦

(χ, g)◦(χ′,g′)=(χχ′ω(g,g′),g·g′)

and indicate byGˆωthe obtained group. The map

Gˆω(χ, g)→χUg=:V(χ,g)

is aunitary representationofGˆω. If the initial representation is normalized,Gˆω
is said to be acentral extension ofGby means ofU(1)[11; 5]. Indeed, the
elements (χ, e),χ∈U(1), commute with all the elements ofGˆωand thus they
belong to the center of the group.

Remark 2.3.57.These types of unitary representations of central extensions play
a remarkable role inphysics. SometimesGˆωwith a particular choice forωis seen
as thetruegroup of symmetries at quantum level, whenGis theclassical groupof
symmetries. There is a very important case. IfGis theGalilean group– the group
of transformations between inertial reference frames in classical physics, viewed as
activetransformations – as clarified by Bargmann[ 5 ]the only physically relevant
unitary projective representations in QM are just the ones which arenotequivalent
to unitary representations! The multiplicators embody the information about the
massof the system. This phenomenon gives also rise to a famous superselection
structure in the Hilbert space of quantum systems admitting the Galilean group as
a symmetry group, known asBargmann’s superselection rule[ 5 ].

To conclude, we just state a technically important result[ 5 ]which introduces the
one-parameter strongly continuous unitary groups as crucial tool in QM.

Theorem 2.3.58.Letγ:Rr→Urbe a continuous unitary projective repre-
sentation of the additive topological groupRon the complex Hilbert spaceH.The
following facts hold.