162 From Classical Mechanics to Quantum Field Theory. A Tutorial
TheactionofthissymmetryontheobservableXkturns out to be
su∗(Xk)=UuXkUu−^1 =Xk+uδk 1 I, u∈R.2.3.6.2 Groups of quantum symmetries
As in (1) in the example above, very often in physics one deals with groups of
symmetries. In other words, there is a certain groupG, with unit elementeand
group product·, and one associates each elementg∈Gto a symmetrysg(if
Kadison or Wigner is immaterial here, in view of the above discussion). In turn,
sgis associated to an operatorUg, unitary or antiunitary.
Remark 2.3.51.In the rest of this section, we assume that all theUgare unitary.
It would be nice to fix these operatorsUgin order that the mapGg→Ugbe a
unitary representationofGonH,thatis
Ue=I, UgUg′=Ug·g′ g,g′∈G (2.92)The identities (2.91) found in (1) in example 2.3.50 shows that it is possible at
least in certain cases. In general the requirement (2.92) does not hold. What we
know is thatUg·g′equalsUgUg′justup to phases:
UgUg′Ug−·g^1 ′=ω(g,g′)I withω(g,g′)∈U(1) for allg,g′∈G. (2.93)Forg=ethis identity gives in particular
Ue=ω(e, e)I. (2.94)The numbersω(g,g′) are calledmultipliers. They cannot be completely arbitrary,
indeed associativity of composition of operators (Ug 1 Ug 2 )Ug 3 =Ug 1 (Ug 2 Ug 3 ) yields
the identity
ω(g 1 ,g 2 )ω(g 1 ·g 2 ,g 3 )=ω(g 1 ,g 2 ·g 3 )ω(g 2 ,g 3 ),g 1 ,g 2 ,g 3 ∈G (2.95)which also implies
ω(g,e)=ω(e, g)=ω(g′,e),ω(g,g−^1 )=ω(g−^1 ,g),g,g′∈G. (2.96)Definition 2.3.52. IfGis a group, a mapGg→Ug –wheretheUg are
unitary operators in the complex Hilbert spaceH–isnamedaunitary projective
representationofGonHif (2.93) holds (so that also (2.94) and (2.95) are
valid). Moreover,
(i)two unitary projective representationGg→UgandGg→Ug′ are
said to beequivalentifUg′=χgUg,whereχg∈U(1)for everyg∈G.