Mathematical Foundations of Quantum Mechanics 163That is the same as requiring that there are numbersχh∈U(1),ifh∈G,
such that
ω′(g,g′)=χg·g′
χgχg′ω(g,g′) ∀g,g′∈G (2.97)with obvious notation;
(ii)a unitary projective representation withω(e, e)=ω(g,e)=ω(e, g)=1for
everyg∈Gis said to benormalized.Remark 2.3.53.
(a)It is easily proven that every unitary projective representation is always
equivalent to a normalized representation.
(b)It is clear that two projective unitary representations are equivalent if and
only if they are made of the same Wigner (or Kadison) symmetries.
(c)In case of superselection rules, continuous symmetries representing a con-
nected topological group do not swap different coherent sectors when acting on pure
states[ 5 ].
(d)One may wonder if it is possible to construct a group representationG
g→Vgwhere the operatorsVgmay be both unitary or antiunitary. If everyg∈G
canbewrittenasg=h·hfor somehdepending ong– and this is the case ifGis
a connected Lie group – all the operatorsUgmust be unitary becauseUg=UhUhis
necessarily linear no matter ifUhis linear or anti linear. The presence of arbitrary
phases does not change the result.
Given a unitary projective representation, a technical problem is to check if it is
equivalent to a unitary representation, because unitary representations are much
simpler to handle. This is a difficult problem[11; 5]which is tackled especially
whenGis atopological group(orLie group) and the representation satisfies the
following naturalcontinuity property
Definition 2.3.54. A unitary projective representation of the topological group
G,Gg→Ugon the Hilbert spaceHis said to becontinuousif the map
Gg→|〈ψ,Ugφ〉|is continuous for everyψ,φ∈H.
The notion of continuity defined above is natural as it regards continuity of proba-
bility transitions. A well-known co-homological condition assuring that a unitary
projective representation of Lie groups is equivalent to a unitary one is due to
Bargmann[24; 5].
Theorem 2.3.55(Bargmann’s criterion). LetGbe a connected and simply con-
nected (real finite dimensional) Lie group with Lie algebrag. Every continuous