Subatomic Physics

200 Additive Conservation Laws
The operator F is an observable; it represents a physical quantity. Its expectation
values must be real in order to correspond to measured values, and F consequently
must be Hermitian,
F † = F. (7.10)
Note the difference between F and U which is a transformation operator. The latter
is unitary and changes one wave function into another one, as in Eq. (7.7).
In general, transformation operators are not Hermitian and consequently do
not correspond to observables. However, there exist exceptions, and to discuss
these we note that nature contains two types of transformations, continuous and
noncontinuous ones. The continuous ones connect smoothly to the unit operator;
the noncontinuous ones do not. Among the latter category we find the operators
that are simultaneously unitary and Hermitian. Consider, for instance, the parity
operation (space inversion) which changes x into −x and represents a mirroring
at the origin. Such an operation is obviously not continuous; it is impossible to
mirror “just a little bit.” Mirroring is either done or not done. If space inversion is
performed twice, the original situation is regained; noncontinuous operators often
have this property:
U 2 h =1. (7.11)
As can be seen from Eqs. (7.8) and (7.10), Uh then is unitary and Hermitian and
it is an observable.
A well-known example of a continuous transformation is the ordinary rotation.
A rotation about a given axis can occur through any arbitrary angle, α, andα can
be made as small as desired. In general, a continuous transformation can always be
made so small that its operator approaches the unit operator. The operator U for
a continuous transformation can be written in the form
U = e iɛF
(7.12)
where ɛ is a real parameter and where F is called the generator of U. The action
of such an exponential operator on a wave function ψ is defined by
Uψ = e iɛF �
�
(iɛF )2
ψ ≡ 1+iɛF + + ··· ψ.
2!
As a rule exp(iɛF ) �= exp(−iɛF † )andU is not Hermitian. However, the unitarity
condition, Eq. (7.8), yields (if [F, F † ]=0)
or
exp(−iɛF † )exp(iɛF )=exp[iɛ(F − F † )] = 1
F † = F. (7.13)