3.1 Orbits of Lie Group Actions on P(H)
37
y(t) := U(exp(t 1 ξ 1 + t 2 ξ 2 + · · · + t n ξ n ))x ∈ O z
(3.1.4)
corresponds to the point t ∈ R
n
. We would like to interpret physically the coordinates as possible values of ‘quantities’ ξ j (where the choice of lengths of vectors ξ j
corresponds to a choice of units). If we, however, take such a point of view that only
the expectation values
F x (ξ ) := T r(P x X ξ ), ξ ∈ g,
(3.1.5)
of quantal observables X ξ in states x ∈ P(H) are measurable, then, for a general orbit
O z and a group G, not all values t ∈ R
n (neither all t in any open neighbourhood
of 0 ∈ R
n ) are physically distinguishable. From this point of view the most natural
coordinates of x ∈ PA G are just the values F x (ξ ) for a conveniently chosen subset
of ξ ∈ g. These values need not distinguish points of a neighbourhood of x ∈ O z :
d
dt
t=0
F exp(tη)·x (ξ ) = F x ([ξ, η]), ξ, η ∈ g,
(3.1.6)
(compare (2.3.21)), and the derivative might be zero for some nonvanishing η ∈ m
◦
x
and for all ξ ∈ g. If it is the case, then the derivative in (3.1.6) vanish on the whole
curve t → exp(tη) · x (t ∈ R). This is easily seen with a help of the next Lemma,
cf. Proposition 3.1.6:
3.1.4 Lemma. For all g ∈ G and ξ ∈ g, we have:
U (g)X ξ U (g
−1
) = X Ad(g)ξ ,
(3.1.7)
where the adjoint representation Ad of G is defined in 2.3.9.
Proof. According to the definition of Ad, the curve t → g exp(tξ)g
−1 at the identity
e of G determines the tangent vector Ad(g)ξ ∈ T e G, and this one, in turn, according
to the definition of the Lie algebra g, determines a unique curve t → exp(t Ad(g)ξ )
in G at e. Hence,
g exp(tξ) g
−1
= exp(t Ad(g)ξ ) ∀t ∈ R, g ∈ G, ξ ∈ g.
(3.1.8)
From the definition (2.3.20) of the generators X ξ of the representation U (G), we
then obtain
U (g) exp(−it X ξ )U (g
−1
) = U (g exp(tξ)g
−1
) = U (exp(t Ad(g)ξ )), (3.1.9)
and after differentiation at t = 0 we obtain (3.1.7).
3.1.5 Suppose now that F x ([ξ, η]) = 0 for all ξ ∈ g at some x ∈ O z . Substitution
of exp(tη) · x to the place of x gives according to the preceding lemma:
37
y(t) := U(exp(t 1 ξ 1 + t 2 ξ 2 + · · · + t n ξ n ))x ∈ O z
(3.1.4)
corresponds to the point t ∈ R
n
. We would like to interpret physically the coordinates as possible values of ‘quantities’ ξ j (where the choice of lengths of vectors ξ j
corresponds to a choice of units). If we, however, take such a point of view that only
the expectation values
F x (ξ ) := T r(P x X ξ ), ξ ∈ g,
(3.1.5)
of quantal observables X ξ in states x ∈ P(H) are measurable, then, for a general orbit
O z and a group G, not all values t ∈ R
n (neither all t in any open neighbourhood
of 0 ∈ R
n ) are physically distinguishable. From this point of view the most natural
coordinates of x ∈ PA G are just the values F x (ξ ) for a conveniently chosen subset
of ξ ∈ g. These values need not distinguish points of a neighbourhood of x ∈ O z :
d
dt
t=0
F exp(tη)·x (ξ ) = F x ([ξ, η]), ξ, η ∈ g,
(3.1.6)
(compare (2.3.21)), and the derivative might be zero for some nonvanishing η ∈ m
◦
x
and for all ξ ∈ g. If it is the case, then the derivative in (3.1.6) vanish on the whole
curve t → exp(tη) · x (t ∈ R). This is easily seen with a help of the next Lemma,
cf. Proposition 3.1.6:
3.1.4 Lemma. For all g ∈ G and ξ ∈ g, we have:
U (g)X ξ U (g
−1
) = X Ad(g)ξ ,
(3.1.7)
where the adjoint representation Ad of G is defined in 2.3.9.
Proof. According to the definition of Ad, the curve t → g exp(tξ)g
−1 at the identity
e of G determines the tangent vector Ad(g)ξ ∈ T e G, and this one, in turn, according
to the definition of the Lie algebra g, determines a unique curve t → exp(t Ad(g)ξ )
in G at e. Hence,
g exp(tξ) g
−1
= exp(t Ad(g)ξ ) ∀t ∈ R, g ∈ G, ξ ∈ g.
(3.1.8)
From the definition (2.3.20) of the generators X ξ of the representation U (G), we
then obtain
U (g) exp(−it X ξ )U (g
−1
) = U (g exp(tξ)g
−1
) = U (exp(t Ad(g)ξ )), (3.1.9)
and after differentiation at t = 0 we obtain (3.1.7).
3.1.5 Suppose now that F x ([ξ, η]) = 0 for all ξ ∈ g at some x ∈ O z . Substitution
of exp(tη) · x to the place of x gives according to the preceding lemma: