2.2 Symplectic Structure
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2.2.5 According to a theorem by Wigner, any bijective transformation F of P(H)
which conserves the ‘transition probabilities’, i.e.:
T r(P x P y ) = T r(F(P x )F(P y )), ∀x, y ∈ H, x = 0 = y,
(2.2.6)
can be extended to a transformation F of H onto itself, which is either unitary or
antiunitary, compare [53, 3.2.1 and 3.2.14]:
T r(P F x P Fy ) = T r(P x P y ).
(2.2.7)
Such transformations conserve also distances and the metric Q, see 2.1.2 and 2.1.9.
Bijections of P(H) onto itself conserving the metric Q will be called the Wigner
maps.
On the other hand, antiunitary transformations F of H do not conserve the symplectic form : F ∗ = −. Transformations F of P(H) conserving are called
symplectic transformations.
2.2.6 Lemma. Let F be any symplectic transformation of P(H) the restriction of
which to T x P(H) for any x ∈ P(H) (i.e. the mappings F ∗ : T x P(H) → T Fx P(H))
are complex linear with respect to the complex structure J, cf. 2.2.1. Then F can be
extended to a unitary transformation F ∈ L(H ).
Proof. Symplecticity and complex linearity of F give
Q x (v, w) = − x (v, J w) = − Fx (F ∗ v, J F ∗ w) = Q Fx (F ∗ v, F ∗ w),
(2.2.8)
i.e. Q = F ∗ Q, what implies the invariance of distances:
d(Fx, Fy) = d(x, y),
which in turn implies the invariance of T r(P x P y ). Hence F can be extended either to
a unitary or to an antiunitary transformation. Since antiunitary transformations have
nonsymplectic projections in P(H), extension F of F must be unitary.
2.2.7 Proposition. Any symplectic isometry F : P(H) → P(H) is an analytic diffeomorphism of P(H).
Proof. F is a symplectic Wigner map, hence extendable to a unitary F ∈ L(H ).
With the help of the charts x , analyticity follows for the projection U of any unitary
U ∈ L(H ). The same considerations apply to the inverse map F
−1 , and the assertion
follows.