From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 53

and is such that:

ŴT(z+z′)=̂WT(z)̂WT(z′)exp{−ıωT(z,z′)/ 2 }. (1.271)

Also, as before, one can define the infinitesimal generators:

̂WT(λz)=exp{iλĜ(z)}, (1.272)

which satisfy the commutation relations:
[
Ĝ(z),Ĝ(z′)

]

=−ıωT(z,z′). (1.273)

This observation allows us to consider Weyl systems for a vector space with an
arbitrary and translationally invariant symplectic structureω. Indeed, by Darboux
theorem, there always exists an invertible linear transformationTthat mapsω 0
inω,T :(S,ω)→(S,ω 0 ). Denoting then withW :(S,ω 0 )→U(H)theWeyl
map with respect to the standard symplectic form, we can define a Weyl system
for (S,ω) by setting:W◦T=WTor, more explicitly:

̂WT(z)≡̂W(Tz). (1.274)

In physics, a conspicuous example of a one-parameter group of symplectic trans-
formations is provided by the time evolution of a Hamiltonian system, of which
we now give some simple examples^24 , leaving the details of the calculations to the
reader.

Example 1.3.7. Free particle evolution.
For a free particle of massm, the one-parameter group is given by: (q,p)→
(q+tp/m, p) and can be represented by the matrix:
∣∣
∣∣q(t)
p(t)

∣∣

∣∣=F(t)

∣∣

∣∣q

p

∣∣

∣∣,F(t)=

∣∣

∣∣^1 t/m

01

∣∣

∣∣,F(t)F(t′)=F(t+t′). (1.275)

Then, one finds:

Ŵt(q,p)=Ŵ(q(t),p(t)) = exp

{

(ı/)

[

q(t)P̂+p(t)Q̂

]}

≡exp

{

(ı/)

[

qP̂t+pQ̂t

]}

, (1.276)

with

P̂t=P,̂ Q̂t=Q̂+tP/m.̂ (1.277)

(^24) Other examples may be found in[ 14 ].