From Classical Mechanics to Quantum Field Theory

50 From Classical Mechanics to Quantum Field Theory. A Tutorial

Thus, one has:

̂W(q,p)=Û(q)V̂(p)exp{−ıqp/} (1.249)

or explicitly:
(
̂W(q,p)ψ

)

(x)=exp{ıp[x+q/2]/}ψ(x+q). (1.250)

Also, the infinitesimal generators are given, in the appropriate domains, by:
(
Qψ̂

)

(x)=xψ(x)

(

Pψ̂

)

(x)=−ıdψ

dx

. (1.251)

Finally, a generic matrix element of̂W(q,p) is given by:

〈

φ,̂W(q,p)ψ

〉

=exp{ıqp/ 2 }

+∫∞

−∞

dxφ(x)∗exp{ıpx/}ψ(x+q), (1.252)

where

〈

φ,̂W(q,p)ψ

〉

is square-integrable for allφ, ψ∈L^2 (R), as it can be seen
from the fact that:
∥∥
∥

〈

φ,̂W(q,p)ψ

〉∥∥

∥

2

=

∫ dqdp

2 π

∣∣

∣

〈

φ,Ŵ(q,p)ψ

〉∣∣

∣

2

=‖φ‖^2 ‖ψ‖^2. (1.253)

Explicitly, if we use a generalized basis of plane-waves:{|k〉=eıkx/

√

2 π},weget
the formula:
〈
k′|̂W(q,p)|k

〉

=δ(k−k′+p/)eıq(k+k

′)/ 2

, (1.254)

which will be useful in the following.
The construction outlined in the previous example can be extended to build
a concrete realization of a Weyl system inthe general case ofa symplectic vector
space (S,ω) which decomposes as the direct sumS=S 1 ⊕S 2 of the two Lagrangian
subspacesS 1 ,S 2 .IfU:S 1 →H,V:S 2 →Hare unitary, irreducible and strongly
continuous representations ofS 1 andS 2 respectively on a separable Hilbert space
Hwhich satisfy the additional constraint:

Û(z 1 )V̂(z 2 )=V̂(z 2 )Û(z 1 )exp{ıω((z 1 ,0),(0,z 2 ))/}, (1.255)

we can define the Weyl system as:

̂W(z)=Û(z 1 )V̂(z 2 )exp{−ıω((z 1 ,0),(0,z 2 ))/ 2 }. (1.256)