From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 51

Thanks to the Von Neumann’s theorem[ 34 ], we can affirm then that there exists
a unitary mapT:H→L^2 (Rn,dμ) such that:
(
TÛ(q)T−^1 ψ

)

(x)=ψ(x+q)

(

TV̂(p)T−^1 ψ

)

(x)=eıx·pψ(x)

, (1.257)

where, as before, we have setz 1 =(q,0),z 2 =(0,p) and denotedÛ(z 1 ),V̂(z 2 )as
Û(q),V̂(p) respectively. This also shows that all the representations of the Weyl
commutation relations are unitarily equivalent to the Schr ̈odinger representation
and hence are unitarily equivalent among themselves.

1.3.3.2 Linear transformations

We start by observing that a linear transformationT:S→Sthat preserves the
symplectic structure,ω(Tz,Tz′)=ω(z,z′),∀z,z′∈S, induces a map:

̂WT:S→H

z→̂WT(z)≡Ŵ(Tz)

, (1.258)

such that:

ŴT(z+z′)=̂WT(z)̂WT(z′)exp{−ıω(z,z′)/ 2 }, (1.259)

as it can be easily proved. HenceŴT is also a Weyl system, which, by von
Neumann’s theorem, it is unitarily equivalent toŴ. Thus, to the mapTthere is
associated an automorphismÛT∈U(H) such that:

ŴT(z)=ÛT†

(

̂W(z)

)

ÛT. (1.260)

Example 1.3.6. Fourier transform.
The action of the (unitary) Fourier transform

F:L^2 (R)→L^2 (R)

ψ(x)→ψ ̃(p)=

∫∞

−∞

√dx

2 π

ψ(x)e−ıp·x

(1.261)

translates on the Weyl operators as the transformation:
( ̃
eıxP̂ψ

)

(p)=eıxpψ ̃(p) ⇒

(

P̂ψ ̃

)

(p)=pψ ̃(p)

( ̃

eıp^0 Q̂ψ

)

(p)=ψ ̃(p−p 0 )⇒

(

Q̂ψ ̃

)

(p)=ıdψ ̃(p)/dp

, (1.262)