From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 59

(2) Setting:Ô=Q̂, we find at once:

Ω−^1

(

Q̂

)

(q,p)=q. (1.312)

(3) Consider nowÔ=|φ〉〈ψ|, which is the simplest example of a finite-rank

operator. Then it is immediate to see that:

Ω−^1 (|φ〉〈ψ|)(q,p)=

∫∞

−∞

dξeıpξ/φ(q+ξ/2)ψ∗(q−ξ/2). (1.313)

It is also easy to check formula (1.310):

∫

dp

2 π

e{−ıp(x−x

′)/}

Ω−^1

(

Ô

)(x+x′

2

,p

)

=φ(x)ψ∗(x′)=〈x|φ〉〈ψ|x′〉.

(4) We can now proceed to consider a self-adjoint operator with discrete spec-

trum: Ô|φn〉=λn|φn〉, with〈φn|φm〉=δnm,

∑

n|φn〉〈φn|=I.Thenwe

can write:

Ω−^1

(

Ô

)

(q,p)=

∑

n

λn

∫

dξeıpξ/φn(q+ξ/2)φ∗n(q+ξ/2). (1.314)

The most interesting consequence of what was seen above is the fact that
the Weyl and Wigner mapsestablish a bijection[ 19 ]between Hilbert-Schmidt
operators and square-integrable functions on phase space, which is also strongly
bicontinuous.
Indeed the following theorem holds^27 :
f will be square-integrable if and only ifΩ(f)is Hilbert-Schmidt. Similarly,

Ω−^1

(

Â

)

will be square-integrable if and only ifÂis Hilbert-Schmidt.
We notice that, since [Fs(η,ξ)]∗=Fs(−η,−ξ)and̂W†(ξ,η)=̂W(−ξ,−η),
the Weyl and Wigner maps preserve conjugation:

Ω(f∗)=Ω(f)†, Ω−^1

(

Ô†

)

=Ω−^1

(

Ô

)∗

, (1.315)

so guaranteeing thatfis real iff Ω (f) is at least a symmetric operator.
Before looking at some examples, we also observe that Eq. (1.310) implies

Trx

[

Ô

]

≡

∫

dx〈x|O|x〉=

∫

dqdp

2 π

Ω−^1

(

Ô

)

(q,p), (1.316)

(^27) See[ 14 ]for a proof.