From Classical Mechanics to Quantum Field Theory

48 From Classical Mechanics to Quantum Field Theory. A Tutorial

AWeyl systemis a map:

W:S→U(H)

z→̂W(z), (1.230)

withŴ(z)Ŵ†(z)=̂W†(z)̂W(z)=̂I, such that:

(i)W is strongly continuous;

(ii) for anyz,z′∈S:

Ŵ(z+z′)=̂W(z)̂W(z′)exp{−ıω(z,z′)/ 2 }. (1.231)

In other words, aWeyl mapprovides a projective (i.e. up to a factor) unitary
representation of the vector spaceS, thought of as the group manifold of the
translation group, in the Hilbert spaceH.
From (1.231), it is easy to derive that:
̂W(0) =̂I, ̂W†(z)=̂W(−z) (1.232)

and

Ŵ(z)Ŵ(z′)=Ŵ(z′)Ŵ(z)exp{ıω(z,z′)/},∀z,z′. (1.233)

Suppose now thatSsplits into the direct sum of two Lagrangian subspacesS=
S 1 ⊕S 2 , so that any vectorzcan be written asz=(z 1 ,0) + (0,z 2 ),z 1 ∈S 1 ,
z 2 ∈S 2. The restrictions ofWto the Lagrangian subspaces:

U=W|S 1 :S 1 →H,

V=W|S 2 :S 2 →H (1.234)

yield faithful abelian representations of the corresponding Lagrangian subspaces:

Û(z 1 +z 1 ′)=Û(z 1 )Û(z 1 ′),z 1 ,z 1 ′∈S 1

V̂(z 2 +z′ 2 )=V̂(z 2 )V̂(z′ 2 ),z 2 ,z 2 ′∈S 2 , (1.235)

which satisfy:

Û(z 1 )V̂(z 2 )=V̂(z 2 )Û(z 1 )exp{ıω((z 1 ,0),(0,z 2 ))/}. (1.236)

Vice versa, it is simple to show that two faithful representationsUandV of two
transversal Lagrangian subspaces of a symplectic vector spaceSsatisfying (1.236),
yield a Weyl map by setting:

z→Ŵ(z)=Û(z 1 )V̂(z 2 )exp{−ıω((z 1 ,0),(0,z 2 ))/ 2 }. (1.237)

We also notice that

{

̂W(αz)

}

α∈R

is a strongly continuous one-parameter group

of unitaries since, from (1.231) we have

̂W(αz)Ŵ(βz)=̂W((α+β)z),α,β∈R. (1.238)