A Short Course on Quantum Mechanics and Methods of Quantization 27The commutator between any two elements of the algebra is easily found to be
given by:
[x, y]=B(x, y)e 3 withB(x, y)=x 1 y 2 −x 2 y 1. (1.123)Notice thatB(x, y) is the standard symplectic form on the plane of coordinates
(x 1 ,x 2 ), i.e. (P,Q).
The Heisenberg-WeylW 1 group is, as usual, obtained by exponentiation. Ex-
plicitly, using the complex representation (1.122), one has:
g=ex=eısID(α) with D(α)≡eαa†−α∗a
(1.124)and
D(α)D(β)=eıIm (αβ∗)
D(α+β)=e^2 ıIm (αβ∗)
D(β)D(α) (1.125)a formula that can be proved via the Baker-Campbell-Hausdorff formula^16.
The latter expression shows that the operatorseısID(α) give a representation of
the groupW 1 ,fixedby(s, α), withsreal andαcomplex. The groupW 1 is nilpotent
and its center is given by the subgroup generated by the elements with (s,0), whose
most general representation is fixed by a real numberλvia: Tλ((s,0)) =eısλI.
Anyλ= 0 gives a non-equivalent infinite-dimensional representation ofW 1 ,and
any Unitary Irreducible Representation (UIR) is of this kind[ 33 ]^17.
1.3.1 Coherent states and Bargmann-Fock representation
In this section, we will show explicit realizations of the representations of the Weyl
group, by means of the so called coherent states, introduced first by Glauber and
Sudarshan; the interested reader may find more on this subject and a thorough
overview of applications in[ 24 ].
1.3.1.1 Definition and basic properties
Let us consider a Hilbert H, on which we have defined a couple of bosonic
creation/annihilation operators: [a, a†]=I. As in (1.29), we denote with|n〉
(n=0, 1 , 2 , ...) the orthonormal states of the Fock basis, which are eigenstates of
the number operatorN=a†a, with eigenvaluesn:N|n〉=n|n〉. In the following,
(^16) This formula reads:eA+B=eAeBe−[A,B]/ (^2) =eBeAe[A,B]/ (^2) wheneverA, Bare such that
[ 17 A, B]=cI(c∈C).
In the particular caseλ= 0 the unitary representation is one-dimensional and fixed by a pair
of real numbersμ, νvia:T(g)=eμx^1 +νx^2.