5 DISEÑO COMPLETO DEL CONVERTIDOR
5.4.1 D ISEÑO DE LA INDUCTANCIA
In the classical case,I ≡0 and the little group consists of the entire homogeneous subgroup. As the stabiliser group is the semidirect product of the little group with the normal subgroup, in this case the stabiliser group is the entire Weyl-Heisenberg group. Therefore, the Mackey induction method does not need to be used, and the unitary irreducible representations τ of the group are, from Equation 2.38, the tensor product of the unitary irreducible representations σ = eiXiαi of the
homogeneous group with the unitary irreducible representations ξ =eiPiai of the
normal subgroup:
τ =σ⊗ξ
=eiXiαi+iPiai. (2.50)
These representations act on the Hilbert space:
Hτ =Hσξ ⊗Hξ
=C⊗C
=C.
Note that in the tensor product, the dimension of the product is equal to the product of the dimensions.
Also note that the representations of Equation 2.50 are the representations of a translation group of order 2n. That isT(2n), where configuration space consists of an N-tuple fN, N = 1,2, ...,2n, fi = αi and fi+n = ai. In other words, the unitary irreducible representationτ is a mapping fromR2nto the circle group [30].
This is an abelian group, and so the Lie algebra is trivial.
2.6.2.2 Quantum Case
In the quantum case, I 6= 0 and the little group consists of solely the identity element of the homogeneous group. The stabiliser group therefore is the semidirect product of this element with the normal subgroup:
Gξ={0}
nT(n+ 1)∼=T(n+ 1).
The unitary irreducible representations are defined by the construction of Equation 2.42. Define X to be the quotient space of the Weyl-Heisenberg group and the stabiliser group: X=H(n)/T(n+ 1)∼=T(n). Then Θ is the coset representative function that takes elements of x∈X to H(n):
Choose Θ(x) = (αi,0,0). Then x = (αi,0,0)N, where N = T(n + 1). For g−1 = (−βi,−ai,−c), g−1x= (−βi,−ai,−c)(αi,0,0)N = (αi−βi,−ai,−c−1 2a·α)N = (αi−βi,0,0)(0,−ai,−c−a·α−1 2a·β)N = (αi−βi,0,0)N ⇒Θ(g−1x) = (αi−βi,0,0), and Θ−1(x) = Θ(x−1) = (−αi,0,0). Therefore, Θ−1(x)gΘ(g−1x) = (−αi,0,0)(βi, ai, c)(αi−βi,0,0) = (βi −αi, ai, c+ 12α·a)(αi−βi,0,0) = (0, ai, c+α·a− 1 2a·β)
From Equation 2.42, the representations of the quantum case of the Heisenberg group are given by:
(τ(g)ψ)(x) = τξ(Θ(x)−1gΘ(g−1x))ψ(g−1x) =τξ(0, ai, c+α·a− 1
2a·β)ψ((α
i−βi,0,0)N)
The representations τξ are the representations of the stabiliser group, T(n+ 1).
From Equation 2.38, these representations are the tensor product of the represen- tations σ of the homogeneous group and the representations of the little group,
ξ. In this case, this reduces to simply the representations of the little group. As elements x of the quotient space X are dependent only upon the parameter
αi ∈ T1(n),x≡(αi,0,0) will be replaced by simply αi. This means thatg−1x will be replaced by αi−βi. Then the representations are:
(τ(g)ψ)(αi) : = ˜ψ(αi)
=τξ(0, ai, c+α·a−1
2a·β)ψ(α
i−βi)
=eiAiai+iC(c+α·a− 12a·β)ψ(αi−βi),
where Ai and C are the character labels. Now,
ψ(αi−βi) =e−βi∂αi∂ ψ(αi),
by a simple Taylor expansion of ψ(αi − βi) centred at αi (this can be seen by evaluating the left-hand-side and right-hand-side of the equation to a few terms - each side of the equation matches the other term by term) and so
˜
A simple calculation5 shows that the Baker-Campbell-Hausdorff formula for the Heisenberg algebra, with non-commuting generatorsX andY, is given exactly by:
eXeY =eX+Y+12[X,Y]. (2.52)
In fact, this is true for any set of operators {X, Y}which all commute with their commutator: [X,[X, Y]] = [Y,[X, Y]] = 0, which means Equation 2.52 can be applied to Equation 2.51, which is made up of linear combinations of operators which all commute with their commutators. SettingX =i[Aiai+C(c+α·a−12a·β)]
and Y =iβii ∂ ∂αi: [X, Y] = =− Aiai+C(c+α·a−12a·β), βii∂α∂i =−[Aiai+C(c+α·a− 12a·β)]βii∂α∂i +β ii ∂ ∂αi[Aia i+C(c+α·a− 1 2a·β)] =iβiCai. And so 2.51 becomes: ˜ ψ(αi) =ei[Aiai+C(c+α·a−12a·β)+βii ∂ ∂αi+ 1 2β iCa i]ψ(αi) =ei[ai(Ai+Cαi)+Cc+iβi ∂ ∂αi]ψ(αi) (2.53) =ei(aiPi+cI+βiXi)ψ(αi), (2.54) where: hα|Pi|ψi: =Ai+Cαiψ(α) hα|I|ψi: =Cψ(α) hα|Xi|ψi: =i ∂ ∂αiψ(α).
Without loss of generality,Ai can be taken to 0 and C to the identity. Then:
hα|Pi|ψi: =αiψ(α) (2.55)
hα|I|ψi: =ψ(α) (2.56)
hα|Xi|ψi: =i
∂
∂αiψ(α). (2.57)
These operators satisfy the requisite commutation relations of Equation 2.48. Note that T1(n) was initially chosen to be the homogeneous subgroup; this is the “α-representation”. Alternatively, T2(n) could have been chosen as the homogeneous subgroup, with the normal subgroup therefore being T1(n)⊗ T(1). With this choice, the unitary irreducible representations of the Heisenberg group in the “a-representation” would be:
(τ(g)ψ)(ai) =eiXiαi+iI(c−a·α+12α·b)ψ(ai−bi)
=eαiai+c−ibi ∂∂aiψ(a)
=ei(biPi+cI+αiXi)ψ(ai), (2.58)
where: ha|Pi|ψi: =−i ∂ ∂aiψ(a) ha|I|ψi: =ψ(a) ha|Xi|ψi: =aψ(a).
The Heisenberg group is a central extension of T(n) which is the subgroup described by the classical case of Section 2.6.2.1. The central charge is given by the partial derivative of the general group element with respect to the parameter
c. This is described in more detail in Chapter 7. The Hilbert space in the quantum case is given by:
Hτ =L2(Rn+1,C, µ). (2.59)