Geometric Quantization, Parallel Transport and the Fourier Transform

Abstract

In quantum mechanics, the momentum space and position space wave functions are related by the Fourier transform. We investigate how the Fourier transform arises in the context of geometric quantization. We consider a Hilbert space bundle H over the space J of compatible complex structures on a symplectic vector space. This bundle is equipped with a projectively flat connection. We show that parallel transport along a geodesic in the bundle H → J is a rescaled orthogonal projection or Bogoliubov transformation. We then construct the kernel for the integral parallel transport operator. Finally, by extending geodesics to the boundary (for which the metaplectic correction is essential), we obtain the Segal-Bargmann and Fourier transforms as parallel transport in suitable limits.A complex structure J ∈ End(V ) is compatible with the symplectic form ω if ω(J · , J · ) = ω( · , · ) and ω( · , J · ) > 0. Given such a J, the complexification of V decomposes asLet J be the set of all compatible complex structures on V . J can be identified, as follows, with the Siegel upper half-spaceWe associate a compatible complex structure J ∈ J to a point Ω ∈ H n so that V (1,0) J (1,0) J over J. Let K → J be the dual determinant bundle with fiber K J = n (V (1,0) J ) * . Since J is contractible, there is a unique (up to equivalence) square root bundle √ K → J such that √ K ⊗ √ K = K. This square root bundle is known as the bundle of half-forms. We define the corrected quantum Hilbert space bundleĤ → J asĤ = H ⊗ √ K. The fiberĤ J = H J ⊗ √ K J is called the corrected quantum Hilbert space. Including the bundle of half-forms is known as the metaplectic correction. Symplectic and metaplectic group actionsGiven a vector space V with a symplectic form ω, the symplectic group Sp(V, ω) is the set of linear transformations on V preserving ω. Upon choosing a set of linear symplectic coordinates {x i , y i } i=1,...,n , the group Sp(V, ω) is isomorphic toThe group Sp(V, ω) acts on the set J of compatible complex structures by g : J → gJg −1 . The corresponding action on positive complex Lagrangian subspaces is g : V (1,0) J → gV(1,0) J Corollary 3.12 Let J ∈ J and let L, L ′ ∈ S be a transverse pair. Then for anyψ ∈Ĥ J , in the above topology on E, we have lim J ′ →LÛ

Complex time evolution in geometric quantization and generalized coherent state transforms

“…In the second case, the need for half-forms is less evident. Nonetheless, it is commonly believed that the half-form correction is necessary also in the Kähler case and many arguments have been presented in its favor: the half-form correction renders the BKS pairing map unitary in the quantization of vector spaces with translation invariant complex structures [ADW91,KW06] and of Abelian varieties [BMN10] and allows for a transparent explanation of the vacuum energy shift in the Kähler quantization of symplectic toric varieties with toric Kähler structures [KMN10].…”

“…This bundle has a natural connection (generalizing the connections of Axelrod-Della Pietra-Witten [ADW91] and Hitchin [Hit90]) which is given by projecting the trivial connection in the trivial bundle whose fiber is the space of all square-integrable sections of the prequantum line bundle onto the almost-holomorphic subbundle. In the case that the symplectic manifold is a symplectic vector space and one restricts to translation invariant complex structures, this connection is known to be projectively flat [ADW91,KW06]. On the other hand, if one considers the full family of almost complex structures, Foth and Uribe have shown that the connection is never projectively flat, even semiclassically [FU07].…”

Geometric Phases in the Quantisation of Bosons and Fermions

“…The real Lagrangian subspaces are on the boundary (at infinity) of the space of complex structures. When the geodesic is extended to infinity, the parallel transport yields the Segal-Bargmann and Fourier transforms (see [8]). …”

“…The rest of the topological boundary consists of polarisations that are partly real and partly complex. For any J 0 ∈ J ω and L ∈ L ω , there is a geodesic {J t } in J ω from J 0 such that lim t→+∞ J t = L. We have [5] Geometric phases in quantisations 225 (see [8] for half-form quantisation. The result for half-density quantisation is then straightforward).…”

“…Wu [8] where α ω (J 1 , J 2 , J 3 ) is called the generalised Maslov index. The above formula also holds for half-density quantisation.…”