Data Loading...

Geometric phases in the quantisation of bosons and fermions Flipbook PDF

Geometric phases in the quantisation of bosons and fermions Siye Wu HKU-IMR preprint IMR2010:#16 October 2010 (revised:

100 Views
51 Downloads
FLIP PDF 358.4KB

HKU-IMR preprint IMR2010:#16 October 2010 (revised: November 2010)

Geometric phases in the quantisation of bosons and fermions Siye Wu∗

Abstract After reviewing geometric quantisation of linear bosonic and fermionic systems, we study the holonomy of the projectively flat connection on the bundle of Hilbert spaces over the space of compatible complex structures and relate it to the Maslov index and its various generalisations. We also consider bosonic and fermionic harmonic oscillators parametrised by compatible complex structures and compare Berry’s phase with the above holonomy. Keywords: geometric quantisation, Hermitian symmetric spaces, Maslov index, Berry’s phase MSC(2010): Primary 53D50, Secondary 32M15, 53D12, 81Q70

Dedicated to Alan Carey, on the occasion of his 60 th birthday

1

Introduction

In geometric quantisation, quantum Hilbert space is constructed from the classical phase space (a symplectic manifold) together with a choice of polarisation. An important question is whether the Hilbert spaces from different polarisations can be naturally identified. For linear bosonic systems, there is a projectively flat connection on the bundle of Hilbert spaces over the space of compatible linear complex structures [1]. This identifies vectors in various Hilbert spaces up to a phase. In [8], parallel transport along geodesics in the space of polarisations was calculated and shown to agree with the Bogoliubov transformation [17, 18] and other definitions of intertwining operators [14, 11]. The real Lagrangian subspaces are on the boundary (at the infinity) of the space of complex structures. When the geodesic is extended to infinity, the parallel transport yields the Segal-Bargmann and Fourier transforms [8]. For linear fermionic systems, pre-quantisation and quantisation was considered [9, 17] and the space of compatible complex structures is a compact Hermitian symmetric space. The bundle of Hilbert spaces again admits a projectively flat connection whose curvature is proportional to the standard K¨ahler form on the base [22]. The parallel transport along the geodesics in the space of polarisations yields intertwining operators between various constructions of the spinor representation [22]. In this paper, we study the holonomy of these projectively flat connections and explore its geometric significance. In the bosonic case, the holonomy along a geodesic triangle is related to the generalised Maslov index in [11]. When the vertices of the triangle approach three mutually transverse Lagrangian subspaces at infinity, the holonomy becomes the composition of three Fourier transformations, which is known to be related to the triple Maslov index of Kashiwara [10]. Thus we get interesting formulae for the Maslov index and its generalisation in terms of integrations of curvature on a surface bounded by three geodesics. In the fermionic case, the holonomy along a geodesic triangle is related to the orthogonal counterpart of the Maslov index [12] and we obtain similar results using the holonomy. We also consider bosonic and fermionic harmonic oscillators whose Hamiltonians are parametrised by compatible complex structures. As the parameter changes adiabatically, the energy eigenstates acquire a ∗ Department

of Mathematics, University of Hong Kong, Pokfulam, Hong Kong, China Email address: [email protected]

1

geometric phase called Berry’s phase [2], which we study using the its relation [20] with the universal connection [13]. We find that Berry’s phase on the vacuum vector is inverse to the holonomy of the projectively flat connection discussed above. However, the connection responsible for (non-Abelian) Berry’s phase when the energy eigenvalue is degenerate is not projectively flat. The rest of the paper is organised as follows. In §2, we review the work on geometric quantisation of linear bosonic and fermionic systems. In §3, we study the holonomy of the projectively flat connection on the bundle of Hilbert spaces over the space of compatible complex structures and relate it to the triple Maslov index and its various generalisations. In §4, we consider bosonic and fermionic harmonic oscillators parametrised by compatible complex structures and compare Berry’s phase with the holonomy in §3.

2 2.1

Quantisation of bosonic and fermionic systems Quantisation of linear bosonic systems

Let (V, ω) be a symplectic vector space of dimension√2n. The pre-quantum line bundle ` → V is a complex line bundle with a connection whose curvature is ω/ −1 . The pre-quantum Hilbert space H0 is the space of L2 -integrable sections (with respect to the Liouville measure) of `. It can be identified with L2 (V, C) upon choosing a trivialisation of `. A complex structure J on V is compatible with ω if ω(J·, J·) = ω(·, ·) and ω(·, J·) > 0. Each such J determines a complex polarisation: a complex Lagrangian subspace VJ1,0 of V C . The corresponding quantum Hilbert space is HJ = {ψ ∈ H0 | ∇x ψ = 0, ∀x ∈ VJ0,1 }. The Heisenberg algebra (generated by V subject to the canonical commutation relation) acts on HJ by an irreducible representation. The space Jω of compatible complex structures is a non-compact Hermitian symmetric space isomorphic to Sp(2n, R)/U (n). Fixing J0 ∈ Jω and choosing a unitary basis of V , the space Jω can be identified with a ¯ > 0. bounded symmetric domain parametrised by n × n complex symmetric matrices Z such that I − ZZ 1,0 (The subspace VJ is the graph of Z under this basis.) The natural K¨ahler form on Jω is √ √ ¯ −1 d Z ∧ (In − ZZ) ¯ −1 d Z). ¯ trV 1,0 (d J ∧ d J) = −1 tr((1 − Z Z) σω = − −1 4 J

Since HJ is a subspace of H0 for each J ∈ Jω , there is a bundle of Hilbert spaces H → Jω whose fibre over J is HJ . The trivial connection on the product bundle Jω × H0 projects orthogonally to a natural, projectively flat connection on H whose curvature is [1] √ (2.1) FH = (σω /2 −1 ) idH = − 18 trV 1,0 (d J ∧ d J) idH , J

where idH is the section of End(H) which is the identity operator on HJ at J ∈ Jω . Parallel transport in the bundle H identifies, up to a phase, states in the quantum Hilbert spaces HJ constructed from various polarisations J. Since Jω is contractible and non-positively curved, there is a unique geodesic γJ1 J2 from J1 to J2 for any J1 , J2 ∈ Jω . The parallel transport UJ2 J1 : HJ1 → HJ2 along γJ1 J2 was calculated in [8]. For example, the parallel transport of the coherent state √ 1 cα α, x1,0 J1 (x) = exp[ −1 ω(¯ J1 ) − 4 ω(x, J1 x)], 0,1 C where α ∈ VJ1,0 is a parameter and x = x1,0 = VJ1,0 ⊕ VJ0,1 , is [8, 22] J1 + xJ1 ∈ V according to V 1 1 1 J1 +J2 (UJH2 J1 cα J1 )(x) = det 2

−1/4

1

e− 4 ω(x,J2 x) exp

1 2

ω x1,0 ¯, J2 − α

J1 +J2 −1 1,0 (xJ2 2

−α ¯)

.

The operator UJ2 J1 is, up to a rescaling by positive constant, the orthogonal projection from HJ1 to HJ2 in H0 [8]. Therefore UJ2 J1 coincides with the Bogoliubov transformation defined in [17, 18]. It also agrees with the operator studied in [14, 11] that intertwines the two equivalent irreducible representations HJ1 and HJ2 of the Heisenberg algebra.

2

We now include metaplectic correction. Let V → Jω be a vector bundle whose fibre over J ∈ Jω is VJ1,0 . This is a sub-bundle of the product bundle Jω × V C , and the trivial connection on the latter projects to one on V. Its curvature is 1 V 1 ¯ −1 d Z¯ ∧ (1 − Z Z) ¯ −1 d Z(1 − ZZ) ¯ −1 (1, −Z). ¯ F = − 4 (d J ∧ d J) V = − (1 − ZZ) (2.2) Z Vn 1,0 ∗ Consider the line bundle K = det V∗ ; the fibre over J ∈ Jω is KJ = (V√ J ) . The induced connection on K −1 σω . Since Jω is contractible, K is compatible with the Hermitian structure and its curvature is F = √ √ √ there is a unique line bundle K → Jω such that ( K)⊗2 = K. A half-form on VJ1,0 is an element of KJ . √ ˆ J = HJ ⊗ KJ . As J varies, they The Hilbert space of half-form quantisation (with the polarisation J) is H √ ˆ ˆ = H ⊗ K over Jω , i.e., the curvature FH ˆ J can be form a flat bundle H = 0 [18, 8]. Thus all the fibres H canonically identified. For any J , J ∈ J , there is a natural non-degenerate sesquilinear pairing between 1 2 ω p p ˆ J and H ˆ J . The parallel transport UˆJ J : H ˆJ → H ˆ J is in fact the KJ1 and KJ2 and hence between H 1 2 2 1 1 2 operator determined by the pairing between them [8]. Next, we consider half-density quantisation. We associate to the vector space VJ1,0 a real line |KJ | on which a linear transformation A ∈ EndC (VJ1,0 ) acts as multiplication by | det A|−1 . An element of |KJ | is p p 1,0 1,0 called a density on VJ . A half-density on VJ is an element of |KJ |, a linepsuch that ( |KJ |)⊗2 = |KJ | and on which thep linear transformation A acts by | det A|−1/2 . The lines |KJ | ( |KJ |, respectively) form real line bundles |K| ( p|K|, respectively) over Jω , which are naturally flat. √ In a good open covering, the transition Jω , the functions of |K| ( |K|, respectively) are the norm of those p of K ( K, respectively). For any J ∈p ˜ J = HJ ⊗ |KJ |. They form a bundle H ˜ = H ⊗ |K| of Hilbert space of half-density quantisation is H Hilbert spaces over Jω . It has a natural projectively flat connection with curvature √ ˜ FH = FH = σω /2 −1 . p p For any J1 , J2 ∈ Jω , there is a natural non-degenerate pairing between |KJ1 | and |KJ2 | and hence ˜ J and H ˜ J . The parallel transport U˜J J : H ˜J → H ˜ J is also the operator determined by the between H 1 2 2 1 1 2 pairing between them. A real Lagrangian subspace L ⊂ V is a real polarisation selecting the sections of ` that are covariantly constant along L; such a section can be identified with p a complex-valued function on V /L. Let KL = √ Vn (V /L)∗ . We have, respectively, the spaces KL and |KL | of half-forms and half-densities on V /L, and ˆ L and H ˜ L of half-form and half-density quantisation. The spaces H ˆ L and H ˜ L have the Hilbert spaces H natural inner products and are irreducible representations of the Heisenberg algebra. If J ∈ Jω , there is a ˜L → H ˜ J (or BˆJL : H ˆL → H ˆ J ) that intertwines the two equivalent Segal-Bargmann transformation B˜JL : H irreducible representations. Let L1 , L2 ⊂ V be two transverse Lagrangian subspaces. Then the intertwining ˜L → H ˜ L (or FˆL L : H ˆL → H ˆ L ) is a Fourier transformation [10]. operator F˜L2 L1 : H 1 2 2 1 1 2 The real Lagrangian subspaces in V form the Shilov boundary Lω of Jω (as a bounded domain). The rest of the topological boundary consists of polarisations that are partly real and partly complex. For any J0 ∈ Jω and L ∈ Lω , there is a geodesic {Jt } in Jω from J0 such that limt→+∞ Jt = L. We have lim U˜Jt J0 = (B˜J0 L )−1 ,

lim UˆJt J0 = (BˆJ0 L )−1 .

t→+∞

t→+∞

(See [8] for half-form quantisation; the result for half-density quantisation is then straightforward.) Two Lagrangian subspaces L+ , L− ∈ Lω are transverse if and only if there is a geodesic {Jt } in Jω such that limt→±∞ Jt = L± [8]. In this case, we have lim U˜Jt J−t = F˜L+ L− ,

lim UˆJt J−t = FˆL+ L− .

t→+∞

t→+∞

The above limits are in the sense of tempered distributions on V [8].

2.2

Quantisation of linear fermionic systems

Let (V, g) be an oriented Euclidean vector space of dimension 2n. Despite the of an honest preVn absence quantum line bundle, the pre-quantum Hilbert space can be taken as H0 = (V C )∗ , on which covariant

3

derivative operators act [9, 17, 22]. A compatible complex structure J on (V, g) is one that is compatible with the orientation of V and such that g(J·, J·) = g(·, ·). Each such J determines a polarisation: a maximally isotropic complex subspace VJ1,0 of V . The corresponding quantum Hilbert space is HJ = {ψ ∈ H0 | ∇x ψ = 0, ∀x ∈ VJ0,1 }. The Clifford algebra (generated by V subject to the canonical anti-commutation relation) acts on HJ by an irreducible representation. In fact, up to a fermionic Gaussian factor, HJ agrees with the standard construction of the spinor representation [17, 22]. The space Jg of compatible complex structures on (V, g) is a compact Hermitian symmetric space isomorphic to SO(2n)/U (n). Fixing J0 ∈ Jg and choosing a unitary basis of V , the complement of the cut locus of J0 in Jg , which is an open dense subset, can be parametrised by n × n complex skew-symmetric matrices Z. (Again, the subspace VJ1,0 corresponds to the graph of Z.) The natural K¨ahler form on Jg is √

σg =

−1 4

√ ¯ −1 d Z ∧ (In − ZZ) ¯ −1 d Z). ¯ trV 1,0 (d J ∧ d J) = − −1 tr((1 − Z Z) J

Since HJ is a subspace of H0 for each J ∈ Jg , there is a bundle of Hilbert spaces H → Jg whose fibre over J is HJ . The trivial connection on the product bundle Jg × H0 projects orthogonally to a natural connection on H. Just like the bosonic case, the connection is projectively flat and the curvature is [22] √ (2.3) FH = (σg /2 −1 ) idH = 18 trV 1,0 (d J ∧ d J) idH . J

Parallel transport in the bundle H identifies, up to a phase, states in the quantum Hilbert spaces HJ from various polarisations J. Unlike Jω , the space Jg is compact and non-negatively curved. Geodesics from J1 to J2 , where J1 , J2 ∈ Jg , are not unique. However, if J2 is not in the cut locus of J1 , there is a unique length-minimising geodesic γJ2 J1 from J1 to J2 . The parallel transport UJ2 J1 : HJ1 → HJ2 along γJ1 J2 was calculated in [22]. For the coherent state √ 1,0 cα ¯ ) + −1 J1 (θ) = exp[−g(θJ1 , α 4 g(J1 θ, θ)], where θ = θJ1,0 + θJ0,1 is a fermionic vector in V and α ∈ VJ1,0 is a fermionic parameter, the parallel transport 1 1 1 from J1 to J2 is [22] J1 +J2 (UJH1 J0 cα J0 )(θ) = det 2

1/4

e

√ −1 4

g(J2 θ,θ)

exp

√−1 2

g θJ1,0 −α ¯, 2

J1 +J2 −1 1,0 (θJ2 2

−α ¯)

.

The operator UJ2 J1 is, up to a rescaling by positive constant, the orthogonal projection from HJ1 to HJ2 in H0 [22]. Like the bosonic case, UJ2 J1 coincides with the Bogoliubov transformation defined in [17]. It is the operator that intertwines the two equivalent irreducible representations HJ1 and HJ2 of the Clifford algebra. Metaplectic correction of fermionic was introduced in [22]. Consider the line bundle K−1 = det V Vn systems 1,0 −1 whose fibre over J ∈ Jg is KJ = VJ . The natural connection on K−1 is compatible with the Hermitian √ K−1 structure and its curvature = −1 σg√. Since Jg is simply connected and since c1 (K) is even, there √ is F ⊗2 is a unique line bundle K → Jg such that q ( K) = K [22]. The Hilbert space of half-form quantisation ˆ J = HJ ⊗ K−1 [22]. (Notice the opposite power of K as in the bosonic case.) (with the polarisation J) is H J √ ˆ ˆJ ˆ When J varies, they form a flat bundle H = H ⊗ K over Jg , i.e., FH = 0 [22]. Thus all the fibres H (that is, the spinor representation spaces from variousppolarisations) p can be canonically identified. For any J1 , J2 ∈ Jg , the natural sesquilinear pairing between KJ1 and KJ2 is non-degenerate if and only if J1 ˆJ and J2 are not in the cut locus of each other [22]. In this case, there is a sesquilinear pairing between H 1 ˆ ˆ ˆ ˆ and HJ2 . The parallel transport UJ2 J1 : HJ1 → HJ2 along the length-minimising geodesic γJ2 J1 from J1 to J2 is in fact the operator determined by the pairing between them [22]. Half-density quantisation can also be established in the fermionic setting. Associated to the vector space 1,0 VJ1,0 is a real line |K−1 J | on which a linear transformation A ∈ EndC (VJ ) acts as multiplication by | det A|. An element of |K−1 a fermionic density of VJ1,0 . A fermionic half-density on VJ1,0 is an element of J | is called q q ⊗2 p |K−1 |K−1 = |K−1 | det A|. J |, a line such that J | J | and on which the linear transformation A acts by

4

q p −1 The lines |KJ−1 | ( |K−1 | ( |K−1 |, respectively) over Jg , which J |, respectively) form real line bundles |K q ˜ J = HJ ⊗ |K−1 |. are naturally flat. For any J ∈ Jg , the Hilbert space of half-density quantisation is H J p −1 ˜ They form a bundle H = H ⊗ |K | of Hilbert spaces over Jg . It has a natural projectively flat connection with curvature √ ˜ FH = FH = σg /2 −1 . q When J1 and J2 are not in the cut locus of each other, there is a non-degenerate pairing between |K−1 J1 | q ˜ ˜ ˜ ˜ ˜ and |K−1 J2 | and hence between HJ1 and HJ2 . The parallel transport UJ2 J1 : HJ1 → HJ2 along the lengthminimising geodesic γJ2 J1 from J1 to J2 is also the operator determined by the pairing between them.

3 3.1

Holonomy of the bundle of Hilbert spaces Bosonic systems: Maslov index and its generalisation

˜ L from half-density We recall that if L is a real Lagrangian subspace of V , we have a Hilbert space H quantisation. For two transverse real Lagrangian subspaces L1 , L2 ∈ Lω , the Fourier transform operator ˜L → H ˜ L intertwines the two equivalent irreducible representations of the Heisenberg algebra. F˜L2 L1 : H 1 2 ˜ → Jω ˜ The operator FL2 L1 is also the limit, in a certain sense, of the parallel transport in the bundle H along a geodesic in Jω extending to L1 and L2 [8]. Suppose there are three mutually transverse Lagrangian subspaces L1 , L2 , L3 ∈ Lω , then we have [10] F˜L1 L3 ◦ F˜L3 L1 ◦ F˜L2 L1 = exp[

√

−1 π 4

αω (L1 , L2 , L3 )] idH ˜L ,

(3.1)

1

where αω (L1 , L2 , L3 ) is the triple Maslov index of Kashiwara (see [10]). It is defined as the signature of the quadratic form ω(x1 , x2 ) + ω(x2 , x3 ) + ω(x3 , x1 ) on L1 ⊕ L2 ⊕ L3 , where xi ∈ Li (i = 1, 2, 3). The triple Maslov index takes integer values in [−n, n] and satisfies the properties [10] that for any mutually transverse Lagrangian subspaces L1 , L2 , L3 , L4 ∈ Lω , (a) αω (gL1 , gL2 , gL3 ) = αω (L1 , L2 , L3 ), ∀g ∈ Sp(V, ω); (b) αω (L1 , L2 , L3 ) = αω (L2 , L3 , L1 ) = −αω (L2 , L1 , L3 ); (c) αω (L1 , L2 , L3 ) + αω (L2 , L4 , L3 ) + αω (L3 , L4 , L1 ) + αω (L4 , L2 , L1 ) = 0. The last property is a cocycle condition on αω . For each complex structure J ∈ Jω , we have a quantum Hilbert space HJ . Given any two complex structures J1 , J2 ∈ Jω , the parallel transport UJ2 J1 : HJ1 → HJ2 in the bundle H → Jω along the geodesic γJ2 J1 from J1 to J2 is equal to the intertwining operator between the representations of the Heisenberg algebra on HJ1 and HJ2 [8]. We can use the same notation UJ2 J1 for the latter. For any three complex structures J1 , J2 , J3 ∈ Jω , we have [11] √

UJ1 J3 ◦ UJ3 J2 ◦ UJ2 J1 = exp[

−1 π 4

αω (J1 , J2 , J3 )] idHJ1 ,

(3.2)

where αω (J1 , J2 , J3 ) is called the generalised Maslov index. The above formula also holds for half-density quantisation. Representing Ji by symmetric matrices Zi (i = 1, 2, 3), we have [11] αω (J1 , J2 , J3 ) = − π2 [arg det(In − Z¯1 Z2 ) + arg det(In − Z¯2 Z3 ) + arg det(In − Z¯3 Z1 )].

(3.3)

¯ 0 ) = 0 when Since Jω is contractible, the function “arg” can be defined continuously so that arg det(In − ZZ 0 either Z or Z is zero. We also note that the Bergman kernel function of the domain can be expressed ¯ 0 ) [7]. The generalised Maslov index takes real values and satisfies the explicitly in terms of det(In − ZZ properties [11] that for any J1 , J2 , J3 , J4 ∈ Jω , (a) αω (gJ1 g −1 , gJ2 g −1 , gJ3 g −1 ) = αω (J1 , J2 , J3 ), ∀g ∈ Sp(V, ω); (b) αω (J1 , J2 , J3 ) = αω (J2 , J3 , J1 ) = −αω (J2 , J1 , J3 ); (c) αω (J1 , J2 , J3 ) + αω (J2 , J4 , J3 ) + αω (J3 , J4 , J1 ) + αω (J4 , J2 , J1 ) = 0. The last property means that αω is a 2-cocycle on the Sp(V, ω)-space Jω with values in R [11].

5

Our observation is that (3.2) is the holonomy of the projectively flat bundle H → Jω along a loop that √consists of three geodesics γJ2 J1 , γJ3 J2 , γJ1 J3 . Using the curvature (2.1), the holonomy is equal to R σ ], where ∆ is an oriented surface bounded by the three geodesics. This implies that exp[ −1 2 ∆ ω Z 2 αω (J1 , J2 , J3 ) = σω . (3.4) π ∆ (It is clear that the equality holds modulo 8π; the additive constant is 0 by continuity when ∆ shrinks to a point.) A direct proof of this result is also possible. We write σω = d φω , where √ ¯ φω = −1 (∂¯ − ∂) log det(In − ZZ). It is easy to show that Z

φω = − arg det(In − Z¯1 Z2 )

(3.5)

γJ2 J1

using for example the transformation of both sides under the symplectic group [6]. Formula (3.4) then follows from (3.3) and Stokes’ theorem. Therefore |α(J1 , J2 , J3 )| ≤ n by [6]. The properties of αω listed above also follow easily from (3.4). We want to establish the analogue of (3.3) for the triple Maslov index αω (L1 , L2 , L3 ). A real Lagrangian subspace L ∈ Lω , being on the Shilov boundary of Jω , is represented by an n × n complex symmetric matrix that is unitary. Two Lagrangian subspaces L1 , L2 are transverse if and only if their corresponding matrices Z1 , Z2 satisfy det(In − Z¯1 Z2 ) 6= 0. (The vanishing of this determinant implies the existence of a non-zero vector v ∈ Cn such that Z1 v = Z2 v, and hence a non-zero vector in L1 ∩L2 . The converse is also true.) When the Lagrangian subspaces L1 , L2 , L3 (parametrised by Z1 , Z2 , Z3 , respectively) are mutually transverse, we claim that (3.6) αω (L1 , L2 , L3 ) = − π2 [arg det(In − Z¯1 Z2 ) + arg det(In − Z¯2 Z3 ) + arg det(In − Z¯3 Z1 )]. R Consequently, the triple Maslov index can be expressed as the limit of the integration ∆ σω when the vertices of ∆ approach the Shilov boundary from the interior Jω . This result, together with its generalisation to Hermitian symmetric tube domains, appeared in [21]. See also [4] for the case n = 1 and [5] for the general case. However, the surface ∆ itself (as well as its boundaries and vertices) moves in this limit procedure. To show (3.6), we observe that both sides are invariant under the symplectic group Sp(V, ω). Since Sp(V, ω) acts transitively on transverse pairs of Lagrangian subspaces, we can assume, without loss of generality, that L1 , L2 are represented by Z1 √ = −In , Z2 = In , respectively. To bring L3 or Z3 to a canonical form, we make a Cayley transform Z 7→ Ω = −1 (In − Z)(In + Z)−1 . The image of Jω is Siegel’s upper-half space that consists of n × n complex symmetric matrices with a positive-definite imaginary part [15]. The Shilov boundary Lω is mapped to the set of real symmetric matrices plus a √ stratum of real codimension 1 at infinity. For example, the above L1 , L2 are represented by Ω1 = lima→+∞ −1 aIn , Ω2 = 0, respectively, and Ω 3 is finite and invertible since L3 is transverse to both L1 and L2 . A symplectic group element A B ∈ Sp(2n, R) acts on Ω by a fractional linear transformation Ω 7→ (AΩ + B)(CΩ + D)−1 . The C

D

subgroups that preserves Z1,2 = ∓In or Ω1 , Ω2 consists of elements such that B√ = C = 0, D = TA−1 . I −1 I 0 0 r r Using such an element, it is possible to bring Ω3 to the form 0 −I or Z3 to , where √ n−r

0

− −1 In−r

0 ≤ r ≤ n. By a simple calculation, both sides of (3.6) are equal to n − 2r. It is possible to express the triple Maslov index αω (L1 , L2 , L3 ) as a generalised integral over a fixed surface. Let ∆ be a surface with boundary in Jω . Suppose L0 ∈ Lω is in the boundary of ∆ (when Jω is regarded as a bounded domain). We say that ∆ is admissible at L0 if (a) there are two geodesics γ1 , γ2 in Jω parametrised by t that is affine with respect to arc length on each geodesic such that limt→+∞ γ1 (t) = limt→+∞ γ2 (t) = L0 , and (b) there exists T S > 0 such that the boundary of ∆ in Jω contains γ1 (t), γ2 (t) for all t ≥ T and such that ∆ contains ∆L0 ,T = t≥T γL0 ,t , where γL0 ,t is the geodesic segment joining γ1 (t) and γ2 (t). Let L0 , γ1 (t), γ2 (t) be represented by symmetric matrices Z0 , Z1 (t), Z2 (t), respectively. Then a straightforward calculation shows that lim [arg det(In − Z¯0 Z1 (t)) + arg det(In − Z1 (t)Z2 (t)) + arg det(In − Z2 (t)Z0 )] = 0.

t→+∞

6

(3.7)

Roughly, this means that as t → +∞, the “symplectic area” of ∆L0 ,t (despite the lack of its definition as ∆L0 ,t is unbounded) goes to 0. Now let L1 , L2 , L3 ∈ Lω be three mutually transverse Lagrangian subspaces joined by geodesics γL2 L1 , γL3 L2 , γL1 L3 in Jω . Suppose ∆ is a surface in Jω bounded by these three geodesics, which make ∆ admissible at L1 , L2 , L3 . For sufficiently large t, let ∆(t) = ∆\(∆L1 ,t ∪ ∆L2 ,t ∪ ∆L3 ,t ). Then ∆(t) is a bounded region in ∆ whose boundary is the union of six geodesic segments. Using (3.5), (3.6) and (3.7), we obtain Z 2 lim σω . αω (L1 , L2 , L3 ) = π t→+∞ ∆(t) In this way, the integration on the right-hand side is over a region ∆(t) that expands (as t → +∞) in a fixed surface ∆. Alternatively, we can integrate σω over the part of ∆ that is outside three horospheres that recede to L1 , L2 , L3 , respectively, in the limit.

3.2

Fermionic systems: orthogonal analogue of Maslov index

For a fermionic system whose phase space is the Euclidean space (V, g) of dimension 2n, the space Jg of polarisations is compact. The bundle of Hilbert spaces H → Jg admits a natural projectively flat connection. Suppose J1 , J2 ∈ Jg and J2 is not in the cut locus of J1 (and vice versa). Let γJ2 J1 be the unique lengthminimising geodesic from J1 to J2 . Then the parallel transport UJ2 J1 along γJ2 J1 is the unitary intertwining operator between the equivalent representations of the Clifford algebra on HJ1 and HJ2 [22]. Suppose J1 , J2 , J3 ∈ Jg such that Ji is not in the cut locus of Jj whenever i 6= j. Then as in the bosonic case (3.2), we have the identity UJ1 J3 ◦ UJ3 J2 ◦ UJ2 J1 = χg (J1 , J2 , J3 ) idHJ1 , (3.8) where χg (J1 , J2 , J3 ) ∈ U (1). This follows from the irreducibility of HJ1 as a representation of the Clifford algebra. The phase χg itself was constructed in [12] as a 2-cocycle on the SO(V, g)-space Jg with values in U (1). It satisfies the properties (a) χg (gJ1 g −1 , gJ2 g −1 , gJ3 g −1 ) = χg (J1 , J2 , J3 ), ∀g ∈ SO(V, g); (b) χg (J1 , J2 , J3 ) = χg (J2 , J3 , J1 ) = χg (J2 , J1 , J3 )−1 ; (c) χg (J1 , J2 , J3 )χg (J2 , J4 , J3 )χg (J3 , J4 , J1 )χg (J4 , J2 , J1 ) = 1 for any J1 , J2 , J3 , J4 ∈ Jg such that Ji is not in the cut locus of Jj (i 6= j). This is the orthogonal or spinorial counterpart of the generalised Maslov index in [11]. We observe that the phase χg (J1 , J2 , J3 ) is the holonomy of the projectively flat connection on H along ˜ → Jg of the three geodesics γJ2 J1 , γJ3 J2 and γJ1 J3 . (Here and below, we can replace H by the bundle H half-density quantisation without changing the holonomy.) Since Jg is simply connected, there is an oriented surface ∆ whose boundary consists of the three geodesics. Using the curvature (2.3), we get √

χg (J1 , J2 , J3 ) = exp[

−1 2

R ∆

σg ].

(3.9)

We note that the exponent on the right-hand side is not well defined, as the integration over ∆ changes when ∆ is replaced by another surface with the same boundary but not homotopic to ∆. However, the exponential does not depend on the choice of ∆. This is consistent with the fact that σg is a closed 2-form whose periods are in 4πZ [22]. We can choose J0 ∈ Jg such that J1 , J2 , J3 are in the complement of its cut locus. Then the phase χg (J1 , J2 , J3 ) can be lifted to be real valued, i.e., √

χg (J1 , J2 , J3 ) = exp[

−1 π 4

αg (J1 , J2 , J3 )],

where αg (J1 , J2 , J3 ) ∈ R depends smoothly on J1 , J2 , J3 and αg (J1 , J2 , J3 ) = 0 whenever any two of J1 , J2 , J3 coincide. This is because the complement of a cut locus is contractible. We note however that αg does depend on the choice of J0 . If we further choose ∆ as a surface that lies entirely in the complement and whose boundary consists of the three geodesics joining J1 , J2 , J3 , then formula (3.9) for χg (J1 , J2 , J3 ) can be lifted to Z 2 σg , (3.10) αg (J1 , J2 , J3 ) = π ∆

7

an identity in R that is formally similar to (3.4). When J1 , J2 , J3 , J4 ∈ Jg are not in the cut locus of each other and suppose they are not in the cut locus of J0 either, we have (a) αg (gJ1 g −1 , gJ2 g −1 , gJ3 g −1 ) = αg (J1 , J2 , J3 ), ∀g ∈ SO(V, g); (b) αg (J1 , J2 , J3 ) = αg (J2 , J3 , J1 ) = −αg (J2 , J1 , J3 ); (c) αg (J1 , J2 , J3 ) + αg (J2 , J4 , J3 ) + αg (J3 , J4 , J1 ) + αg (J4 , J2 , J1 ) = 0. In (a), αg (gJ1 g −1 , gJ2 g −1 , gJ3 g −1 ) is defined because gJ1 g −1 , gJ2 g −1 , gJ3 g −1 are not in the cut locus of gJ0 g −1 . Using the parametrisation of J1 , J2 , J3 by skew-symmetric matrices Z1 , Z2 , Z3 , respectively, we can write the K¨ ahler form on the complement of the cut locus as σg = d φg , where √ ¯ log det(In − ZZ). ¯ φg = −1 (∂ − ∂) ¯ So log det(In − ZZ) is essentially the K¨ ahler potential on the compliment of a cut locus. As in the bosonic case, we get a formula (cf. [12]) αg (L1 , L2 , L3 ) =

4

2 π

[arg det(In − Z¯1 Z2 ) + arg det(In − Z¯2 Z3 ) + arg det(In − Z¯3 Z1 )].

Comparison with Berry’s phase

When the Hamiltonian of a quantum mechanical system undergoes an adiabatic change, an energy eigenstate of the initial Hamiltonian evolves to that of the new Hamiltonian, multiplied by a phase. If the change is in a cycle, the phase contains a dynamical part and Berry’s geometric phase [2]. The latter was related to the holonomy of bundles over the space of parameters [16]. Berry’s phase was generalised to the non-Abelian setting when the energy levels are possibly degenerate [19]. In [20], it was shown that non-Abelian Berry’s phase is related to universal connection [13] in the following way. Let B be the space of parameters of the system. Assume that an energy eigenvalue varies smoothly over B and its degeneracy r does not change. This defines a map from B to the Grassmannian of r-planes in the quantum Hilbert space. Over the latter there is a universal connection in the tautological vector bundle [13] and Berry’s phase is the holonomy of its pull-back along a cyclic path in B [20]. The universal connection is defined by orthogonal projection of the trivial connection [13]. This is mathematically similar to the construction of the projectively flat connection in the bundle of Hilbert spaces over the space of polarisations (§2). There is however an important conceptual difference. Although polarisations provide many ways to construct the quantum Hilbert space, they are not physical parameters of the system. On the contrary, our purpose was to show that physics is independent of the choice of polarisations by identifying the Hilbert spaces through parallel transport. However, Berry’s phase is physical and can be measured experimentally. It occurs when the physical parameters of the system changes, regardless of the polarisation chosen. It is the latter that concerns us now, even though the physical systems considered below are parametrised exactly by Jω and Jg . We start with a linear bosonic system whose phase space is a symplectic vector space (V, ω) of dimension 2n. We fix the quantum Hilbert space Hb . Consider the Hamiltonian of harmonic oscillator HJ =

1 2

ω(·, J·) ∈ Sym2 (V ∗ ),

ˆJ where J ∈ Jω is now a physical parameter. Its quantisation is a positive-definite self-adjoint operator H acting on Hb , which has a Fock space decomposition as follows. Recall that the creation and annihilation operators are the quantised linear functionals on VJ1,0 and VJ0,1 , respectively. For any polynomial f ∈ Sym(VJ1,0 )∗ , its quantisation fˆ acts on Hb . (Operator ordering is not a problem here as the creation ˆ J and let operators commute with each other.) Let |0iJ be the vacuum state of H (k) HJ = Symk (VJ1,0 )∗ |0iJ = { fˆ|0iJ | f ∈ Symk (VJ1,0 )∗ } (k)

for any k ∈ N = {0, 1, 2, . . . }. Each HJ n+k−1 and as a Hilbert space, k

ˆ J with energy k + n/2. We have dimC H(k) = is an eigenspace of H J Hb =

M k∈N

8

(k)

HJ .

(k) ˆ J form a vector bundle As the parameter J ∈ Jω varies, the eigenspaces HJ (for a fixed k ∈ N) of H (k) H over Jω . Each bundle H has a natural connection by orthogonal projection from the product bundle Jω × Hb . This is also the pull-back of the universal connection from the Grassmannian. Therefore under an adiabatic cyclic evolution in Jω , Berry’s phase in the kth eigenspace is the holonomy of the bundle H(k) . √ −1 (0) (0) H is a line bundle whose fibre over J ∈ Jω is spanned by |0iJ ∈ Hb . Its curvature is F = 2 σω ; note the opposite sign from (2.1). Therefore Berry’s phase of the vacuum√state is inverse to the holonomy considered in §3.1. The bundle H(0) with its connection √ is isomorphic to K. For a general k, the bundle H(k) with its connection is isomorphic to Symk (V∗ ) ⊗ K; the latter has a connection induced from V, whose curvature is given by (2.2). In particular, if n = 1, then H(k) ∼ = K⊗(k+1/2) and therefore its curvature is (cf. [3]) √ F(k) = (k + 21 ) −1 σω . (k)

However, unless k = 0 or n = 1 when H(k) is a line bundle, the connection on H(k) is not projectively flat. Berry’s phase also appears in fermionic systems. Suppose the phase space is a Euclidean space (V, g) of dimension 2n. Let Hf be the quantum Hilbert space. We consider a fermionic harmonic oscillator whose Hamiltonian is V2 ∗ (V ), HJ = 21 g(J·, ·) ∈ ˆ J is a self-adjoint operator on Hf . As in the where J ∈ Jg is again a physical parameter. Its quantisation H bosonic case, the creation and annihilation operators are the quantised linear functionals on VJ1,0 and VJ0,1 , Vk 1,0 ∗ respectively. For any f ∈ (VJ ) , the quantum operator fˆ acts on Hf . Let |0iJ be the vacuum state and, for any k ∈ Z with 0 ≤ k ≤ n, let (k)

HJ = (k)

Then each HJ

Vk

Vk 1,0 ∗ (VJ1,0 )∗ |0iJ = { fˆ|0iJ | f ∈ (VJ ) }.

ˆ J with energy k − n/2. We have dimC H(k) = is an eigenspace of H J Hf =

n M

n k

and

(k)

HJ .

k=0 (k) ˆ J form a vector bundle H(k) over Jg . The natural connection on The eigenspaces HJ (for a fixed k) of H H by orthogonal projection from the product bundle Jg × Hf coincides with the pull-back of the universal connection from the Grassmannian. Therefore under an adiabatic cyclic evolution in Jg , Berry’s phase in the kth eigenspace is the holonomy of the√bundle H(k) . H(0) is a line bundle whose fibre over J ∈ Jg is spanned Berry’s phase of the by |0iJ ∈ Hf . Its curvature is F(0) = −1 2 σg ; note again the opposite sign from (2.3). So √ (0) −1 . For 0 ≤ k ≤ n, K vacuum state is inverse to the holonomy in §3.2. The line bundle H is isomorphic to √ Vk ∗ (k) −1 the bundle H with its connection is isomorphic to (V ) ⊗ K , which has a connection induced from √ √ −1 (0) (n) V. In particular, H(n) ∼ K a line bundle that is dual to H and its curvature is F = − σ = g . In 2 (k) general cases (k 6= 0, n), the connection on H is not projectively flat. (k)

Acknowledgments. Part of the work was done around 2000 while the author was at the University of Adelaide. In particular, §3.1 and the generalisation to arbitrary Hermitian symmetric spaces were reported at various conferences [21]. The author thanks his former colleagues, conference organisers and participants for their continuing interest and encouragement. He also thanks the referee for helpful comments.

References [1] S. Axelrod, S. Della Pietra and E. Witten, Geometric quantization of Chern-Simons gauge theory, J. Diff. Geom. 33 (1991) 787-902 [2] M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. Roy. Soc. London Ser. A 392 (1984) 45-57

9

[3] M. V. Berry, Classical adiabatic angles and quantal adiabatic phase, J. Phys. A18 (1985) 15-27 [4] J. L. Clerc and B. 6Orsted, The Maslov index revisited, Transform. Groups 6 (2001) 303-320 [5] J. L. Clerc and B. 6Orsted, The Gromov norm of the Kaehler class and the Maslov index, Asian J. Math. 7 (2003) 269-295; Corrigendum, ibid. 8 (2004) 391-393 [6] A. Dominic and D. Toledo, The Gromov norm of the Kaehler class of symmetric domains, Math. Ann. 276 (1987) 425-432 [7] L.-K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Science Press, Peking (1958); English translation, Amer. Math. Soc., Providence, RI (1963) [8] W. D. Kirwin and S. Wu, Geometric quantization, parallel transport and the Fourier transform, Commun. Math. Phys. 266 (2006) 577-594, arXiv:math.SG/0409555 [9] B. Kostant, Graded manifolds, graded Lie theory, and prequantization, in: Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn, 1975), Lecture Notes in Math., 570, eds. K. Bleuler and A. Reetz, Springer, Berlin (1977), pp. 177-306 [10] G. Lion and M. Vergne, The Shale-Weil representations and the Maslov index, in: The Weil representation, Maslov index and theta series, Prog. in Math. 6, Birkh¨auser, Boston, MA (1980), part I [11] B. Magneron, Spineurs symplectiques purs et indice de Maslov de plan Lagrangiens positifs, J. Funct. Anal. 59 (1984) 90-122 [12] B. Magneron, Spineurs purs et description cohomologique des groupes spinoriels, J. Algebra 112 (1988) 349-369 [13] M. S. Narasimhan and S. Ramanan, Existence of universal connections, Amer. J. Math. 83 (1961) 563-572 [14] I. Satake, Fock representations and theta-functions, in: Advances in the theory of Riemann surfaces, eds. L. V. Ahlfors et al, Princeton Univ. Press, Princeton, NJ (1971), pp. 393-405 [15] C. L. Siegel, Symplectic geometry, Amer. J. Math. 65 (1943) 1-86 [16] B. Simon, Holonomy, the quantum adiabatic theorem, and Berry’s phase, Phys. Rev. Lett. 51 (1983) 2167-2170 [17] N. M. J. Woodhouse, Geometric quantization and the Bogoliubov transformation, Proc. Royal Soc. London A 378 (1981) 119-139 [18] N. M. J. Woodhouse, Geometric quantization (2nd ed.), Oxford Univ. Press Inc., Oxford and New York (1992) [19] F. Wilczek and A. Zee, Appearance of gauge structure in simple dynamical systems, Phys. Rev. Lett. 52 (1984) 2111-2114 [20] S. Wu, Quantum adiabatic theorem and universal holonomy, Lett. Math. Phys. 16 (1988) 339-345 [21] S. Wu, Projective flatness in geometric quantization and the Maslov index, talks at conferences in the Fields Institute (Toronto, 2001), Univ. of Colorado (Boulder, 2001), AMS meeting (Irvine, 2001); Hermitian symmetric spaces, Shilov boundary, and Maslov index, talks at the conference on representation theory (Sydney, 2002) and the ICM (Beijing, 2002) [22] S. Wu, Projective flatness in the quantization of bosons and fermions, preprint HKU-IMR2010:#11, August (2010), arXiv:1008.5333[math.SG]

10