QUANTUM MECHANICS AND NON-ABELIAN THETA FUNCTIONS ...

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10RĂZVAN GELCA AND ALEJANDRO URIBEare of the formf(x,y) = exp 2πi(mx + ny), m,n ∈ Z.We quantize them using Weyl quantization. In the holomorphic polarizationWeyl quantization is defined as follows (see [5]): A fundamental domain ofthe torus is the unit square [0,1]×[0,1] (this is done in the (x,y) coordinates,in the complex plane it is actually a parallelogram). The value of a thetafunction is completely determined by its values on this unit square. TheHilbert space of classical theta functions can be isometrically embeddedinto L 2 ([0,1] × [0,1]) with the inner product√ ∫ 2 1∫ 1〈f,g〉 = f(x,y)g(x,y)e −2Nπy2 dxdyN00(the constant in front of the integral is chosen so that the specified thetaseries have norm 1). The operator associated to a function f on the torus isthe Toeplitz operator with symbol e − ∆4N f, where ∆ is the Laplacian definedas∆f = 1 ( ∂ 2 )f2π ∂x 2 + ∂2 f∂y 2 .This operator maps a classical theta function g to the orthogonal projectiononto the Hilbert space of classical theta functions of (e − ∆4N f)g.The Weyl quantization of the exponentials gives rise to the Schrödingerrepresentation of the Heisenberg group with integer entries H(Z) onto thespace of theta functions. We define this Heisenberg group aswith multiplicationH(Z) = {(p,q,k), p,q,k ∈ Z}(p,q,k)(p ′ ,q ′ ,k ′ ) = (p + p ′ ,q + q ′ ,k + k ′ + (pq ′ − qp ′ )).As before, we denote exp(pP +qQ+kE) = (p,q,k). Then exp(pP +qQ+kE)is the Weyl quantization of the function e πiN k exp 2πi(qx+py). A few computationsthat can be found in [9] show that the quantization of exponentialsyields the action of this Heisenberg group on theta functions given byIn particularexp(pP)θ τ j = θ τ j+p,exp(pP + qQ + kE)θ τ jπi= e− N pq−2πi Nexp(qQ)θ τ j = e −2πi N qj θ τ j ,qj+πiN k θ τ j+p .exp(kE)θ τ j = e πiN k θ τ j .Here the indices of theta functions are taken mod N. Note that the thetaseries are the eigenvectors of exp(Q) and they can be generated from θ0 τ byθj τ = exp(jP)θ0 τ for all j = 1,2,... ,N − 1. The rigid structure specifiesP and Q, once they are known, the theta series are characterized by theseproperties (up to a multiplication of all of them by the same constant). Thisis how the rigid structure determines the theta series.We want to factor out by the exponentials that act as identity operators.
QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 11Proposition 3.1. The set of elements in H(Z) that act on theta functionsas identity operators is a normal subgroup consisting of the Nth powers ofelements of the form exp(pP + qQ + kE) with k even. The quotient groupis a finite Heisenberg group.Proof. Ifexp(pP + qQ + kE)θ τ jπi= e− N pq−2πi πiqj+ N N k θj+pτequals θj τ for all j, then p is a multiple of N. Setting j = N 2, we concludethat k is a multiple of 2N. Finally, because −2q(j + N 2) must be an multipleof 2N for all j, it follows that q must be a multiple of N. It is not hard tosee that any such element is the Nth power of an element with the thirdentry even. The computation(exp(pP + qQ + kE)) N θ τ jN(N−1)= e−2πijq−2πi 2Npq−N πipq πik+N N N θj+Nτ= e −2πijq−πiNpq+πik θ τ j = e πik θ τ jshows that the Nth powers of elements with third entry even act as theidentity operator. This proves the first part of the statement.Next, define the finite Heisenberg groupby the multiplication ruleH(Z N ) = {(p,q,k), p,q ∈ Z N ,k ∈ Z 2N }(p,q,k)(p ′ ,q ′ ,k ′ ) = (p + p ′ ,q + q ′ ,k + k ′ + pq ′ ).It is straightforward to check that the mapφ((p,q,k)) = (p,q,k + pq)is a group homomorphism whose kernel is the set of elements of the formexp(pP +qQ+kE) with p and q divisible by N and k divisible by 2N. Hencethe desired quotient group is isomorphic to a finite Heisenberg group. □From now on we will denote by H(Z N ) the quotient of the Heisenberg withinteger entries defined in the previous proposition. Whenever we mentionthe finite Heisenberg group, we have in mind this group. We have seenthat indeed, this group is isomorphic to an actual finite Heisenberg group,but we do not work with the latter, because for example the bilinear form((p,q),(p ′ ,q ′ )) → pq ′ which defines multiplication is not invariant undersymplectomorphisms. Thus whenever we work with an exponential of theform exp(pP +qQ+kE), we think of it as the equivalence class of the element(p,q,k) ∈ H(Z).Theorem (Stone-von Neumann) The Schrödinger representation of H(Z N )is the unique irreducible unitary representation of this group with the propertythat exp(kE) acts as e πiN k Id for all k ∈ Z.
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