Classical Equivalence and Quantum Equivalence of Magnetic Fields on Flat Tori

Abstract

Let M be a real 2m-torus equipped with a translation-invariant metric h and a translation-invariant symplectic form w; the latter we interpret as a magnetic field on M. The Hamiltonian flow of half the norm-squared function induced by h on T^*M (the "kinetic energy") with respect to the twisted symplectic form w_{T^*M}+ \pi^*w describes the trajectories of a particle moving on M under the influence of the magnetic field w. If [w] is an integral cohomology class, then we can study the geometric quantization of the symplectic manifold (T^*M,w_{T^*M}+\pi^*w) with the kinetic energy Hamiltonian. We say that the quantizations of two such tori (M_1,h_1,w_1) and (M_2,h_2,w_2) are quantum equivalent if their quantum spectra, i.e., the spectra of the associated quantum Hamiltonian operators, coincide; these quantum Hamiltonian operators are proportional to the h_j-induced bundle Laplacians on powers of the Hermitian line bundle on M with Chern class [w]. In this paper, we construct continuous families {(M,h_t)}_t of mutually nonisospectral flat tori (M,h_t), each endowed with a translation-invariant symplectic structure w, such that the associated classical Hamiltonian systems are pairwise equivalent. If w represents an integer cohomology class, then the (M,h_t,w) also have the same quantum spectra. We show moreover that for any translation-invariant metric h and any translation-invariant symplectic structure w on M that represents an integer cohomology class, the associated quantum spectrum determines whether (M,h,w) is Kaehler, and that all translation-invariant Kaehler structures (h,w) of given volume on M have the same quantum spectra. Finally, we construct pairs of magnetic fields (M,h,w_1), (M,h,w_2) having the same quantum spectra but nonsymplectomorphic classical phase spaces. In some of these examples the pairs consist of Kaehler manifolds.