Classical Equivalence and Quantum Equivalence of Magnetic Fields on Flat Tori

Details Classical Equivalence and Quantum Equivalence of Magnetic Fields on Flat Tori Classical Equivalence and Quantum Equivalence of Magnetic Fields on Flat Tori

2011-08-25

arxiv.org/abs/1108.5113v1

Mathematics

Differential Geometry

14 pages

Scientific paper

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.

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