Abstract
Using algebraic topology, the appearance of the Quantum Adiabatic Phase over various parameter manifolds is investigated. The relation with nontrivial gauge bundles (both abelian and non-abelian) is studied and it is shown that the phase appears as a result of homotopically non-trivial mappings, induced by the Hamiltonian in the space of wave-functions. The cohomological picture is developed and some topological considerations concerning field theory anomalies in the Hamiltonian picture are presented. A proof of the Nielsen-Ninomiya theorem is given inspired from the notion of the adiabatic phase.
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References
Berry, M.: Proc. R. Soc. Lond. A392, 45 (1984)
Simon, B.: Holonomy, the quantum adiabatic theorem, and Berry's phase. Phys. Rev. Lett.51, 2167 (1983)
Wilczek, F., Zee, A.: Appearance of gauge structure in simple dynamical systems. Phys. Rev. Lett.52, 2111 (1984)
Moody, J., Shapere, A., Wilczek, F.: ITP preprint NSF-ITP-85-78
Kuratsuji, H., Iida, S.: Kyoto, Univ. preprint, KUNS-770
Nelson, P., Alvarez-Gaumé L.: Hamiltonian interpretation of anomalies. Commun. Math. Phys.99, 103 (1985)
Sonoda, H.: The Wess-Zumino term and the Hamiltonian formulation for anomalies. Phys. Lett.156B, 220 (1985)
Sonoda H.: Berry's phase in chiral gauge theories. Nucl. Phys. B266, 410 (1986)
Niemi, A.J., Semenoff, G.W.: Quantum holonomy and the chiral gauge anomaly. Phys. Rev. Lett.55, 927 (1985)
Steenrod, N.: The topology of fibre bundles. Princeton, NJ: Princeton University Press 1951
Whitehead, G.: Elements of homotopy theory. Berlin, Heidelberg, New York: Springer 1978
Nielsen, H.B., Ninomiya, M.: Absence of neutrinos on a lattice (I). Proof by homotopy theory. Nucl. Phys. B185, 20 (1981)
Thouless, D.J., Kohmoto, M., Nightingale, M.P., den Nijs, M.: Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett.49, 405 (1982)
Avron, J.E., Seiler, R., Simon, B.: Homotopy and quantization in condensed matter physics. Phys. Rev. Lett.51, 51 (1983)
Alvarez-Gaumé, L., Ginsparg, P.: The topological meaning of non-abelian anomalies. Nucl. Phys. B243, 449 (1984)
Feng, Jn, Preskill, J.: Caltech preprint CALT-68-1278
Witten, E.: AnSU(2) anomaly. Phys. Lett.117B, 324 (1982)
Witten, E.: Talk at the: Symposium on anomalies, geometry and topology.
Bardeen, W.A., White, A.R. (eds.): Global gravitational anomalies. Commun. Math. Phys.100, 197 (1985)
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Communicated by K. Osterwalder
Work supported in part by the U.S. Department of Energy under contract DEAC 03-81-ER 40050
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Kiritsis, E. A topological investigation of the Quantum Adiabatic Phase. Commun.Math. Phys. 111, 417–437 (1987). https://doi.org/10.1007/BF01238907
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DOI: https://doi.org/10.1007/BF01238907