Loops, projective invariants, and the realization of the Borromean topological link in quantum mechanics
Tóm tắt
All the typical global quantum mechanical observables are complex relative phases obtained by interference phenomena. They are described by means of some global geometric phase factor, which is thought of as the “memory” of a quantum system undergoing a “cyclic evolution” after coming back to its original physical state. The origin of a geometric phase factor can be traced to the local phase invariance of the transition probability assignment in quantum mechanics. Beyond this invariance, transition probabilities also remain invariant under the operation of complex conjugation. Most important, geometric phase factors distinguish between unitary and antiunitary transformations in terms of complex conjugation. These two types of invariance functions as an anchor point to investigate the role of loops and based loops in the state space of a quantum system as well as their links and interrelations. We show that arbitrary transition probabilities can be calculated using projective invariants of loops in the space of rays. The case of the double slit experiment serves as a model for this purpose. We also represent the action of one-parameter unitary groups in terms of oppositely oriented-based loops at a fixed ray. In this context, we explain the relation among observables, local Boolean frames of projectors, and one-parameter unitary groups. Next, we exploit the non-commutative group structure of oriented-based loops in 3-d space and demonstrate that it carries the topological semantics of a Borromean link. Finally, we prove that there exists a representation of this group structure in terms of one-parameter unitary groups that realizes the topological linking properties of the Borromean link.
Tài liệu tham khảo
Berry, M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45 (1984)
Simon, B.: Holonomy, the quantum adiabatic theorem, and Berry’s phase. Phys. Rev. Lett. 51, 2167 (1983)
Aharonov, Y., Anandan, J.: Phase change during a cyclic quantum evolution. Phys. Rev. Lett. 58(16), 1593–1596 (1987)
Wilczek, F., Shapere, A.: Geometric phases in physics. World Scientific, Singapore (1989)
Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485–491 (1959)
Aharonov, Y., Bohm, D.: Further considerations on electromagnetic potentials in the quantum theory. Phys. Rev. 123, 1511–1524 (1961)
von Müller, A.: The Forgotten Present. In: von Müller, A., Filk, T. (eds.) Re-thinking Time at the Interface of Physics and Philosophy, pp. 1–46. Springer, Heidelberg (2015)
Filk, T., von Müller, A.: A categorical framework for quantum theory. Ann. der Phys. 522(11), 783–801 (2010)
Aravind, P.K.: Borromean entanglement of the GHZ state. In: Cohen, R.S., Horne, M., Stachel, J. (eds.) Quantum potentiality, entanglement and passion-at-a- distance, essays for abner shimony, pp. 53–59. Kluwer, Dordrecht (1997)
Cromwell, P., Beltrami, E., Rampichini, M.: The Borromean Rings. Math. Intell. 20(1), 53–62 (1998)
Debrunner, H.: Links of Brunnian type. Duke Math. J. 28, 17–23 (1961)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Lindström, B., Zetterström, H.-O.: Borromean circles are impossible. Am. Math. Mon. 98(4), 340–341 (1991)
Kawauchi, A.: A Survey of Knot Theory. Birkhäuser, New York (1996)
Greenberger, D., Horne, M., Shimony, A., Zeilinger, A.: Bell’s theorem without inequalities. Am. J. Phys. 58, 1131–1143 (1990)
Watanabe, S.: Symmetry of physical laws. Part III. Prediction and retrodiction. Rev. Mod. Phys. 27(2), 179 (1955)
Aharonov, Y., Bergmann, P.J., Lebowitz, J.L.: Time symmetry in the quantum process of measurement. Phys. Rev. B. 134(6), 1410–1416 (1964)
Aharonov, Y., Cohen, E., Gruss, E., Landsberger, T.: Measurement and collapse within the two-state vector formalism. Quantum Stud. Math. Found. 1(1–2), 133–146 (2014)
Vaidman, L.: Weak-measurement elements of reality. Found. Phys. 26(7), 895–906 (1996)
Aharonov, Y., Albert, D.Z., Vaidman, L.: How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60(14), 1351–1354 (1988)
Feynman, R.P., Hibbs, A.R., Styer, D.F.: Quantum mechanics and path integrals, emended edn. Dover Books on Physics, USA (2010)
Schwinger, J.: Quantum kinematics and dynamics. Westview Press, USA (2000)
Zafiris, E.: The Global Symmetry Group of Quantum Spectral Beams and Geometric Phase Factors. Adv. Math. Phys. (2015). doi:10.1155/2015/124393 (Article ID:124393)
Urbantke, H.K.: The Hopf fibration-seven times in physics. J. Geom. Phys. 46(2), 125–150 (2003)
Bargmann, V.: Note on Wigner’s theorem on symmetry operations. J. Math. Phys. 5, 862–868 (1964)
Stone, M.H.: On one-parameter unitary groups in Hilbert Space. Ann. Math. 33(3), 643–648 (1932)
Epperson, M., Zafiris, E.: Foundations of relational realism: a topological approach to quantum mechanics and the philosophy of nature. Lexington Books, Lanham (2013)
Zafiris, E.: Generalized Topological Covering Systems on Quantum Events’ Structures. J. Phys. A Math. General 39, 1485 (2006)
Omnés, R.: The interpretation of quantum mechanics. Princeton University Press, Princeton (1994)
Selesnick, S.A.: Quanta, logic and spacetime, 2nd edn. World Scientific, Singapore (2003)
Zafiris, E.: Sheaf-theoretic representation of quantum measure algebras. J. Math. Phys. 47, 092103 (2006)
Zafiris, E., Karakostas, V.: A categorial semantic representation of quantum event structures. Found. Phys. 43, 1090–1123 (2013)
Choi, M.-D.: The full C* algebra of the free group on two generators. Pac. J. Math. 87(1), 41–48 (1980)
Mac Lane, S.: Categories for the Working Mathematician, Graduate Texts in Mathematics, 2nd edn., vol. 5. Springer, New York (1998)
Cohen, J.: \(C^{\star }\)-algebras without idempotents. J. Funct. Anal. 33, 211–216 (1979)
Johnstone, P.T.: Stone spaces. Cambridge University Press, Cambridge (1986)