Beyond the Dirac Phase Factor: Dynamical Quantum Phase-Nonlocalities in the Schrödinger Picture

Author(s)
Konstantinos Moulopoulos

ABSTRACT

Generalized solutions of the standard gauge transformation equations are presented and discussed in physical terms. They go beyond the usual Dirac phase factors and they exhibit nonlocal quantal behavior, with the well-known Relativistic Causality of classical fields affecting directly the phases of wavefunctions in the Schrödinger Picture. These nonlocal phase behaviors, apparently overlooked in path-integral approaches, give a natural account of the dynamical nonlocality character of the various (even static) Aharonov-Bohm phenomena, while at the same time they seem to respect Causality. For particles passing through nonvanishing magnetic or electric fields they lead to cancellations of Aharonov-Bohm phases at the observation point, generalizing earlier semiclassical experimental observations (of Werner & Brill) to delocalized (spread-out) quantum states. This leads to a correction of previously unnoticed sign-errors in the literature, and to a natural explanation of the deeper reason why certain time-dependent semiclassical arguments are consistent with static results in purely quantal Aharonov-Bohm configurations. These nonlocalities also provide a remedy for misleading results propagating in the literature (concerning an uncritical use of Dirac phase factors, that persists since the time of Feynman’s work on path integrals). They are shown to conspire in such a way as to exactly cancel the instantaneous Aharonov-Bohm phase and recover Relativistic Causality in earlier “paradoxes” (such as the van Kampen thought-experiment), and to also complete Peshkin’s discussion of the electric Aharonov-Bohm effect in a causal manner. The present formulation offers a direct way to address time-dependent single- vs double-slit experiments and the associated causal issues—issues that have recently attracted attention, with respect to the inability of current theories to address them.

Generalized solutions of the standard gauge transformation equations are presented and discussed in physical terms. They go beyond the usual Dirac phase factors and they exhibit nonlocal quantal behavior, with the well-known Relativistic Causality of classical fields affecting directly the phases of wavefunctions in the Schrödinger Picture. These nonlocal phase behaviors, apparently overlooked in path-integral approaches, give a natural account of the dynamical nonlocality character of the various (even static) Aharonov-Bohm phenomena, while at the same time they seem to respect Causality. For particles passing through nonvanishing magnetic or electric fields they lead to cancellations of Aharonov-Bohm phases at the observation point, generalizing earlier semiclassical experimental observations (of Werner & Brill) to delocalized (spread-out) quantum states. This leads to a correction of previously unnoticed sign-errors in the literature, and to a natural explanation of the deeper reason why certain time-dependent semiclassical arguments are consistent with static results in purely quantal Aharonov-Bohm configurations. These nonlocalities also provide a remedy for misleading results propagating in the literature (concerning an uncritical use of Dirac phase factors, that persists since the time of Feynman’s work on path integrals). They are shown to conspire in such a way as to exactly cancel the instantaneous Aharonov-Bohm phase and recover Relativistic Causality in earlier “paradoxes” (such as the van Kampen thought-experiment), and to also complete Peshkin’s discussion of the electric Aharonov-Bohm effect in a causal manner. The present formulation offers a direct way to address time-dependent single- vs double-slit experiments and the associated causal issues—issues that have recently attracted attention, with respect to the inability of current theories to address them.

Cite this paper

nullK. Moulopoulos, "Beyond the Dirac Phase Factor: Dynamical Quantum Phase-Nonlocalities in the Schrödinger Picture,"*Journal of Modern Physics*, Vol. 2 No. 11, 2011, pp. 1250-1271. doi: 10.4236/jmp.2011.211156.

nullK. Moulopoulos, "Beyond the Dirac Phase Factor: Dynamical Quantum Phase-Nonlocalities in the Schrödinger Picture,"

References

[1] Y. Aharonov and D. Bohm, “Significance of Electromagnetic Potentials in the Quantum Theory,” Physical Review, Vol. 115, No. 3, 1959, pp. 485-491. doi:10.1103/PhysRev.115.485

[2] S. Popescu, “Dynamical Quantum Nonlocality,” Nature Physics, Vol. 6, 2010, pp. 151-153. doi:10.1038/nphys1619

[3] M. Peshkin and A. Tonomura, “The Aharonov-Bohm Effect,” Lecture Notes in Physics, Vol. 340, 1989, Part I, Appendix B, Springer-Verlag, Berlin, pp. 27-28.

[4] H. R. Brown and P. R. Holland, “The Galilean Covariance of Quantum Mechanics in the Case of External Fields,” American Journal of Physics, Vol. 67, 1999, pp. 204-214. doi:10.1119/1.19227

[5] K. Moulopoulos, “Nonlocal Phases of Local Quantum Mechanical Wavefunctions in Static and Time-Dependent Aharonov-Bohm Experiments,” Journal of Physics A: Mathematical and Theoretical, Vol. 43, No. 35, 2010, Article ID: 354019, pp. 1-32.

[6] J. Tollaksen, Y. Aharonov, A. Casher, T. Kaufherr and S. Nussimov, “Quantum Interference Experiments, Modular Variables and Weak Measurements,” New Journal of Physics, Vol. 12, 2010, Article ID: 013023, pp. 1-29.

[7] G. P. He, “Flexible Scheme for Measuring Experimentally the Speed of the Response of Quantum States to the Change of the Boundary Condition,” November 2009, pp. 1-6. http://arxiv.org/PS_cache/arxiv/pdf/0907/0907.1974v2.pdf

[8] T. T. Wu and C. N. Yang, “Concept of Nonintegrable Phase Factors and Global Formulation of Gauge Fields,” Physical Review D, Vol. 12, No. 12, 1975, pp. 3845- 3857. doi:10.1103/PhysRevD.12.3845

[9] P. D. Noerdlinger, “Elimination of the Electromagnetic Potentials,” Il Nuovo Cimento, Vol. 23, 1962, pp. 158- 167. doi:10.1007/BF02733550

[10] F. G. Werner and D. R. Brill, “Significance of Electromagnetic Potentials in the Quantum Theory in the Interpretation of Electron Interferometer Fringe Observations,” Physical Review Letters, Vol. 4, No. 7, 1960, pp. 344-347. doi:10.1103/PhysRevLett.4.344

[11] R. P. Feynman, R. B. Leighton and M. Sands, “The Feynman Lectures on Physics,” Addison-Wesley, Vol. II, Chapter 15, 1964, p. 13.

[12] B. Felsager, “Geometry, Particles, and Fields”, Springer- Verlag, Berlin, 1998, p. 55. doi:10.1007/978-1-4612-0631-6

[13] H. Batelaan and A. Tonomura, “The Aharonov-Bohm Effects: Variations on a Subtle Theme,” Physics Today, Vol. 62, No. 9, 2009, pp. 38-43. doi:10.1063/1.3226854

[14] M. P. Silverman, “Quantum Superposition,” Springer- Verlag, Berlin, 2008, pp. 13, 19.

[15] N. G. van Kampen, “Can the Aharonov-Bohm Effect Transmit Signals Faster than Light?” Physics Letters, Vol. 106A, 1984, pp. 5-6.

[16] P. G. Luan and C. S. Tang, “Charged Particle Motion in a Time-Dependent Flux-Driven Ring: An Exactly Solvable Model,” Journal of Physics: Condensed Matter, Vol. 19, 2007, Article ID: 176224, pp. 1-10.

[17] H. Erlichson, “Aharonov-Bohm Quantum Effects on Charged Particles in Field-Free Regions,” American Jour- nal of Physics, Vol. 38, 1970, pp. 162-173. doi:10.1119/1.1976266

[18] R. A. Brown and D. Home, “Locality and Causality in Time-Dependent Aharonov-Bohm Interference,” Il Nuovo Cimento, Vol. 107B, 1992, pp. 303-316. doi:10.1007/BF02728492

[19] T. Troudet, “Aharonov-Bohm Effect versus Causality?” Physics Letters, Vol. 111A, 1985, pp. 274-276.

[20] H. Lyre, “Aharonov-Bohm Effect,” In: M. Greenberger, K. Hentschel and F. Weinert, Eds., Compendium of Quan- tum Physics, Springer-Verlag, Berlin, 2009, pp. 1-3. doi:10.1007/978-3-540-70626-7_1

[21] M. Z. Hasan and C. L. Kane, “Colloquium: Topological Insulators,” Reviews of Modern Physics, Vol. 82, No. 4, 2010, pp. 3045-3067. doi:10.1103/RevModPhys.82.3045

[22] A. Shapere and F. Wilczek, “Geometric Phases in Physics,” World Scientific, Singapore, 1989.

[1] Y. Aharonov and D. Bohm, “Significance of Electromagnetic Potentials in the Quantum Theory,” Physical Review, Vol. 115, No. 3, 1959, pp. 485-491. doi:10.1103/PhysRev.115.485

[2] S. Popescu, “Dynamical Quantum Nonlocality,” Nature Physics, Vol. 6, 2010, pp. 151-153. doi:10.1038/nphys1619

[3] M. Peshkin and A. Tonomura, “The Aharonov-Bohm Effect,” Lecture Notes in Physics, Vol. 340, 1989, Part I, Appendix B, Springer-Verlag, Berlin, pp. 27-28.

[4] H. R. Brown and P. R. Holland, “The Galilean Covariance of Quantum Mechanics in the Case of External Fields,” American Journal of Physics, Vol. 67, 1999, pp. 204-214. doi:10.1119/1.19227

[5] K. Moulopoulos, “Nonlocal Phases of Local Quantum Mechanical Wavefunctions in Static and Time-Dependent Aharonov-Bohm Experiments,” Journal of Physics A: Mathematical and Theoretical, Vol. 43, No. 35, 2010, Article ID: 354019, pp. 1-32.

[6] J. Tollaksen, Y. Aharonov, A. Casher, T. Kaufherr and S. Nussimov, “Quantum Interference Experiments, Modular Variables and Weak Measurements,” New Journal of Physics, Vol. 12, 2010, Article ID: 013023, pp. 1-29.

[7] G. P. He, “Flexible Scheme for Measuring Experimentally the Speed of the Response of Quantum States to the Change of the Boundary Condition,” November 2009, pp. 1-6. http://arxiv.org/PS_cache/arxiv/pdf/0907/0907.1974v2.pdf

[8] T. T. Wu and C. N. Yang, “Concept of Nonintegrable Phase Factors and Global Formulation of Gauge Fields,” Physical Review D, Vol. 12, No. 12, 1975, pp. 3845- 3857. doi:10.1103/PhysRevD.12.3845

[9] P. D. Noerdlinger, “Elimination of the Electromagnetic Potentials,” Il Nuovo Cimento, Vol. 23, 1962, pp. 158- 167. doi:10.1007/BF02733550

[10] F. G. Werner and D. R. Brill, “Significance of Electromagnetic Potentials in the Quantum Theory in the Interpretation of Electron Interferometer Fringe Observations,” Physical Review Letters, Vol. 4, No. 7, 1960, pp. 344-347. doi:10.1103/PhysRevLett.4.344

[11] R. P. Feynman, R. B. Leighton and M. Sands, “The Feynman Lectures on Physics,” Addison-Wesley, Vol. II, Chapter 15, 1964, p. 13.

[12] B. Felsager, “Geometry, Particles, and Fields”, Springer- Verlag, Berlin, 1998, p. 55. doi:10.1007/978-1-4612-0631-6

[13] H. Batelaan and A. Tonomura, “The Aharonov-Bohm Effects: Variations on a Subtle Theme,” Physics Today, Vol. 62, No. 9, 2009, pp. 38-43. doi:10.1063/1.3226854

[14] M. P. Silverman, “Quantum Superposition,” Springer- Verlag, Berlin, 2008, pp. 13, 19.

[15] N. G. van Kampen, “Can the Aharonov-Bohm Effect Transmit Signals Faster than Light?” Physics Letters, Vol. 106A, 1984, pp. 5-6.

[16] P. G. Luan and C. S. Tang, “Charged Particle Motion in a Time-Dependent Flux-Driven Ring: An Exactly Solvable Model,” Journal of Physics: Condensed Matter, Vol. 19, 2007, Article ID: 176224, pp. 1-10.

[17] H. Erlichson, “Aharonov-Bohm Quantum Effects on Charged Particles in Field-Free Regions,” American Jour- nal of Physics, Vol. 38, 1970, pp. 162-173. doi:10.1119/1.1976266

[18] R. A. Brown and D. Home, “Locality and Causality in Time-Dependent Aharonov-Bohm Interference,” Il Nuovo Cimento, Vol. 107B, 1992, pp. 303-316. doi:10.1007/BF02728492

[19] T. Troudet, “Aharonov-Bohm Effect versus Causality?” Physics Letters, Vol. 111A, 1985, pp. 274-276.

[20] H. Lyre, “Aharonov-Bohm Effect,” In: M. Greenberger, K. Hentschel and F. Weinert, Eds., Compendium of Quan- tum Physics, Springer-Verlag, Berlin, 2009, pp. 1-3. doi:10.1007/978-3-540-70626-7_1

[21] M. Z. Hasan and C. L. Kane, “Colloquium: Topological Insulators,” Reviews of Modern Physics, Vol. 82, No. 4, 2010, pp. 3045-3067. doi:10.1103/RevModPhys.82.3045

[22] A. Shapere and F. Wilczek, “Geometric Phases in Physics,” World Scientific, Singapore, 1989.