请教jqsphy先生一个问题。-挑战相对论-西陆网

[7楼] 作者:jqsphy 发表时间: 2002/06/01 12:19

这一篇后半部分是我的关于相对论的工作的总结（拓扑对偶荷与它的相对论动力学以及它的引力理论）
Hyperbolical geometric quantum phase and topological dual mass Jian Qi Shen1 and Hong Yi Zhu2 1. Zhejiang Institute of Modern Physics & Department of Physics of Zhejiang University, Hangzhou 310027, P. R. China 2.State Key Laboratory of Modern Optical Instrumentation, Center for Optical and Electromagnetic Research, College of Information Science and Engineering, Zhejiang University, Hangzhou 310027, P. R. China Both geometric phase1,2 of wave function in Quantum Mechanics and gravitomagnetic charge (topological dual charge of mass) in the general theory of relativity reveal Nature’s geometric or global properties. Differing from the dynamical phase, geometric phase is dependent only on the geometric nature of the pathway along which the quantum system evolves3,4. Here we show the existence of the hyperbolical geometric quantum phase that is different from the ordinary trigonometric geometric quantum phase5. Gravitomagnetic charge (dual mass) is the gravitational analogue of magnetic monopole in Electrodynamics6, but, as we will show here, it possesses more interesting and significant features, e.g., it may constitute the dual matter that has different gravitational properties compared with mass. In order to describe the space-time curvature due to the topological dual mass, we construct the dual Einstein’s tensor. Further investigation shows that gravitomagnetic potentials caused by dual mass are respectively analogous to the trigonometric and hyperbolic geometric phase. The study of the geometric phase and dual mass provides a valuable insight into the time evolution of quantum systems and the topological properties in General Relativity. Geometric phase exists in time-dependent quantum systems or systems whose Hamiltonian possesses evolution parameters7,8. As is well known, the dynamical phase of wave function in Quantum Mechanics depends on dynamical quantities such as energy, frequency, coupling coefficients and velocity of a particle or a quantum system, while the geometric phase is immediately independent of these physical quantities. When Berry found that the wave function would give rise to a non-integral phase (Berry’s phase) in quantum adiabatic process1, geometric phase attracts attentions of many physicists in various fields such as gravity theory9-10, differential geometry4, atomic and molecular physics11, nuclear physics12, quantum optics13-15, condensed matter physics16-19 and molecular-reaction chemistry11 as well. In many simple quantum systems such as an electron possessing intrinsic magnetic moment interacting with a time-dependent magnetic field (or a neutron spin interacting with the Earth’s rotation20), a photon propagating inside the curved optical fiber5,15, and the time-dependent Jaynes-Cummings model describing the interaction of the two-level atom with a radiation field21, geometric phase is often proportional to , which equals the solid angle subtended by the curved with respect to the origin of parameter space (Fig. 1). This, therefore, implies that geometric phase differs from dynamical phase and it involves global and topological properties of the time evolution of quantum systems. In addition to this trigonometric geometric phase, there exists the so-called hyperbolical geometric phase that is expressed by with the hyperbolical cosine in some time-dependent quantum systems, e.g., the two-level atomic system with electric dipole-dipole interaction and the harmonic-oscillator system22. It is verified that the generators of the Hamiltonians of these quantum systems form the Lie algebra. Further analysis indicates that quantum systems, which possess the non-compact Lie algebraic structure (whose group parameters can be taken to be infinity) will present the hyperbolical geometric phase, while quantum systems with compact Lie algebraic structure will give rise to the trigonometric geometric phase. Since Lorentz group in the special theory of relativity is also a non-compact group, this leads us to consider the topological properties associated with space-time. We take into account the gravitational analogue of magnetic charge6, i.e., gravitomagnetic charge that is the source of gravitomagnetic field just as the case that mass (gravitoelectric charge) is the source of gravitoelectric field (i. e., Newtonian gravitational field in the sense of weak-approximation). In this sense, gravitomagnetic charge is also termed dual mass. It should be noted that the concept of the ordinary mass is of no physical significance for the gravitomagnetic charge; it is of interest to investigate the relativistic dynamics and gravitational effects as well as geometric properties of this topological dual mass (should such exist). From the point of view of differential geometry, matter may be classified into two categories: gravitoelectric matter and gravitomagnetic matter. The former category possesses mass and constitutes the familiar physical world, while the latter possesses dual mass that would cause the non-analytical property of space-time metric. Einstein’s field equation of gravitation in general theory of relativity governs the couplings of gravitoelectric matter (which possesses mass) to gravity (space-time); accordingly, we should have a field equation governing the interaction of dual matter with gravity. By making use of the variational principle, the gravitational field equation of gravitomagnetic matter can be obtained where the dual Einstein’s tensor is denoted by with and being the four-dimensional Levi-Civita completely antisymmetric tensor and the Riemann curvature tensor that describes the space-time curvature, respectively. By exactly solving this field equation, one can show that the topological property of the solution can be illustrated in Figure 2. Such property is in analogy with that of the geometric quantum phase in the time-dependent spin-gravity coupling (i.e., the interaction between a spinning particle with gravitimagnetic field20) and other quantum adiabatic processes. The topological properties of gravitomagnetic charge (dual mass) may be shown in terms of the global features of geometric quantum phase (Table 1). It follows from Figure 2 that the expression of gravitomagnetic potential, , due to dual mass is exactly analogous to that of the trigonometric geometric phase. In the similar fashion, it is readily verified that the gravitomagnetic potential, , is similar to that of the hyperbolical geometric phase. This feature originates from the fact that Lorentz group is a non-compact group. Although there is no evidence for the existence of this topological dual mass at present, it is still essential to consider this topological or global phenomenon in General Relativity. It is believed that there would exist formation (or creation) mechanism of gravitomagnetic charge in the gravitational interaction, just as some prevalent theories provide the theoretical mechanism of existence of magnetic monopole in various gauge interactions23,24. Magnetic monopole in electrodynamics and gauge field theory has been discussed and sought after for decades, and the existence of the ’t Hooft-Polyakov monopole solution23 has spurred new interest of both theorists and experimentalists24-26. As the topological gravitomagnetic charge in the curved space-time, dual mass is believed to give rise to such interesting situation similar to that of magnetic monopole. If it is indeed present in universe, dual mass will also lead to significant consequences in astrophysics and cosmology. We emphasize that although the gravitomagnetic vector potential produced by the gravitomagnetic charge is the classical solution to the field equation, this kind of topological gravitomagnetic monopoles may arise not as fundamental entities in gravity theory, e.g., it will behave like a topological soliton. Gravitomagnetic charge has some interesting relativistic quantum gravitational effects20,27, e.g., the gravitational Meissner effect, which may serve as an interpretation of the smallness of the observed cosmological constant. In accordance with quantum field theory, vacuum possesses infinite zero-point energy density due to the vacuum quantum fluctuations; whereas according to Einstein’s theory of General Relativity, infinite vacuum energy density yields the divergent curvature of space-time, namely, the space-time of vacuum is extremely curved. Apparently it is in contradiction with the practical fact, since it follows from experimental observations that the space-time of vacuum is asymptotically flat. In the context of quantum field theory a cosmological constant corresponds to the energy density associated with the vacuum and then the divergent cosmological constant may result from the infinite energy density of vacuum quantum fluctuations. However, a diverse set of observations suggests that the universe possesses a nonzero but very small cosmological constant28-31. How to give a natural interpretation for the above paradox? Here, provided that vacuum matter is perfect fluid, which leads to the formal similarities between the weak-gravity equation in perfect fluid and the London’s electrodynamics of superconductivity, we suggest a potential explanation by using the canceling mechanism via gravitational Meissner effect: the gravitoelectric field (Newtonian field of gravity) produced by the gravitoelectric charge (mass) of the vacuum quantum fluctuations is exactly canceled by the gravitoelectric field due to the induced current of the gravitomagnetic charge of the vacuum quantum fluctuations; the gravitomagnetic field produced by the gravitomagnetic charge (dual mass) of the vacuum quantum fluctuations is exactly canceled by the gravitomagnetic field due to the induced current of the gravitoelectric charge (mass current) of the vacuum quantum fluctuations. Thus, at least in the framework of weak-field approximation, the extreme space-time curvature of vacuum caused by the large amount of the vacuum energy does not arise, and the gravitational effects of cosmological constant is eliminated by the contributions of the gravitomagnetic charge (dual mass). If gravitational Meissner effect is of really physical significance, then it is necessary to apply this effect to the early universe where quantum and inflationary cosmologies dominate the evolution of the universe. Study of the geometric property in quantum regimes is an interesting and valuable direction. Since it reveals the global and topological properties of evolution of quantum systems, geometric phase has many applications in various branches of physics, say, in the coupling of neutron spin to the Earth’s rotation27,32, a potential application may be suggested where the information on the Earth’s variations of rotating frequency will be obtained by measuring the geometric phase of the oppositely polarized neutrons through the neutron-gravity interferometer experiment20. The topological charge in curved space-time also deserves further investigation, since it reflects plentiful global or geometric properties hidden in the gravity theory. It is believed that both theoretical and experimental interest in this direction may enables people to understand the global phenomena of the physical world. 1. Berry, M.V. Quantal phase factor accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45-57 (1984). 2. Berry, M.V. Interpreting the anholonomy of coiled light. Nature 326, 277-278 (1987). 3. Robinson, L. An optical measurement of Berry’s phase. Science 234, 424-426 (1986). 4. Simon, B. Holonomy, the quantum adiabatic theorem, and Berry’s phase. Phys. Rev. Lett. 51, 2167-2170 (1983). 5. Chiao, R. & Wu, Y. S. Manifestations of Berry’s phase for the photon. Phys. Rev. Lett. 57, 933-936 (1986). 6. Dirac, P. A. M. Quantized singularities in the electromagnetic field. Proc. R. Soc. Lond. A 133, 60-71 (1931). 7. Gao, X. C., Xu, J. B. & Qian, T. Z. Geometric phase and the generalized invariant formulation. Phys. Rev. A 44, 7016-7021 (1991). 8. Yang, L. G. & Yan, F. L. The area theorem of the Berry’s phase for the time-dependent externally driven system. Phys. Lett. A 265, 326-330 (2000). 9. Furtado, C. & Bezerra, V. B. Gravitational Berry’s quantum phase. Phys. Rev. D 62, 045003-(1-5) (2000). 10. Shen, J. Q., Zhu, H. Y. & Li, J. Exact solutions to the interaction between neutron spin and gravitation by using invariant theory. Acta Phys. Sini. 50, 1884-1887 (2001). 11. Wu, Y. S. M. & Kuppermann, A. Prediction of the effect of the geometric phase on product rotational state distributions and integral cross sections. Chem. Phys. Lett. 201, 178-186 (1993) and references therein. 12. Wagh, A. G. et al. Neutron polarimetric separation of geometric and dynamical phases. Phys. Lett. A 268, 209-216 (2000). 13. Sanders, B. C. et al. Geometric phase of three-level systems in interferometry. Phys. Rev. Lett. 86, 369-372 (2001). 14. Gong, L. F., Li, Q. & Chen, Y. L. Nonlinear behavior of geometric phases in transformations. Phys. Lett. A 251, 387-393 (1999). 15. Shen, J. Q., Zhu, H. Y. & Shi, S. L. Polarization of single-mode photon propagating inside the fiber and its geometric phase factor. Acta Phys. Sini. 51, 536-540 (2002). 16. Exner, P. & Geyler, V. A. Berry phase for a potential well transported in a homogeneous magnetic field. Phys. Lett. A 276, 16-18 (2000). 17. Yan. F. L., Yang, L. G. & Li, B. Z. Formal exact solution for the Heisenberg spin system in a time-dependent magnetic field and Aharonov-Anandan phase. Phys. Lett. A 251, 289-293 (1999). 18. Taguchi, Y. et al. Spin chirality, Berry phase, and Anomalous Hall effect in a frustrated ferromagnet. Science 291, 2573-2576 (2001). 19. Falci, G. et al. Detection of geometric phases in superconducting nanocircuits. Nature 407, 355-358 (2000). 20. Shen, J. Q., Zhu, H. Y. & Shi, S. L. Gravitomagnetic field and time-dependent spin-rotation coupling. Phys. Scr. 65, 465-468 (2002). 21. Shen, J. Q., Zhu, H. Y. & Mao, H. An approach to exact solutions of the time-dependent supersymmetric two-level three-photon Jaynes-Cummings model. J. Phys. Soc. Jpn. 71, 1205-1209 (2002). 22. Shen, J. Q., Zhu, H. Y. & Chen, P. Exact solutions of time-dependent three-generator systems. quant-ph/0205170 . 23. Hooft G. ’t. Magnetic monopoles in unified gauge theories. Nucl. Phys. B 79, 276-284 (1974). 24. Craigie, N. S. et al. Monopoles in Quantum Field theory: Proceedings of the Monopole meeting (World Scientific Publishing, Singapore, 1982). 25. Barriola, M. & Vilenkin, A. Gravitational field of a global monopole. Phys. Rev. Lett. 63, 341-343 (1989). 26. Chakraborty, S. Motion of test particles around monopoles. Gen. Rel. Grav. 28, 1115-1119 (1996). 27. Mashhoon, B. gravitational couplings of intrinsic spin. Class. Quant. Grav. 17, 2399-2410 (2000). 28. Weinberg, S. The cosmological constant problem. Rev. Mod. Phys. 61, 1-23 (1989). 29. Datta, D. P. Can vacuum energy gravitate? Gen. Rel. Grav. 27, 341-346 (1995). 30. Krauss, L. M. & Turner, M. The cosmological constant is back. Gen. Rel. Grav. 27, 1137-1144 (1995). 31. Kakushadze, Z. Why the cosmological constant problem is hard. Phys. Lett. B 488, 402-409 (2000). 32. Mashhoon, B. On the spin-rotation-gravity coupling. Gen. Rel. Grav. 31, 681-691 (1999).

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