Exactly solvable path integral for open cavities in terms of quasinormal modes

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 61, Issue 3, 2000, Pages 2367-2375

  van den Brink, Alec Maassen ^a  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

[No Author keywords available]

Indexed keywords

PHASE SPACE METHODS; QUANTUM THEORY;

CANONICAL QUANTIZATION; EIGENSTATES; EVOLUTION OPERATOR; OPEN CAVITY; PATH INTEGRAL; PHASE SPACES; QUASINORMAL MODES; SCALAR FIELDS;

HIGH ENERGY PHYSICS;

EID: 0007264584     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.61.2367     Document Type: Article

References (35)

1. R. Lang, M.O. Scully, and W.E. Lamb, Phys. Rev. A 7, 1788 (1973)
2. R. Lang and M.O. Scully, Opt. Commun. 9, 331 (1973)
3. Opt. Commun.H. Dekker, 10, 114 (1974)
4. H. DekkerPhysica C 83, 183 (1976)
5. A.J. Campillo, J.D. Eversole, and H.-B. Lin, Phys. Rev. Lett. 67, 437 (1991).
6. P. Meystre and M. Sargent III, Elements of Quantum Optics (Springer, Berlin, 1991).
7. G. Schön and A.D. Zaikin, Phys. Rep. 198, 237 (1990).
8. U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 1993).
9. S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, Oxford, 1983).
10. E.S.C. Ching, P.T. Leung, W.M. Suen, and K. Young, Phys. Rev. D 54, 3778 (1996).
11. H. Dekker, Phys. Rep. 80, 1 (1981).
12. P.T. Leung, A. Maassen van den Brink, and K. Young, in Frontiers in Quantum Physics, proceedings of the International Conference, Kuala Lumpur, edited by S. C. Lim, R. Abd-Shukor, and K. H. Kwek (Springer, Singapore, 1998), p. 214.
13. K.C. Ho, P.T. Leung, A. Maassen van den Brink, and K. Young, Phys. Rev. E 58, 2965 (1998).
14. E.S.C. Ching, P.T. Leung, A. Maassen van den Brink, W.M. Suen, S.S. Tong, and K. Young, Rev. Mod. Phys. 70, 1545 (1998).
15. R.W.F. van der Plank and L.G. Suttorp, Phys. Rev. A 53, 1791 (1996).
16. R.P. Feynman and F.L. Vernon, Ann. Phys. (N.Y.) 24, 118 (1963)
17. Ann. Phys. (N.Y.)A.O. Caldeira and A.J. Leggett, 149, 374 (1983).
18. Since (Formula presented) limits the singularity of (Formula presented) to at most a (Formula presented) function, (Formula presented) itself is continuous. Stronger singularities would also leave equations such as Eq. (2.4) undefined distributionally.
19. A. Maassen van den Brink and K. Young (unpublished).
20. A. Maassen van den Brink and K. Young, e-print math-ph/9905019.
21. Since the Euclidean phase-space action does not contain a term (Formula presented) in the absence of mass regularization, there is no Gaussian (Formula presented) cutoff and the source term should be purely imaginary if the path integral is to have a fighting chance of being definable. Due to the overall factor i in the bilinear map, the choice in Eq. (3.2) satisfies this condition for real (Formula presented).
22. L.S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).
23. G. Roepstorff, Path Integral Approach to Quantum Physics (Springer, Berlin, 1994).
24. When including interactions as contemplated in Sec. VI, phase-space integration should not lead to problems, since at least in the perturbative regime the interacting path integral merely is a formal tool for arriving at the diagram expansion.
25. E.g., H. Dekker, Phys. Lett. 104A, 72 (1984).
26. The QNM expansion is thus applied to all two-component fields, since the path integral is not restricted to only outgoing ones, see below Eq. (3.4). For a justification, see Refs. 9 (where one faces the analogous issue for operator fields) and 14.
27. However, from this one may not incorrectly conclude that the effective action vanishes in the conservative limit. Rather, if (Formula presented) also (Formula presented) in the action of Eq. (4.2); in the limit the diagonal contribution of these terms only yields the action of the closed cavity in terms of its NMs. See also Ref. 9, the end of Sec. VII, and Appendix A.
28. A subtlety in Eq. (4.3) is that (Formula presented) in general, making the integration in (Formula presented) space b dependent. However, one can verify (most systematically by splitting the (Formula presented) integrals into real and imaginary parts and subsequently using contour methods) that this does not affect the result: the full b dependence in Eq. (4.3) is the one indicated explicitly.
29. With the standard shorthand (Formula presented) (and (Formula presented), one can use the (Formula presented) obeying (Formula presented), formally as if they were independent variables.
30. L.H. Ryder, Quantum Field Theory (Cambridge University Press, Cambridge, 1986).
31. A.A. Abrikosov, L.P. Gor’kov, and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1975).
32. K.C. Ho, P.T. Leung, A. Maassen van den Brink, and K. Young, in Proceedings of the APPC7 Conference, edited by H.S. Chen (Science Press, Beijing, 1999), p. 433.
33. To streamline a lengthy argument in Ref. 9, set (Formula presented) as in Eq. (6.6) (all cross references are in Ref. 9). A key point of Ref. 9 is that (Formula presented) has a QNM expansion (2.8) just as its classical counterpart, so (Formula presented). Equations. (6.2), (6.3), and (6.7) now at once yield Eq. (6.8) for (Formula presented) While this calculation no longer uses Appendix C, the tensor expansion presented there remains useful for reference.
34. The (Formula presented) here and G of Ref. 15 have opposite signs, since the usual sign for the quantum propagator is not convenient in classical field theory. The signs of (Formula presented) also differ.
35. P.T. Leung, S.Y. Liu, and K. Young, Phys. Rev. A 49, 3057 (1994).