This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
This form is strictly speaking only correct when neglecting repetitions of periodic orbits, for an expression taking them into account, see [8]. The contributions of repeated pairs are however damped exponentially like \(\hbox{e}^{-\lambda T/2}\) with their overall duration \(T\) compared to ones from pseudo orbits traversed once, as the number is the same in both cases, however the stability factors of the repeated ones are reduced. We can thus neglect them anyway when calculating the contributions of pseudo-orbit pairs.
- 3.
An improved version considers a smooth classically small cut off at \(T_H/2\).
- 4.
As we are considering positive as well as negative action differences in the \(s,u\)-integrals, the sign change of the actions is not important.
- 5.
The \(T_0\) could have been introduced always when evaluating sums over orbits by sum rules, but in all cases considered up to now the semiclassical limit \(\hbar\rightarrow 0\) and the limit \(T_0\rightarrow 0\) commuted. This is not the case here, the resulting integrals do not exist for \(T_0=0.\)
- 6.
As can be easily checked it is not possible to construct diagrams with a larger number of links where the number of links and of \((s,u)\)-coordinates needed to characterise the encounter configuration differs by one.
References
Heusler, S., Müller, S., Altland, A., Braun, P., Haake, F.: Phys. Rev. Lett. 98, 044103 (2007). arXiv:0610053 (2006)
Keating, J.P., Müller, S.: Proc. R. Soc. Lond. A 463, 3241 (2007)
Waltner, D., Heusler, S., Urbina, J.-D., Richter, K.: J. Phys. A 42, 292001(F) (2009)
Huber, H.: Math. Ann. 138, 1 (1959)
Voros, A.: Commun. Math. Phys. 110, 439 (1987)
Steiner, F.: Phys. Lett. B 188, 447 (1987)
Voros, A.: J. Phys. A 21, 685 (1988)
Berry, M.V., Keating, J.P.: J. Phys. A 23, 4839 (1990)
Georgeot, B., Prange, R.: Phys. Rev. Lett. 74, 2851 (1995)
Sieber, M., Steiner, F.: Phys. Rev. Lett. 67, 1941 (1991)
Müller, S., Heusler, S., Altland, A., Braun, P., Haake, F.: New J. Phys. 11, 103025 (2009)
Müller, S., Heusler, S., Braun, P., Haake, F., Altland, A.: Phys. Rev. E 72, 046207 (2005)
Müller, S.: Periodic-Orbit Approach to Universality in Quantum Chaos. Ph.D. thesis, Universität Duisburg Essen (2005)
Kettemann, S., Klakow, D., Smilansky, U.: J. Phys. A 30, 3643 (1997)
Waltner, D., Gutiérrez, M., Urbina, J.-D., Richter, K.: Unpublished (2009)
Heusler, S.: Private communication (2009)
Efetov, K.B.: Adv. Phys. 32, 53 (1983)
Edwards, S.F., Anderson, P.W.: J. Phys. F 5, 965 (1975)
Veerbaarschot, J.J.M., Zirnbauer, M.: J. Phys. A 18, 1093 (1985)
Turek, M., Richter, K.: J. Phys. A 36, L455 (2003)
Turek, M.: Semiclassics beyond the diagonal approximation. Ph.D. thesis, Universität Regensburg (2004)
Heusler, S., Müller, S., Braun, P., Haake, F.: Phys. Rev. Lett. 96, 066804 (2006)
Müller, S., Heusler, S., Braun, P., Haake, F.: New J. Phys. 9, 12 (2007)
Richter, K.: Europhys. Letters 29, 7 (1995)
Kuipers, J., Sieber, M.: Phys. Rev. E 77, 046219 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Waltner, D. (2012). Semiclassical Analogues to Field-Theoretical Effects. In: Semiclassical Approach to Mesoscopic Systems. Springer Tracts in Modern Physics, vol 245. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24528-2_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-24528-2_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24527-5
Online ISBN: 978-3-642-24528-2
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)