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The Path Integral on the Poincaré Disc, the Poincaré Upper Half‐Plane and on the Hyperbolic Strip
Author(s) -
Grosche Cristian
Publication year - 1990
Publication title -
fortschritte der physik/progress of physics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.469
H-Index - 71
eISSN - 1521-3978
pISSN - 0015-8208
DOI - 10.1002/prop.2190380704
Subject(s) - path integral formulation , mathematics , feynman diagram , mathematical analysis , hyperbolic geometry , upper half plane , plane (geometry) , complex plane , poincaré conjecture , mathematical physics , fourier transform , volume integral , path (computing) , line integral , physics , quantum mechanics , quantum , integral equation , geometry , differential geometry , computer science , programming language
In this paper rigorous path integral treatments are presented of free motion on the Poincaré disc, the Poincaré upper half‐plane and on the hyperbolic strip, three spaces which are analytically equivalent to each other. Whereas the path integral treatments on the disc and on the strip are new, two further path integral treatments are discussed for the Poincaré upper half‐plane to the existing one. All the calculations are mainly based on Fourier‐expansions of the Feynman kernels which can be easily performed. The remaining path integrals on D, U and S can be reduced to the path integral problems on the pseudosphere Δ 2 , Liouville quantum mechanics and the modified Pöschl‐Teller potential problem, respectively, where the results by means of path integrals are known. The corresponding normalized wave‐functions and the energy‐spectrum are derived. The energy‐spectra are the same in all three spaces and read E = (1/2 m ) ( p 2 + 1/4) ( p ‐momentum). The “zero‐point” energy E 0 = 1/8 m is discussed which can be interpreted in terms of the Heisenberg uncertainty relation. The equivalences between the Feynman kernels on Δ 2 , D, U and S are also discussed.