Abstract
The geometro-stochastic quantization of a gauge theory based on the (4,1)-de Sitter group is presented. The theory contains an intrinsic elementary length parameter R of geometric origin taken to be of a size typical for hadron physics. Use is made of a soldered Hilbert bundle ℋ over curved spacetime carrying a phase space representation of SO(4, 1) with the Lorentz subgroup related to a vierbein formulation of gravitation. The typical fiber of ℋ is a resolution kernel Hilbert space ℋ\(_{\bar \eta }^{(\rho )} \) constructed in terms of generalized coherent states\(\bar \eta \) ρ related to the principal series of unitary irreducible representations of SO(4, 1), namely de Sitter horospherical waves for spinless particles characterized by the parameter ρ. The framework is, finally, extended to a quantum field-theoretical formalism by using bundles with Fock space fibers constructed from ℋ\(_{\bar \eta }^{(\rho )} \).
References
W. Heisenberg,Phys. Today 29(3), 32 (1976).
W. Drechsler,Fortschr. Phys. 32, 449 (1984).
W. Drechsler,J. Math. Phys. 26, 41 (1985).
W. Drechsler and W. Thacker,Class. Quantum Gravit. 4, 291 (1987).
W. Drechsler,Class. Quantum Gravit. 6, 623 (1989).
W. Drechsler,Found. Phys. 19, 1479 (1989).
E. Prugovečki,Stochastic Quantum Mechanics and Quantum Spacetime (Reidel, Dordrecht, 1984; corr. printing 1986).
E. Prugovečki,Nuovo Cimento A 97, 597, 837 (1987);100, 827 (1988);101, 853 (1989);102, 881 (1989).
E. Prugovečki,Class. Quantum Gravit. 4, 1659 (1987).
E. Prugovečki,Found. Phys. Lett. 2, 81, 163, 403 (1989).
E. Prugovečki and S. Warlow,Found. Phys. Lett. 2, 409 (1989);Rep. Math. Phys. 28, 105 (1989).
W. Drechsler,Fortschr. Phys. 23, 607 (1976).
W. Drechsler,Found. Phys. 7, 629 (1977).
W. Drechsler and R. Sasaki,Nuovo Cimento 46, 527 (1978).
E. Prugovečki,Found. Phys. 21, 93 (1991).
E. P. Wigner, inQuantum Theory and Measurement, J. A. Wheeler and H. Zurek, eds. (Princeton University Press, Princeton, 1983), pp. 260–314.
E. H. Kronheimer and R. Penrose,Proc. Cambridge Philos. Soc. 63, 481 (1967).
J. Ehlers, E. A. E. Pirani, and A. Schild, inGeneral Relativity, L. O'Raifeartaigh, ed. (Clarendon Press, Oxford, 1972).
J. Ehlers and A. Schild,Commun. Math.Phys. 32, 119 (1973).
N. M. H. Woodhouse,J. Math. Phys. 14, 495 (1973).
J. Ehlers, inGeneral Relativity and Cosmology, B. K. Sachs, ed. (Academic Press, New York, 1973).
A. Einstein,Geometrie und Erfahrung; English translation inReadings on the Philosophy of Science, H. Feigl and M. Brodbeck, eds. (Appleton-Century-Crofts, New York, 1953), pp. 189–194.
H. Salecker and E. P. Wigner,Phys. Rev. 109, 571 (1958).
W. Drechsler,Fortschr. Phys. 38, 63 (1990).
M. Born,Rev. Mod. Phys. 21, 463 (1949).
R. R. Aldingeret al., Phys. Rev. D. 28, 3020 (1983).
A. Böhmet al., Int. J. Mod. Phys. 3, 1103 (1988).
I. M. Gel'fand, M. I. Graev, and N. Ya. Vilenkin,Generalized Functions, Vol. 5 (Academic Press, London, 1966), Chap. V.
S. Helgason,Lie Groups and Symmetric Spaces, in Battelle Rencontres, C. M. De Witt and J. A. Wheeler, eds. (Benjamin, New York, 1968).
S. Helgason,Topics in Harmonic Analysis on Homogeneous Spaces (Birkhäuser, Basel, 1981).
W. Drechsler and E. M. Mayer,Fiber Bundle Techniques in Gauge Theories (Lecture Notes in Physics, Vol. 67) (Springer, Heidelberg, 1977).
A. Perelomov,Generalized Coherent States and Their Applications (Springer, Heidelberg, 1986).
W. Rossmann,J. Funct. Anal. 30, 448 (1978).
R. S. Strichartz,J. Funct. Anal. 12, 341 (1972).
W. Drechsler, inGroup Theoretical Methods in Physics (Proceedings, Austin, 1978), W. Beiglböck, A. Böhm, and E. Takasugi eds. (Lecture Notes in Physics, Vol. 94) (Springer, Heidelberg, 1979), p. 98.
E. Prugovečki,Quantum Geometry (Kluwer, Dordrecht, to appear).
S. Kobayashi,Can. J. Math. 8, 145 (1956).
Ch. Ehresmann,Colloque de Topologie (espaces fibrés), Bruxelles, 1950, p. 29.
H. P. Künzle and C. Duval,Class. Quantum Gravit. 3, 957 (1986).
S. Kobayashi and K. Nomizu,Foundations of Differential Geometry, Vol. I (Wiley, New York, 1963).
S. T. Ali and E. Prugovečki,Acta Appl. Math. 6, 1 (1986).
J. A. Brooke and E. Prugovečki,Nuovo Cimento A 89, 237 (1984).
J. A. Wheeler,Geometrodynamics (Academic Press, New York, 1962).
C. W. Misner, K. S. Thorne, and J. A. Wheeler,Gravitation (Freeman, San Francisco, 1973).
N. N. Bogolubov, A. A. Logunov, and I. T. Todorov,Introduction to Axiomatic Quantum Field Theory (Benjamin, Reading, Massachusetts, 1975).
F. A. Berezin,The Method of Second Quantization (Academic, New York, 1966).
R. P. Feynman and A. R. Hibbs,Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
C. Itzykson and J.-B. Zuber,Quantum Field Theory (McGraw-Hill, New York, 1980).
N. A. Chernikov and E. A. Tagirov,Ann. Inst. H. Poincaré A 9, 109 (1968).
Author information
Authors and Affiliations
Additional information
Supported in part by NSERC Research Grant No. A5206.
Rights and permissions
About this article
Cite this article
Drechsler, W., Prugovečki, E. Geometro-stochastic quantization of a theory for extended elementary objects. Found Phys 21, 513–546 (1991). https://doi.org/10.1007/BF00733257
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00733257