Advertisement

Communications in Mathematical Physics

, Volume 98, Issue 3, pp 369–390 | Cite as

Algebraic quantization of systems with a gauge degeneracy

  • Hendrik B. G. S. Grundling
  • C. A. Hurst
Article

Abstract

Systems with a gauge degeneracy are characterized either by supplementary conditions, or by a set of generators of gauge transformations, or by a set of constraints deriving from Dirac's canonical constraint method. These constraints can be expressed either as conditions on the field algebra ℱ, or on the states on ℱ. In aC*-algebra framework, we show that the state conditions give rise to a factor algebra of a subalgebra of the field algebra ℱ. This factor algebra, ℛ, is free of state conditions. In this formulation we show also that the algebraic conditions can be treated in the same way as the state conditions. The connection between states on ℱ and states on ℛ is investigated further within this framework, as is also the set of transformations which are compatible with the set of constraints. It is also shown that not every set of constraints can give rise to a nontrivial system. Finally as an example, the abstract theory is applied to the electromagnetic field, and this treatment can be generalized to all systems of bosons with linear constraints. The question of dynamics is not discussed.

Keywords

Neural Network Complex System State Condition Electromagnetic Field Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Sagan, H.: Introduction to the calculus of variations. New York: McGraw-Hill 1969Google Scholar
  2. 1a.
    Ewing, G.M.: Calculus of variations with applications. New York: Norton 1969Google Scholar
  3. 1b.
    Choquet-Bruhat, Y., De Witt-Morette, C., Dillard-Bleick, M.: Analysis, manifolds and physics. Amsterdam: North-Holland 1982Google Scholar
  4. 2.
    Dirac, P.A.M.: Lectures on quantum mechanics. New York: Belfer Graduate School of Science, Yeshiva University 1964Google Scholar
  5. 2a.
    Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math.2, 129–148 (1950)Google Scholar
  6. 2b.
    Dirac, P.A.M.: The Hamiltonian form of field dynamics. Can. J. Math.3, 1–23 (1951)Google Scholar
  7. 3.
    Kundt, W.: Canonical quantization of gauge invariant field theories. Springer Tracts in Modern Physics, Vol. 40, pp. 107–168. Berlin, Heidelberg, New York: Springer 1966Google Scholar
  8. 4.
    Sundermeyer, K.: Constrained dynamics. In: Lecture Notes in Physics, Vol. 169. Berlin, Heidelberg, New York, Tokyo: Springer 1982Google Scholar
  9. 5.
    Hanson, A.J., Regge, T., Teitelboim, C.: Constrained Hamiltonian systems. Rome: Accademia Nazionale dei Lincei 1976Google Scholar
  10. 6.
    Sudarshan, E.C.G., Mukunda, N.: Classical dynamics: a modern perspective. New York: Wiley 1974Google Scholar
  11. 7.
    Shanmugadhasan, S.: Canonical formalism for degenerate Lagrangians. J. Math. Phys.14, 677–687 (1975)Google Scholar
  12. 8.
    Gotay, M.J., Nester, J.M., Hinds, G.: Presymplectic manifolds and the Dirac-Bergman theory of constraints. J. Math. Phys.19, 2388–2399 (1978)Google Scholar
  13. 9.
    Fadeev, L.D.: The Feynman integral for singular Lagrangians. Theor. Math. Phys.1, 1–12 (1970)Google Scholar
  14. 9a.
    Kerler, W.: Quantum treatment of constrained systems and implications for path integrals. Phys. Lett.76 B, 423–427 (1978)Google Scholar
  15. 10.
    Bergmann, P.G., Goldberg, I.: Dirac bracket transformations in phase space. Phys. Rev.98, 531–538 (1955)Google Scholar
  16. 11.
    Hermann, R.: Lie algebras and quantum mechanics. New York: Benjamin 1970Google Scholar
  17. 11a.
    Boisseau, B., Barrabas, C.: Quantization of the Liouville mechanics for systems with singular Lagrangians. J. Math. Phys.19, 1032–1036 (1978)Google Scholar
  18. 11b.
    Castellani, L., Dominici, D., Longhi, G.: Canonical transformations and quantization of singular Lagrangian systems. Nuovo Cimento48 A, 91–99 (1978)Google Scholar
  19. 12.
    Källen, G.: Quantum electrodynamics. London: Allan and Unwin 1972Google Scholar
  20. 13.
    Chernoff, P.R.: Mathematical obstructions to quantization. Had. J.4, 879–898 (1981)Google Scholar
  21. 14.
    Emch, G.G.: Algebraic methods in statistical mechanics and quantum field theory. New York: Wiley 1972Google Scholar
  22. 15.
    Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. Berlin, Heidelberg, New York: Springer 1979Google Scholar
  23. 16.
    Carey, A.L., Gaffney, J.M., Hurst, C.A.: AC*-algebra formulation of the quantization of the electromagnetic field. J. Math. Phys.18, 629–640 (1977)Google Scholar
  24. 17.
    Pedersen, G.K.:C*-algebras and their automorphism groups. London: Academic Press 1979Google Scholar
  25. 18.
    Dixmier, J.:C*-algebras. Amsterdam: North-Holland 1977Google Scholar
  26. 19.
    Naimark, M.A.: Normed algebras. Groningen: Wolters Noordhoff 1972Google Scholar
  27. 20.
    Kapuscik, E., Uzes, C.A.: Dirac bracket quantization and central force systems. Am. J. Phys.50, 1094–1097 (1982)Google Scholar
  28. 21.
    Kálnay, A.J.: On certain intriguing physical, mathematical and logical aspects concerning quantization. Had. J.4, 1127–1165 (1981)Google Scholar
  29. 22.
    Manuceau, J.:C*-algebre de relations de commutation. Ann. Inst. Henri Poincaré8, 139–161 (1968)Google Scholar
  30. 23.
    Segal, I.E.: Mathematical problems of relativistic physics. Providence, R.I.: American Mathematical Society 1963Google Scholar
  31. 24.
    Gotay, M.J.: On the validity of Dirac's conjecture regarding first class secondary constraints. J. Phys. A16, L 141-L 145 (1983)Google Scholar
  32. 25.
    Sullivan, D., Maitland Wright, J.D.: On lifting automorphisms. Proc. Symp. Pure Math. Am. Math. Soc., Vol. 38. Part 2, pp. 289–290 (1982)Google Scholar
  33. 25a.
    Sullivan, D., Maitland Wright, J.D.: On lifting automorphisms of monotoneσ-completeC*-algebras. Q. J. Math. Oxford (2),32, 371–381 (1981)Google Scholar
  34. 26.
    Jakobczyk, L.: Canonical quantization with indefinite inner product. Preprint No. 592, Institute of Theoretical Physics, University of Wroclaw, Wroclaw, Poland (1983)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Hendrik B. G. S. Grundling
    • 1
  • C. A. Hurst
    • 1
  1. 1.Department of Mathematical PhysicsUniversity of Adelaide

Personalised recommendations