Communications in Mathematical Physics

, Volume 318, Issue 3, pp 717–766 | Cite as

QCD on an Infinite Lattice

Article

Abstract

We construct a mathematically well–defined framework for the kinematics of Hamiltonian QCD on an infinite lattice in \({\mathbb{R}^3}\) , and it is done in a C*-algebraic context. This is based on the finite lattice model for Hamiltonian QCD developed by Kijowski, Rudolph e.a.. To extend this model to an infinite lattice, we need to take an infinite tensor product of nonunital C*-algebras, which is a nonstandard situation. We use a recent construction for such situations, developed by Grundling and Neeb. Once the field C*-algebra is constructed for the fermions and gauge bosons, we define local and global gauge transformations, and identify the Gauss law constraint. The full field algebra is the crossed product of the previous one with the local gauge transformations. The rest of the paper is concerned with enforcing the Gauss law constraint to obtain the C*-algebra of quantum observables. For this, we use the method of enforcing quantum constraints developed by Grundling and Hurst. In particular, the natural inductive limit structure of the field algebra is a central component of the analysis, and the constraint system defined by the Gauss law constraint is a system of local constraints in the sense of Grundling and Lledo. Using the techniques developed in that area, we solve the full constraint system by first solving the finite (local) systems and then combining the results appropriately. We do not consider dynamics.

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References

  1. 1.
    Akemann C.A., Pedersen G.K., Tomiyama J.: Multipliers of C*-algebras. J. Funct. Anal. 13, 277–301 (1973)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Blackadar, B.: Operator Algebras. Berlin Heidelberg-New York: Springer, 2006Google Scholar
  3. 3.
    Blackadar B.: Infinite tensor products of C*-algebras, Pac. J. Math. 77, 313–334 (1977)MathSciNetGoogle Scholar
  4. 4.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics 1. New York: Springer, 1987Google Scholar
  5. 5.
    Carey A.L., Ruijsenaars S.N.M.: On fermion gauge groups, current algebras and Kac–Moody algebras. Acta Appl. Math. 10, 1–86 (1987)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Costello, P.: The mathematics of the BRST-constraint method. PhD thesis Univ. of New South Wales, 2008, http://arxiv.org/abs/0905.3570v2 [math.OA], 2009
  7. 7.
    Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton NJ: Princeton University Press, 1992Google Scholar
  8. 8.
    Landsman N.P.: Rieffel induction as generalised quantum Marsden–Weinstein reduction. J. Geom. Phys. 15, 285–319 (1995)MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    Giulini D., Marolf D.: On the generality of refined algebraic quantization. Class. Quant. Grav. 16, 2479–2488 (1999)MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Klauder J.: Coherent state quantization of constraint systems. Ann. Physics 254, 419–453 (1997)MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    Faddeev L., Jackiw R.: Hamiltonian reduction of unconstrained and constrained systems. Phys. Rev. Lett. 60, 1692 (1988)MathSciNetADSMATHCrossRefGoogle Scholar
  12. 12.
    Creutz, M.: Quarks, gluons and lattices. Cambridge: Cambridge University Press, 1983Google Scholar
  13. 13.
    Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science: Yeshiva University, 1964Google Scholar
  14. 14.
    Glöckner H.: Direct limit Lie groups and manifolds. J. Math. Kyoto Univ. 43, 1–26 (2003)MATHGoogle Scholar
  15. 15.
    Grundling H., Neeb K-H.: Full regularity for a C*-algebra of the Canonical Commutation Relations, Rev. Math. Phys. 21, 587–613 (2009)MathSciNetMATHGoogle Scholar
  16. 16.
    Grundling H.: Quantum constraints. Rep. Math. Phys. 57, 97–120 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    Grundling H., Hurst C.A.: Algebraic quantization of systems with a gauge degeneracy. Commun. Math. Phys. 98, 369–390 (1985)MathSciNetADSMATHCrossRefGoogle Scholar
  18. 18.
    Grundling, H., Hurst, C.A.: The quantum theory of second class constraints: Kinematics. Commun. Math. Phys. 119, 75–93 (1988) [Erratum: ibid. 122, 527–529 (1989)]Google Scholar
  19. 19.
    Grundling H.: Systems with outer constraints. Gupta–Bleuler electromagnetism as an algebraic field theory. Commun. Math. Phys. 114, 69–91 (1988)MathSciNetMATHGoogle Scholar
  20. 20.
    Grundling H., Lledo F.: Local Quantum Constraints. Rev. Math. Phys. 12, 1159–1218 (2000)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Haag, R.: Local Quantum Physics. Berlin: Springer Verlag, 1992Google Scholar
  22. 22.
    Hannabuss, K.: Some C*-algebras associated to quantum gauge theories. http://arxiv.org/abs/1008.0496v2 [hepth], 2010
  23. 23.
    Huebschmann J., Rudolph G., Schmidt M.: A lattice gauge model for quantum mechanics on a stratified space. Commun. Math. Phys. 286, 459–494 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  24. 24.
    Isham, C.J.: Modern differential geometry for physicists (2nd ed.). Singapore: World Scientific, 1999Google Scholar
  25. 25.
    Jarvis P.D., Kijowski J., Rudolph G.: On the Structure of the Observable Algebra of QCD on the Lattice. J. Phys. A: Math. Gen. 38, 5359–5377 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras II. New York: Academic Press, 1983Google Scholar
  27. 27.
    Kijowski J., Rudolph G.: On the Gauss law and global charge for quantum chromodynamics. J. Math. Phys. 43, 1796–1808 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  28. 28.
    Kijowski J., Rudolph G.: Charge superselection sectors for QCD on the lattice. J. Math. Phys. 46, 032303 (2005)MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    Kijowski J., Rudolph G., Thielman A.: Algebra of Observables and Charge Superselection Sectors for QED on the Lattice. Commun. Math. Phys. 188, 535–564 (1997)ADSMATHCrossRefGoogle Scholar
  30. 30.
    Kijowski J., Rudolph G., Sliwa C.: On the Structure of the Observable Algebra for QED on the Lattice. Lett. Math. Phys. 43, 299–308 (1998)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Kogut J., Susskind L.: Hamiltonian formulation of Wilson’s lattice gauge theories. Phys. Rev. D 11, 395–408 (1975)ADSCrossRefGoogle Scholar
  32. 32.
    Kogut, J.: Three Lectures on Lattice Gauge Theory. CLNS-347 (1976), Lecture Series Presented at the International Summer School, McGill University, June 21–26, 1976Google Scholar
  33. 33.
    Langmann E.: Fermion current algebras and Schwinger terms in (3+1)–dimensions. Commun. Math. Phys. 162, 1–32 (1994)MathSciNetADSMATHCrossRefGoogle Scholar
  34. 34.
    Mickelsson J.: Current algebra representations for 3+1 dimensional Dirac–Yang–Mills theory. Commun. Math. Phys. 117, 261 (1988)MathSciNetADSMATHCrossRefGoogle Scholar
  35. 35.
    Murphy, G.J.: C*-Algebras and Operator Theory. Boston, MA: Academic Press, 1990Google Scholar
  36. 36.
    Napiorkowski K., Woronowicz S.: Operator theory in C*-framework. Rep. on Math. Phys. 31, 353–371 (1992)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Osterwalder K., Seiler E.: Gauge Field Theories on a Lattice. Ann. Phys. 110, 440–471 (1978)MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    Palmer, T.W.: Banach Algebras and the General Theory of C*-algebras. Volume I; Algebras and Banach Algebras, Cambridge: Cambridge Univ. Press, 1994Google Scholar
  39. 39.
    Pedersen, G.K.: C*-Algebras and their Automorphism Groups. London: Academic Press, 1989Google Scholar
  40. 40.
    Raeburn, I.: Dynamical systems and Operator Algebras. In: Proceedings of the Centre for Mathematics and its Applications, Volume 36, p109, 1999, from National Symposium on Functional Analysis, Optimization and Applications, 1998 at The University of Newcastle (the electronic MS is at http://www.math.dartmouth.edu/archive/m123f00/public_html/DynSys5US.pdf)
  41. 41.
    Rieffel M.A.: On the uniqueness of the Heisenberg commutation relations. Duke Math. J. 39, 745–752 (1972)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Rosenberg J.: Appendix to O. Bratteli’s paper on “Crossed products of UHF algebras”. Duke Math. J. 46, 25–26 (1979)MATHCrossRefGoogle Scholar
  43. 43.
    Rudolph G., Schmidt M.: On the algebra of quantum observables for a certain gauge model. J. Math. Phys. 50, 052102 (2009)MathSciNetADSCrossRefGoogle Scholar
  44. 44.
    Seiler, E.: Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics, Lecture Notes in Phys., Vol. 159, Berlin Heidelberg-New York: Springer, 1982Google Scholar
  45. 45.
    Seiler, E.: “Constructive Quantum Field Theory: Fermions”. In: Gauge Theories: Fundamental Interactions and Rigorous Results, eds. P. Dita, V. Georgescu, R. Purice, Bosten, MA: Birkhäuser, 1982Google Scholar
  46. 46.
    Takeda Z.: Inductive limit and infinite direct product of operator algebras. Tohoku Math. J. 7, 67–86 (1955)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Takesaki, M.: Theory of operator algebras I. Springer–Verlag, New York, 1979Google Scholar
  48. 48.
    Takesaki, M.: Theory of Operator Algebras III. Berlin: Springer-Verlag, 2003Google Scholar
  49. 49.
    Varadarajan, V.S.: Geometry of Quantum Theory. Second edition, New York: Springer-Verlag, 1985Google Scholar
  50. 50.
    Wegge-Olsen, N.E.: K–theory and C*-algebras. Oxford: Oxford Science Publications, 1993Google Scholar
  51. 51.
    Williams, D.P.: Crossed products of C*-algebras. Providence, RI: Amer. Math. Soc., 2007Google Scholar
  52. 52.
    Wilson K.G.: Confinement of quarks. Phys. Rev. D10, 2445 (1974)ADSGoogle Scholar
  53. 53.
    Woronowicz S.L.: C*-algebras generated by unbounded elements. Rev. Math. Phys. 7, 481–521 (1995)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of New South WalesSydneyAustralia
  2. 2.Institut für Theoretische PhysikUniversität LeipzigLeipzigGermany

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