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YM2:Continuum expectations, lattice convergence, and lassos

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Abstract

The two dimensional Yang-Mills theory (YM2) is analyzed in both the continuum and the lattice. In the complete axial gauge the continuum theory may be defined in terms of a Lie algebra valued white noise, and parallel translation may be defined by stochastic differential equations. This machinery is used to compute the expectations of gauge invariant functions of the parallel translation operators along a collection of curvesC. The expectation values are expressed as finite dimensional integrals with densities that are products of the heat kernel on the structure group. The time parameters of the heat kernels are determined by the areas enclosed by the collectionC, and the arguments are determined by the crossing topologies of the curves inC. The expectations for the Wilson lattice models have a similar structure, and from this it follows that in the limit of small lattice spacing the lattice expectations converge to the continuum expectations. It is also shown that the lasso variables advocated by L. Gross [36] exist and are sufficient to generate all the measurable functions on the YM2-measure space.

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References

  1. Albeverio, S., Høegh-Krohn, R., Holden, H.: Stochastic multiplicative measures, generalized markov semigroups, and group-valued stochastic processes and fields. J. Funct. Anal.78, 154–184 (1988)

    Google Scholar 

  2. Albeverio, S., Høegh-Krohn, R., Holden, H.: Stochastic Lie group-valued measures and their relations to stochastic curve integrals, gauge-fields and Markov cosurfaces. In: Stochastic processes, mathematics and physics. Lecture Notes in Mathematics, vol.1158. Albeverio, S., Blanchard, P. (eds.). Berlin, Heidelberg, New York: Springer 1986

    Google Scholar 

  3. Albeverio, S., Høegh-Krohn, R., Holden, H.: Random fields with values in Lie groups and Higgs fields. In: Stochastic processes in classical and quantum systems—Lecture Notes in Physics, vol.262. Albeverio, S. et. al. (eds.). Berlin, Heidelberg, New York: Springer 1986

    Google Scholar 

  4. Albeverio, S., Høegh-Krohn, R., Holden, H.: Markov processes on infinite dimensional spaces, Markov fields and Markov cosurfaces. In: Stochastic space-time models and limit theorems. Arnold, L., Kotelenez (eds.). Dordrecht, Boston, Lancaster: D. Reidel 1985

    Google Scholar 

  5. Albeverio, S., Høegh-Krohn, R., Holden, H.: Cosurfaces and gauge fields. In: Stochastic methods and computer techniques in quantum dynamics—Acta Physica Austriaca Supl.XXVI. Wien, New York: Springer 1984

    Google Scholar 

  6. Balian, R., Drouffe, J. M., Itzykson, C.: Gauge fields on a lattice. I. General outlook. Phys. Rev.D10, 3376–3395 (1974); II. Gauge-invariant Ising modelD11 2098–2103 (1975) and III. Strong-coupling expansions and transition points,D11, 2104–2119 (1975)

    Google Scholar 

  7. Balaban, T.: Regularity and decay of lattice Green's functions. Commun. Math. Phys.89, 571–597 (1983)

    Google Scholar 

  8. Balaban, T.: Renormalization group methods in non-abelian gauge theories, Harvard preprint, HUTMP B134

  9. Balaban, T.: Propagators and renormalization transformations for lattice gauge theories. I. Commun. Math. Phys.95, 17–40 (1984)

    Google Scholar 

  10. Balaban, T.: Propagators and renormalization transformations for lattice gauge theories. II. Commun. Math. Phys.96, 223–250 (1984)

    Google Scholar 

  11. Balaban, T.: Recent results in constructing gauge fields. Physica124A, 79–90 (1984)

    Google Scholar 

  12. Balaban, T.: Averaging operations for lattice gauge theories. Commun. Math. Phys.98, 17–51 (1985).

    Google Scholar 

  13. Balaban, T.: Spaces of regular gauge field configurations on a lattice and gauge fixing conditions. Commun. Math. Phys.99, 75–102 (1985)

    Google Scholar 

  14. Balaban, T.: Propagators and renormalization transformations for lattice gauge theories. II. Commun. Math. Phys.96, 223–250 (1984)

    Google Scholar 

  15. Balaban, T.: Ultraviolet stability of three-dimensional lattice pure gauge field theories. Commun. Math. Phys.102, 255–275 (1985)

    Google Scholar 

  16. Balaban, T.: The variational problem and background fields in renormalization group method for lattice gauge theories. Commun. Math. Phys.102, 277–309 (1985)

    Google Scholar 

  17. Balaban, T.: Renormalization group approach to lattice gauge fields theories. I. Generation of effective actions in a small fields approximation and a coupling constant renormalization in four dimensions. Commun. Math. Phys.109, 249–301 (1987)

    Google Scholar 

  18. Borgs, C., Seiler, E.: Lattice Yang-Mills theory at nonzero temperature and the confinement problem. Commun. Math. Phys.91, 329–380 (1983)

    Google Scholar 

  19. Bralić Ninoslav, E., Exact computation of loop averages in two-dimensional Yang-Mills theory. Phys. Rev.D22, 3090–3103 (1980)

    Google Scholar 

  20. Bröcker, Theodor, Dieck, Tammao tom: Representations of Compact Lie Groups. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  21. Driver, Bruce, K.: Classifications of bundle connection pairs by parallel translation and Lassos. J. Funct. Anal.

  22. Driver, Bruce, K.: Convergence of theU 4(1) lattice gauge theory to its continuum limit. Commun. Math. Phys.110, 479–501 (1987)

    Google Scholar 

  23. Dosch, H. G., Müller, V. F.: Lattice gauge theory in two spacetime dimensions. Fortschr. Phys.27, 547–559 (1979)

    Google Scholar 

  24. Federbush, P.: A phase cell approach to Yang-Mills theory, O. Introductory exposition, University of Michigan preprint (1984)

  25. Federbush, P.: A phase cell approach to Yang-Mills theory, I. Small field modes, University of Michigan preprint (1984)

  26. Federbush, P.: A phase cell approach to Yang-Mills theory, IV. General modes and general stability. University of Michigan preprint (1985)

  27. Federbush, P.: A phase cell approach to Yang-Mills theory, V. Stability and interpolation in the small field region, covariance fall-off, University of Michigan preprint (1986)

  28. Federbush, P.: A phase call approach to Yang-Mills theory, VI. Non-abelian lattice-continuum duality, University of Michigan preprint (1986)

  29. Federbush, P.: A phase cell approach to Yang-Mills theory, I. Modes, lattice-continuum duality. Commun. Math. Phys.107, 319–329 (1986)

    Google Scholar 

  30. Federbush, P.: A phase cell approach to Yang-Mills theory, III. Stability, modified renormalization group transformation. Commun. Math. Phys.110, 293–309 (1987)

    Google Scholar 

  31. Federbush, P.: A phase cell approach to Yang-Mills theory, IV. The choice of variables. Commun. Math. Phys.114, 317–343 (1988)

    Google Scholar 

  32. Fröhlich, J.: Statistics of Fields, the Yang-Baxter Equations, and the Theory of Knots and Links. Preprint (1987)

  33. Fröhlich, J.: Some results and comments on quantized gauge fields. In: Recent developments in gauge theories. G. 't Hooft et al. (ed.). New York: Plenum Press 1980

    Google Scholar 

  34. Gross, L., King, C., Sen Gupta, A.: Two dimensional Yang-Mills via stochastic differential equations, preprint (1988)

  35. Gross, L.: Lattice gauge theory; Heuristics and convergence. In: Stochastic processes—Mathematics and physics. Lecture Notes in Mathematics, vol.1158. Albeverio, S., Balanchard Ph., Streit, L. (eds.), Berlin, Heidelberg, New York: Springer 1986

    Google Scholar 

  36. Gross, L.: A poincaré lemma for connection forms. J. Funct. Anal.63, 1–46 (1985)

    Google Scholar 

  37. Gross, L.: Convergence of theU 3(1) lattice gauge theory to its continuum limit. Commun. Math. Phys.92, 137–162 (1983)

    Google Scholar 

  38. Ikeda, N., Shinzo, W.: Stochastic differential equations and diffusion processes. Amsterdam, Oxford, New York: North-Holland 1981

    Google Scholar 

  39. Kazakov, V.: Wilson loop average for an arbitrary contour in two-dimensionalU(N) gauge theory. Nucl. Phys.B179, 283–292 (1981)

    Google Scholar 

  40. Kazakov, V., Kostov, J.: Computation of the Wilson loop functional in two-dimensionalU(∞) lattice gauge theory. Phys. Letts.105B, 453–456 (1981)

    Google Scholar 

  41. Kazakov, V., Kostov, J.: Non-linear strings in two-dimensionalU(∞) gauge theory. Nucl. Phys.B176, 199–215 (1980)

    Google Scholar 

  42. Klimek, S., Kondracki, W.: A construction of two-dimensional quantum chromodynamics. Commun. Math. Phys.113, 389–402 (1987)

    Google Scholar 

  43. Kobayashi, S., Nomizu, K.: Foundations of differential geometry, vol.1. New York: Interscience 1963

    Google Scholar 

  44. Seiler, E.: Gauge Theories as a problem of constructive quantum field theory and statistical mechanics. Lecture Notes in Physics, vol.159. Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  45. Simon, B.: Functional integration and quantum physics. New York: Academic Press 1979

    Google Scholar 

  46. Simon, B.: Trace ideals and their applications. London Mathematical Society Lecture Note Series, vol.35, Cambridge, London, New York, Melbourne: Cambridge University Press 1979

    Google Scholar 

  47. Walsh, J. B.: An introduction to stochastic partial differential equations. In: Ecole d'Ete de Probabilites de Saint-Flour XIV-1984. Lecture Notes in Mathematics, vol.1180. Hennequin, P. L. (ed.). Berlin, Heidelberg, New York: Springer 1986

    Google Scholar 

  48. Wilson, K. G.: Confinement of quarks. Phys. Rev.D10, 2445–2459 (1974)

    Google Scholar 

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  1. Department of Mathematics, University of California, 92093, San Diego, La Jolla, CA, USA

    Bruce K. Driver

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Communicated by K. Gawedzki

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Driver, B.K. YM2:Continuum expectations, lattice convergence, and lassos. Commun.Math. Phys. 123, 575–616 (1989). https://doi.org/10.1007/BF01218586

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