Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 058, 49 pages      arXiv:1510.07445      https://doi.org/10.3842/SIGMA.2016.058

Reflection Positive Stochastic Processes Indexed by Lie Groups

Palle E.T. Jorgensen a, Karl-Hermann Neeb b and Gestur Ólafsson c
a) Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
b) Department Mathematik, FAU Erlangen-Nürnberg, Cauerstrasse 11, 91058-Erlangen, Germany
c) Department of mathematics, Louisiana State University, Baton Rouge, LA 70803, USA

Received October 28, 2015, in final form June 09, 2016; Published online June 21, 2016

Abstract
Reflection positivity originates from one of the Osterwalder-Schrader axioms for constructive quantum field theory. It serves as a bridge between euclidean and relativistic quantum field theory. In mathematics, more specifically, in representation theory, it is related to the Cartan duality of symmetric Lie groups (Lie groups with an involution) and results in a transformation of a unitary representation of a symmetric Lie group to a unitary representation of its Cartan dual. In this article we continue our investigation of representation theoretic aspects of reflection positivity by discussing reflection positive Markov processes indexed by Lie groups, measures on path spaces, and invariant gaussian measures in spaces of distribution vectors. This provides new constructions of reflection positive unitary representations.

Key words: reflection positivity; stochastic process; unitary representations.

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References

  1. Albeverio S., Driver B.K., Gordina M., Vershik A.M., Equivalence of the Brownian motion and energy representations, arXiv:1511.07378.
  2. Alpay D., Jorgensen P.E.T., Stochastic processes induced by singular operators, Numer. Funct. Anal. Optim. 33 (2012), 708-735, arXiv:1109.5273.
  3. Alpay D., Jorgensen P.E.T., Levanony D., A class of Gaussian processes with fractional spectral measures, J. Funct. Anal. 261 (2011), 507-541, arXiv:1009.0233.
  4. Anderson C.C., Defining physics at imaginary time: reflection positivity for certain Riemannian manifolds, Thesis, Harvard University, 2013, available at http://www.math.harvard.edu/theses/senior/anderson/anderson.pdf.
  5. Applebaum D., Stochastic evolution of Yang-Mills connections on the noncommutative two-torus, Lett. Math. Phys. 16 (1988), 93-99.
  6. Banaszczyk W., Additive subgroups of topological vector spaces, Lecture Notes in Math., Vol. 1466, Springer-Verlag, Berlin, 1991.
  7. Bauer H., Probability theory, de Gruyter Studies in Mathematics, Vol. 23, Walter de Gruyter & Co., Berlin, 1996.
  8. Bekka B., de la Harpe P., Valette A., Kazhdan's property (T), New Mathematical Monographs, Vol. 11, Cambridge University Press, Cambridge, 2008.
  9. Bekka B., Mayer M., Ergodic theory and topological dynamics of group actions on homogeneous space, London Mathematical Society Lecture Note Series, Vol. 269, Cambridge University Press, Cambridge, 2000.
  10. Bendikov A., Saloff-Coste L., Brownian motions on compact groups of infinite dimension, in Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), Contemp. Math., Vol. 338, Amer. Math. Soc., Providence, RI, 2003, 41-63.
  11. Bogachev V.I., Gaussian measures, Mathematical Surveys and Monographs, Vol. 62, Amer. Math. Soc., Providence, RI, 1998.
  12. Bratteli O., Robinson D.W., Operator algebras and quantum statistical mechanics. I. $C^*$- and $W^*$-algebras. Symmetry groups. Decomposition of states, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987.
  13. Deitmar A., van Dijk G., Trace class groups, J. Lie Theory 26 (2016), 269-291, arXiv:1501.02375.
  14. van Dijk G., Introduction to harmonic analysis and generalized Gelfand pairs, de Gruyter Studies in Mathematics, Vol. 36, Walter de Gruyter & Co., Berlin, 2009.
  15. van Dijk G., Neeb K.-H., Salmasian H., Zellner C., On the characterization of trace class representations and Schwartz operators, J. Lie Theory 26 (2016), 787-805, arXiv:1512.02451.
  16. Dobbins J.G., Well bounded semigroups in locally compact groups, Math. Z. 148 (1976), 155-167.
  17. Driver B.K., Heat kernels measures and infinite dimensional analysis, in Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), Contemp. Math., Vol. 338, Amer. Math. Soc., Providence, RI, 2003, 101-141.
  18. Fang S., Canonical Brownian motion on the diffeomorphism group of the circle, J. Funct. Anal. 196 (2002), 162-179.
  19. Fröhlich J., Osterwalder K., Seiler E., On virtual representations of symmetric spaces and their analytic continuation, Ann. of Math. 118 (1983), 461-489.
  20. Glimm J., Jaffe A., Quantum physics. A functional integral point of view, Springer-Verlag, New York - Berlin, 1981.
  21. Gordina M., Riemannian geometry of ${\rm Diff}(S^1)/S^1$ revisited, in Stochastic Analysis in Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2008, 19-29, math-ph/0510092.
  22. Gross L., Harmonic functions on loop groups, Astérisque 252 (1998), Exp. No. 846, 5, 271-286.
  23. Hida T., Brownian motion, Applications of Mathematics, Vol. 11, Springer-Verlag, New York - Berlin, 1980.
  24. Hilgert J., A note on Howe's oscillator semigroup, Ann. Inst. Fourier (Grenoble) 39 (1989), 663-688.
  25. Hilgert J., Neeb K.-H., Lie semigroups and their applications, Lecture Notes in Math., Vol. 1552, Springer-Verlag, Berlin, 1993.
  26. Hilgert J., Ólafsson G., Causal symmetric spaces. Geometry and harmonic analysis, Perspectives in Mathematics, Vol. 18, Academic Press, Inc., San Diego, CA, 1997.
  27. Hofmann K.H., Lie algebras with subalgebras of co-dimension one, Illinois J. Math. 9 (1965), 636-643.
  28. Hofmann K.H., Hyperplane subalgebras of real Lie algebras, Geom. Dedicata 36 (1990), 207-224.
  29. Holden H., Øksendal B., Ubøe J., Zhang T., Stochastic partial differential equations. A modeling, white noise functional approach, 2nd ed., Universitext, Springer, New York, 2010.
  30. Howe R., The oscillator semigroup, in The Mathematical Heritage of Hermann Weyl (Durham, NC, 1987), Proc. Sympos. Pure Math., Vol. 48, Amer. Math. Soc., Providence, RI, 1988, 61-132.
  31. Jaffe A., Ritter G., Quantum field theory on curved backgrounds. I. The Euclidean functional integral, Comm. Math. Phys. 270 (2007), 545-572, hep-th/0609003.
  32. Jaffe A., Ritter G., Quantum field theory on curved backgrounds. II. Spacetime symmetries, arXiv:0704.0052.
  33. Jorgensen P.E.T., Analytic continuation of local representations of Lie groups, Pacific J. Math. 125 (1986), 397-408.
  34. Jorgensen P.E.T., Analytic continuation of local representations of symmetric spaces, J. Funct. Anal. 70 (1987), 304-322.
  35. Jorgensen P.E.T., Ólafsson G., Unitary representations of Lie groups with reflection symmetry, J. Funct. Anal. 158 (1998), 26-88, funct-an/9707001.
  36. Jorgensen P.E.T., Ólafsson G., Unitary representations and Osterwalder-Schrader duality, in The Mathematical Legacy of Harish-Chandra (Baltimore, MD, 1998), Proc. Sympos. Pure Math., Vol. 68, Amer. Math. Soc., Providence, RI, 2000, 333-401, math.FA/9908031.
  37. Jørsboe O.G., Equivalence or singularity of Gaussian measures on function spaces, Various Publications Series, Vol. 4, Matematisk Institut, Aarhus Universitet, Aarhus, 1968.
  38. Kakutani S., On equivalence of infinite product measures, Ann. of Math. 49 (1948), 214-224.
  39. Kirillov A.A., Lectures on the orbit method, Graduate Studies in Mathematics, Vol. 64, Amer. Math. Soc., Providence, RI, 2004.
  40. Klein A., Gaussian ${\rm OS}$-positive processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 40 (1977), 115-124.
  41. Klein A., The semigroup characterization of Osterwalder-Schrader path spaces and the construction of Euclidean fields, J. Funct. Anal. 27 (1978), 277-291.
  42. Klein A., Landau L.J., Singular perturbations of positivity preserving semigroups via path space techniques, J. Funct. Anal. 20 (1975), 44-82.
  43. Lawson J.D., Maximal subsemigroups of Lie groups that are total, Proc. Edinburgh Math. Soc. 30 (1987), 479-501.
  44. Lawson J.D., Polar and Ol'shanskii decompositions, J. Reine Angew. Math. 448 (1994), 191-219.
  45. Lüscher M., Mack G., Global conformal invariance in quantum field theory, Comm. Math. Phys. 41 (1975), 203-234.
  46. Merigon S., Neeb K.-H., Ólafsson G., Integrability of unitary representations on reproducing kernel spaces, Represent. Theory 19 (2015), 24-55, arXiv:1406.2681.
  47. Neeb K.-H., On a theorem of S. Banach, J. Lie Theory 7 (1997), 293-300.
  48. Neeb K.-H., Holomorphy and convexity in Lie theory, de Gruyter Expositions in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin, 2000.
  49. Neeb K.-H., Ólafsson G., Reflection positivity and conformal symmetry, J. Funct. Anal. 266 (2014), 2174-2224, arXiv:1206.2039.
  50. Neeb K.-H., Ólafsson G., Reflection positive one-parameter groups and dilations, Complex Anal. Oper. Theory 9 (2015), 653-721, arXiv:1312.6161.
  51. Neeb K.-H., Ólafsson G., Reflection positivity for the circle group, J. Phys. Conf. Ser. 597 (2015), 012004, 16 pages, arXiv:1411.2439.
  52. Neeb K.-H., Ørsted B., Representation in $L^2$-spaces on infinite-dimensional symmetric cones, J. Funct. Anal. 190 (2002), 133-178.
  53. Nelson E., Analytic vectors, Ann. of Math. 70 (1959), 572-615.
  54. Nelson E., Feynman integrals and the Schrödinger equation, J. Math. Phys. 5 (1964), 332-343.
  55. Nelson E., The free Markoff field, J. Funct. Anal. 12 (1973), 211-227.
  56. Ólafsson G., Analytic continuation in representation theory and harmonic analysis, in Global Analysis and Harmonic Analysis (Marseille-Luminy, 1999), Sémin. Congr., Vol. 4, Soc. Math. France, Paris, 2000, 201-233.
  57. Osterwalder K., Schrader R., Axioms for Euclidean Green's functions, Comm. Math. Phys. 31 (1973), 83-112.
  58. Osterwalder K., Schrader R., Axioms for Euclidean Green's functions. II, Comm. Math. Phys. 42 (1975), 281-305.
  59. Poguntke D., Well-bounded semigroups in connected groups, Semigroup Forum 15 (1977), 159-167.
  60. Schrader R., Reflection positivity for the complementary series of ${\rm SL}(2n,{\mathbb C})$, Publ. Res. Inst. Math. Sci. 22 (1986), 119-141.
  61. Schwartz L., Sous-espaces hilbertiens d'espaces vectoriels topologiques et noyaux associés (noyaux reproduisants), J. Analyse Math. 13 (1964), 115-256.
  62. Schwartz L., Radon measures on arbitrary topological spaces and cylindrical measures, Tata Institute of Fundamental Research Studies in Mathematics, Vol. 6, Oxford University Press, London, 1973.
  63. Segal I.E., Ergodic subgroups of the orthogonal group on a real Hilbert space, Ann. of Math. 66 (1957), 297-303.
  64. Simon B., The $P(\phi )_{2}$ Euclidean (quantum) field theory, Princeton Series in Physics, Princeton University Press, Princeton, N.J., 1974.
  65. Trèves F., Topological vector spaces, distributions and kernels, Academic Press, New York - London, 1967.
  66. Varadarajan V.S., Geometry of quantum theory, 2nd ed., Springer-Verlag, New York, 1985.
  67. Warner G., Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der mathematischen Wissenschaften, Vol. 188, Springer-Verlag, New York - Heidelberg, 1972.
  68. Yamasaki Y., Measures on infinite-dimensional spaces, Series in Pure Mathematics, Vol. 5, World Scientific Publishing Co., Singapore, 1985.

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