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A criterion to characterize interacting theories in the Wightman framework

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Abstract

We propose that a criterion to characterize interacting theories in a suitable Wightman framework of relativistic quantum field theories which incorporates a “singularity hypothesis”, which has been conjectured for a long time, is supported by renormalization group theory, but has never been formulated mathematically. The (nonperturbative) wave function renormalization Z occurring in these theories is shown not to be necessarily equal to zero, except if the equal time commutation relations (ETCR) are assumed. Since the ETCR are not justified in general (because the interacting fields cannot in general be restricted to sharp times, as is known from model studies), the condition \(Z=0\) is not of general validity in interacting theories. We conjecture that it characterizes either unstable (composite) particles or the charge-carrying particles, which become infraparticles in the presence of massless particles. In the case of QED, such “dressed” electrons are not expected to be confined, but in QCD, we propose a quark confinement criterion, which follows naturally from lines suggested by the works of Casher, Kogut and Susskind, and Lowenstein and Swieca.

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References

  1. Aschbacher, W.H., Barbaroux, J.M., Faupin, J., Guillot, J.C.: Spectral theory for a mathematical model of the weak interactions: the decay of the intermediate vector bosons. Ann. Henri Poincaré 12, 1539 (2011)

    MathSciNet  MATH  Google Scholar 

  2. Araki, H., Haag, R.: Collision cross sections in terms of local observables. Comm. Math. Phys. 4, 77–91 (1967)

    MathSciNet  Google Scholar 

  3. Licht, A.L.: A generalized asymptotic condition I. Ann. Phys. N. Y. 34, 161 (1965)

    MathSciNet  MATH  Google Scholar 

  4. Araki, H., Munakata, Y., Kawaguchi, M., Goto, T.: Quantum field theory of unstable particles. Progr. Theor. Phys. 17, 419 (1957)

    MathSciNet  MATH  Google Scholar 

  5. Barton, G.: An introduction to advanced field theory. Interscience, Geneva (1963)

    MATH  Google Scholar 

  6. Blanchard, P., Brüning, E.: Mathematical methods in physics-Distributions. Hilbert-space operators and variational methods, Birkhäuser (2003)

    MATH  Google Scholar 

  7. Barnaföldi, G.G., Gogokhia, V.: The origins of mass in qcd and its renormalization. arXiv:1904.07748 (16-4-2019)

  8. Bognar, J.: Indefinite inner product spaces. Springer, Berlin (1974)

    MATH  Google Scholar 

  9. Buchholz, D., Porrmann, M., Stein, U.: Dirac versus Wigner: towards a universal particle concept in quantum field theory. Phys. Lett. B 267, 377–381 (1991)

    MathSciNet  Google Scholar 

  10. Bratelli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics I. Springer, Berlin (1987)

    Google Scholar 

  11. Buchholz, D., Summers, S.J.: Scattering in rqft - fundamental concepts and tools. Encyclopaedia of Mathematical Physics, (2005)

  12. Buchholz, D.: Collision theory of massless bosons. Comm. Math. Phys. 52, 147–173 (1977)

    MathSciNet  Google Scholar 

  13. Buchholz, D.: Gauss’ law and the infraparticle problem. Phys. Lett. B 174, 331 (1986)

    MathSciNet  Google Scholar 

  14. Buchholz, D.: Quarks, gluons, colour: facts or fiction? Nucl. Phys. B 469, 333–356 (1996)

    MathSciNet  MATH  Google Scholar 

  15. Barata, J.C.A., Wreszinski, W.F.: Absence of charged states in the U(1) Higgs lattice gauge theory. Comm. Math. Phys. 103, 637 (1986)

    MathSciNet  MATH  Google Scholar 

  16. Carey, A.L., Ruijsenaars, S.M.J., Wright, J.: The massless Thirring model: positivity of Klaiber’s n point functions. Comm. Math. Phys. 99, 347–364 (1985)

    MathSciNet  Google Scholar 

  17. Casher, A., Kogut, J., Susskind, L.: Vacuum polarization and the quark-parton puzzle. Phys. Rev. Lett. 31, 792 (1973)

    Google Scholar 

  18. Cornwall, J.M.: Dynamical mass generation in continuum quantum electrodynamics. Phys. Rev. D 26, 1453–1478 (1982)

    Google Scholar 

  19. Dirac, P.A.M.: Forms of relativistic dynamics. Rev. Mod. Phys. 21, 392 (1949)

    MathSciNet  MATH  Google Scholar 

  20. Dubin, D.A., Tarski, J.: Interactions of massless spinors in two dimensions. Ann. Phys. 43, 263 (1967)

    Google Scholar 

  21. Farhi, E., Jackiw, R.: Introduction to dynamical gauge symmetry breaking. World Scientific, Singapore (1982)

    Google Scholar 

  22. In: G. Velo and A. S. Wightman, editors, Fundamental problems of gauge field theory. Plenum Press N. Y., (1986)

  23. Guelfand, I.M., Chilov, G.E.: Les Distributions tome I. Dunod, Paris (1962)

    MATH  Google Scholar 

  24. Gogohia, V.: Gluon confinement criterion in qcd. Phys. Lett. B 584, 225–232 (2004)

    Google Scholar 

  25. Haag, R.: Local quantum physics—Fields, particles, algebras. Springer, Berlin (1996)

    MATH  Google Scholar 

  26. Hepp, K.: Théorie de la renormalisation. Springer, Berlin (1969)

    Google Scholar 

  27. Hörmann, G., Jäkel, C.D.: Galilei invariant molecular dynamics. Publ. RIMS Kyoto Univ. 31, 805–827 (1995)

    MathSciNet  MATH  Google Scholar 

  28. Houard, J.C., Jouvet, B.: Étude d’un modèle de champ à constante de rénormalisation nulle. Il Nuovo Cimento 18, 466–481 (1960)

    MATH  Google Scholar 

  29. Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys. 5, 548 (1964)

    MathSciNet  MATH  Google Scholar 

  30. ’t Hooft, G. Reflections on the renormalization procedure for gauge theories. arXiv: 1604.06257v1 [hep-th] 21-04-2016

  31. Itzykson, C., Zuber, J.-B.: Quantum Field Theory. McGraw Hill Book Co., New York (1980)

    MATH  Google Scholar 

  32. Jaekel, C.D., Mund, J.: Canonical interacting quantum fields on two-dimensional de Sitter space. Phys. Lett. B 772, 786–790 (2017)

    MATH  Google Scholar 

  33. Jost, R.: The general theory of quantized fields. American Mathematical Society, Providence (1965)

    MATH  Google Scholar 

  34. Jaffe, A., Witten, E.: Yang-Mills existence and mass gap. http://www.claymath.org/prize-problems

  35. Jäkel, C., Wreszinski, W.F.: Stability of relativistic quantum electrodynamics in the Coulomb gauge. J. Math. Phys. 59, 032303 (2018)

    MathSciNet  MATH  Google Scholar 

  36. Källén, G.: On the definition of the renormalization constants in quantum electrodynamics. Dan. Mat. Fys. Medd. 27, 417 (1953)

    MATH  Google Scholar 

  37. Klaiber, B.: Lectures in theoretical physics, Boulder lectures 1967, p.141. Gordon and Breach, (1968)

  38. Pauli, In W. (ed.): Niels Bohr and the development of physics. Pergamon Press (1955)

  39. Lehmann, H.: Über Eigenschaften von Ausbreitungsfunktionen und Renormierunskonstanten quantisierter Felder. Nuovo Cimento 11, 342 (1954)

    MathSciNet  MATH  Google Scholar 

  40. Lieb, E.H., Loss, M.: Stability of a model of relativistic quantum electrodynamics. Comm. Math. Phys. 228, 561–588 (2002)

    MathSciNet  MATH  Google Scholar 

  41. Lowenstein, J.H., Swieca, J.A.: Quantum electrodynamics in two dimensions. Ann. Phys. 68, 172–195 (1971)

    MathSciNet  Google Scholar 

  42. Lukierski, J.: A field theory describing interacting two-particle systems. Nuovo Cimento A 60, 353 (1969)

    Google Scholar 

  43. Mund, J., Schroer, B., Yngvason, J.: String localised fields and modular localisation. Comm. Math. Phys. 268, 621 (2006)

    MathSciNet  MATH  Google Scholar 

  44. note = arXiv:1906.00940v1 P. Duch, title = Infrared problem in perturbative QFT

  45. Reed, M., Simon, B.: Methods in modern mathematical physics - v.1, Functional Analysis, 1st edn. Academic Press, Cambridge (1972)

    MATH  Google Scholar 

  46. Reed, M., Simon, B.: Methods of modern mathematical physics - v.2, Fourier Analysis, Self-adjointness. Academic Press, Cambridge (1975)

    MATH  Google Scholar 

  47. Raina, A.K., Wanders, G.: The gauge transformations of the Schwinger model. Ann. Phys. 132, 404–426 (1981)

    MathSciNet  Google Scholar 

  48. Srivastava, P.M., Brodsky, S.J.: A unitary and renormalizable theory of the standard model in ghost-free light cone gauge. Phys. Rev. D 66, 045019 (2002)

    Google Scholar 

  49. Schwinger, J.: Gauge invariance and mass. Phys. Rev. 125, 397 (1962)

    MathSciNet  MATH  Google Scholar 

  50. Schroer, B.: Infrateilchen in der Quantenfeldtheorie. Fortschr. der Physik 11, 1 (1963)

    MathSciNet  Google Scholar 

  51. Scharf, G.: Quantum gauge theories: a true ghost story. John Wiley and sons Inc, Hoboken (2001)

    MATH  Google Scholar 

  52. Schroer, B., Stichel, P.: On equal time commutation relations of renormalized currents in perturbation theory. Comm. Math. Phys. 8, 327 (1968)

    MathSciNet  MATH  Google Scholar 

  53. Steinmann, O.: Perturbation expansions in axiomatic field theory, Lecture Notes in Physics, vol. 11. Springer, Berlin (1971)

    Google Scholar 

  54. Steinmann, O.: Perturbative QED in terms of gauge invariant fields. Ann. Phys. (N.Y) 157, 232–254 (1984)

    MathSciNet  Google Scholar 

  55. Strocchi, F.: Gauge problem in quantum field theory III: quantization of Maxwell’s equation and weak local commutativity. Phys. Rev. D 2, 2334 (1970)

    MathSciNet  Google Scholar 

  56. Streater, R.F., Wightman, A.S.: PCT, spin and statistics and all that. Benjamin Inc., Menlo Park (1964)

    MATH  Google Scholar 

  57. Streater, R.F., Wilde, I.F.: Fermion states of a Bose field. Nucl. Phys. B 24, 561 (1970)

    Google Scholar 

  58. Strocchi, F., Wightman, A.S.: Proof of the charge superselection rule in local relativistic quantum field theory. J. Math. Phys. 15, 2198 (1974)

    MathSciNet  Google Scholar 

  59. Swieca, J.A.: Solitons and confinement. Fortschr. der Physik 25, 303 (1977)

    Google Scholar 

  60. Symanzik, K.: Small distance behavior analysis and wilson expansions. Comm. Math. Phys. 23, 49 (1971)

    MathSciNet  MATH  Google Scholar 

  61. Thirring, W.E.: A soluble relativistic field theory. Ann. Phys. 3, 91 (1958)

    MathSciNet  MATH  Google Scholar 

  62. Thirring, W.E., Wess, J.: Solution of a field theory in one space and one time dimension. Ann. Phys. 27, 331 (1964)

    MATH  Google Scholar 

  63. Veltman, M.: Unitary and causality in a renormalizable field theory with unstable particles. Physica 29, 186 (1963)

    MathSciNet  MATH  Google Scholar 

  64. In CERN, editor, Proceedings of the 1962 high energy conference at CERN, (1962)

  65. Weinberg, S.: General theory of broken symmetries. Phys. Rev. D 7, 1068 (1973)

    Google Scholar 

  66. Weinberg, S.: The quantum of fields, vol. 2. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  67. Weinberg, S.: The quantum theory of fields, vol. 1. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  68. Wightman, A.S.: Quantum field theory in terms of vacuum expectation values. Phys. Rev. 101, 860–866 (1956)

    MathSciNet  MATH  Google Scholar 

  69. In Hagen, C.R., Guralnik, G.S., Mathur, V.A. editors, A. S. Wightman, Progress in the foundations of quantum field theory, in Proceedings of the international conference in particles and fields, Rochester. Interscience, (1967)

  70. In Lévy, M.: editor, A. S. Wightman, Introduction to some aspects of quantized fields, in High energy electromagnetic interactions and field theory. Gordon and Breach, (1967)

  71. In A. Zichichi, editor, A. S. Wightman, Should we believe in quantum field theory? in The whys of subnuclear physics. Plenum Press, N. Y., (1979)

  72. Wreszinski, W.F.: Unstable states in a model of nonrelativistic quantum electrodynamics: rate of decay, regeneration by decay products, sojourn time and irreversibility. arXiv:1910.09332 (2019)

  73. Wreszinski, W.F.: Equal time limits of anticommutators in relativistic theories. Nuovo. Cim. A 1, 691–705 (1971)

    Google Scholar 

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Acknowledgements

We are very grateful to the referee. He clarified and corrected several important conceptual issues. We learned a lot from his various objections. He convinced us that our chapter on the classical theory of unstable particles was self-contradictory, and we decided to eliminate it.

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  1. Department de Matemática Aplicada, Universidade de São Paulo (USP), São Paulo, Brazil

    Christian D. Jäkel

  2. Instituto de Fisica, Universidade de São Paulo (USP), São Paulo, Brazil

    Walter F. Wreszinski

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  1. Christian D. Jäkel
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Correspondence to Walter F. Wreszinski.

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Jäkel, C.D., Wreszinski, W.F. A criterion to characterize interacting theories in the Wightman framework. Quantum Stud.: Math. Found. 8, 51–68 (2021). https://doi.org/10.1007/s40509-020-00227-5

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