Inverse Scattering and Local Observable Algebras in Integrable Quantum Field Theories

Abstract

We present a solution method for the inverse scattering problem for integrable two-dimensional relativistic quantum field theories, specified in terms of a given massive single particle spectrum and a factorizing S-matrix. An arbitrary number of massive particles transforming under an arbitrary compact global gauge group is allowed, thereby generalizing previous constructions of scalar theories. The two-particle S-matrix S is assumed to be an analytic solution of the Yang–Baxter equation with standard properties, including unitarity, TCP invariance, and crossing symmetry. Using methods from operator algebras and complex analysis, we identify sufficient criteria on S that imply the solution of the inverse scattering problem. These conditions are shown to be satisfied in particular by so-called diagonal S-matrices, but presumably also in other cases such as the O(N)-invariant nonlinear \({\sigma}\)-models.

This is a preview of subscription content, log in to check access.

References

  1. 1

    Abdalla E., Abdalla C.B., Rothe K.D.: Non-Perturbative Methods in 2-Dimensional Quantum Field Theory. World Scientific, Singapore (2001)

    Google Scholar 

  2. 2

    Alazzawi, S.: Deformations of Quantum Field Theories and the Construction of Interacting Models. Ph.D. Thesis, University of Vienna. arXiv:1503.00897 (2014)

  3. 3

    Araki H.: Mathematical Theory of Quantum Fields. Oxford University Press, Oxford (1999)

    Google Scholar 

  4. 4

    Babujian, H.M., Foerster, A., Karowski, M.: SU(N) and O(N) off-shell nested Bethe ansatz and form factors. Low Dimens. Phys. Gauge Princ. 46 (2011). doi:10.1142/9789814440349_0005

  5. 5

    Babujian, H.M., Foerster, A., Karowski, M. et al.: The form factor program: a review and new results—the nested SU(N) off-shell Bethe ansatz. SIGMA 2, 082 (2006)

  6. 6

    Baumgärtel H., Wollenberg M.: Causal Nets of Operator Algebras. Akademie Verlag, Berlin (1992)

    Google Scholar 

  7. 7

    Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, Cambridge (1982)

    Google Scholar 

  8. 8

    Benfatto G., Falco P., Mastropietro V.: Functional integral construction of the thirring model: axioms verification and massless limit. Commun. Math. Phys. 273, 67–118 (2007)

    ADS  MATH  Article  Google Scholar 

  9. 9

    Benfatto G., Falco P., Mastropietro V.: Massless Sine-Gordon and massive Thirring models: proof of the Coleman’s equivalence. Commun. Math. Phys. 285, 713–762 (2009)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  10. 10

    Bischoff M., Tanimoto Y.: Integrable QFT and Longo–Witten endomorphisms. Ann. H. Poincaré 16(2), 569–608 (2015)

    MATH  MathSciNet  Article  Google Scholar 

  11. 11

    Bisognano J.J., Wichmann E.H.: On the duality condition for a Hermitian scalar field. J. Math. Phys. 16, 985–1007 (1975)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  12. 12

    Bisognano J.J., Wichmann E.H.: On the duality condition for quantum fields. J. Math. Phys. 17(3), 303–321 (1976)

    ADS  MathSciNet  Article  Google Scholar 

  13. 13

    Borchers H.-J., Buchholz D., Schroer B.: Polarization-free generators and the S-matrix. Commun. Math. Phys. 219(1), 125–140 (2001)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  14. 14

    Borchers H.J.: The CPT theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315–332 (1992)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  15. 15

    Bostelmann H., Cadamuro D.: An operator expansion for integrable quantum field theories. J. Phys. A: Math. Theor. 46(9), 095401 (2013)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  16. 16

    Bostelmann H., Cadamuro D.: Characterization of local observables in integrable quantum field theories. Commun. Math. Phys. 337(3), 1199–1240 (2015)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  17. 17

    Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics I. Springer, Berlin (1987)

    Google Scholar 

  18. 18

    Brunetti R., Guido D., Longo R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759–786 (2002)

    MATH  MathSciNet  Article  Google Scholar 

  19. 19

    Buchholz D., D’Antoni C., Longo R.: Nuclear maps and modular structures. I. General properties. J. Funct. Anal. 88, 223–250 (1990)

    MATH  MathSciNet  Article  Google Scholar 

  20. 20

    Buchholz D., D’Antoni C., Longo R.: Nuclear maps and modular structures II: applications to quantum field theory. Commun. Math. Phys. 129(1), 115–138 (1990)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  21. 21

    Buchholz D., Lechner G.: Modular nuclearity and localization. Ann. H. Poincaré 5, 1065–1080 (2004)

    MATH  MathSciNet  Article  Google Scholar 

  22. 22

    Cadamuro D., Tanimoto Y.: Wedge-local fields in integrable models with bound states. Commun. Math. Phys. 340(2), 661–697 (2015)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  23. 23

    Doplicher S., Longo R.: Standard and split inclusions of von Neumann algebras. Invent. Math. 75, 493–536 (1984)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  24. 24

    Epstein H.: Generalization of the “Edge-of-the-Wedge” Theorem. J. Math. Phys. 1(6), 524–531 (1960)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  25. 25

    Epstein, H.: Some analytic properties of scattering amplitudes in quantum field theory. In: Axiomatic Field Theory, vol. 1, p. 1. (1966)

  26. 26

    Fröhlich J.: Quantized “Sine-Gordon” equation with a non-vanishing mass term in two space-time dimensions. Phys. Rev. Lett. 34(13), 833–836 (1975)

    ADS  MathSciNet  Article  Google Scholar 

  27. 27

    Glimm J., Jaffe A.: Quantum Physics. Springer, Berlin (1981)

    Google Scholar 

  28. 28

    Grosse, H., Wulkenhaar, R.: Solvable 4D noncommutative QFT: phase transitions and quest for reflection positivity. arXiv:1406.7755 (2014)

  29. 29

    Haag R.: Local Quantum Physics: Fields, Particles, Algebras. Springer, Berlin (1996)

    Google Scholar 

  30. 30

    Hollands, S., Lechner, G.: SO(d,1)-invariant Yang–Baxter operators and the dS/CFT-correspondence. arXiv:1603.05987 (2016)

  31. 31

    Iagolnitzer D.: Scattering in Quantum Field Theories: The Axiomatic and Constructive Approaches. Princeton University Press, Princeton (1993)

    Google Scholar 

  32. 32

    Jarchow H.: Locally Convex Spaces. Teubner, Stuttgart (1981)

    Google Scholar 

  33. 33

    Jimbo M.: Quantum R-matrix for the generalized Toda system. Commun. Math. Phys. 102(4), 537–547 (1986)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  34. 34

    Kauffman L.: Knots and Physics. World Scientific, Singapore (1993)

    Google Scholar 

  35. 35

    Ketov S.V.: Quantum Non-Linear Sigma-Models. Springer, Berlin (2000)

    Google Scholar 

  36. 36

    Kosaki H.: On the continuity of the map \({\varphi\rightarrow|\varphi|}\) from the predual of a W*-algebra. J. Funct. Anal. 59(1), 123–131 (1984)

    MathSciNet  Article  Google Scholar 

  37. 37

    Lechner G.: Polarization-free quantum fields and interaction. Lett. Math. Phys. 64(2), 137–154 (2003)

    MATH  MathSciNet  Article  Google Scholar 

  38. 38

    Lechner, G.: On the construction of quantum field theories with factorizing S-matrices. Ph.D Thesis, University of Göttingen. arXiv: math-ph/0611050 (2006)

  39. 39

    Lechner G.: Towards the construction of quantum field theories from a factorizing S-matrix. Prog. Math. 251, 175–198 (2007)

    MATH  MathSciNet  Article  Google Scholar 

  40. 40

    Lechner G.: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277, 821–860 (2008)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  41. 41

    Lechner G.: Deformations of quantum field theories and integrable models. Commun. Math. Phys. 312, 265–302 (2012)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  42. 42

    Lechner G.: Algebraic Constructive Quantum Field Theory: Integrable Models and Deformation Techniques, pp. 397–449. Springer, Berlin (2015)

    Google Scholar 

  43. 43

    Lechner G., Sanders K.: Modular nuclearity: a generally covariant perspective. Axioms 5(1), 5 (2016)

    Article  Google Scholar 

  44. 44

    Lechner G., Schützenhofer C.: Towards an operator-algebraic construction of integrable global gauge theories. Ann. H. Poincaré 15, 645–678 (2014)

    MATH  Article  Google Scholar 

  45. 45

    Liguori A., Mintchev M.: Fock representations of quantum fields with generalized statistics. Commun. Math. Phys. 169, 635–652 (1995)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  46. 46

    Liguori A., Mintchev M.: Fock spaces with generalized statistics. Lett. Math. Phys. 33, 283–295 (1995)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  47. 47

    Mund J.: The Bisognano–Wichmann theorem for massive theories. Ann. H. Poincaré 2, 907–926 (2001)

    MATH  MathSciNet  Article  Google Scholar 

  48. 48

    Pietsch A.: Nuclear locally convex spaces. Cambridge University Press, Cambridge (1972)

    Google Scholar 

  49. 49

    Reed M., Simon B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness. Academic Press, Cambridge (1975)

    Google Scholar 

  50. 50

    Reed M., Simon B.: Methods of modern mathematical physics III: scattering theory. Academic Press, Academic Press (1980)

    Google Scholar 

  51. 51

    Schroer B.: Modular localization and the bootstrap-formfactor program. Nucl. Phys. B 499(3), 547–568 (1997)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  52. 52

    Schroer B., Wiesbrock H.-W.: Modular constructions of quantum field theories with interactions. Rev. Math. Phys. 12(02), 301–326 (2000)

    MATH  MathSciNet  Article  Google Scholar 

  53. 53

    Schützenhofer, C.: Multi-particle S-matrix models in \({1+1}\)-dimensions and associated Quantum Field theories. Diploma thesis, University of Vienna (2011)

  54. 54

    Simon B.: Trace Ideals and Their Applications. American Mathematical Society, Providence, RI (2005)

    Google Scholar 

  55. 55

    Smirnov F.A.: Form-factors in completely integrable models of quantum field theory. Adv. Ser. Math. Phys. 14, 1–208 (1992)

    MATH  MathSciNet  Article  Google Scholar 

  56. 56

    Stein E.M., Weiss G.L.: Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton (1971)

    Google Scholar 

  57. 57

    Streater R.F., Wightman A.S.: PCT, spin and statistics, and all that. Princeton University Press, Princeton (1964)

    Google Scholar 

  58. 58

    Tanimoto Y.: Bound state operators and wedge-locality in integrable quantum field theories. SIGMA 12, 100 (2016)

    ADS  MATH  MathSciNet  Google Scholar 

  59. 59

    Zamolodchikov A.B., Zamolodchikov A.B.: Relativistic factorized S-matrix in two-dimensions having O(N) isotopic symmetry. Nucl. Phys. B 133, 525 (1978)

    ADS  MathSciNet  Article  Google Scholar 

  60. 60

    Zamolodchikov A.B., Zamolodchikov A.B.: Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models. Ann. Phys. 120(2), 253–291 (1979)

    ADS  MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sabina Alazzawi.

Additional information

Communicated by Y. Kawahigashi

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alazzawi, S., Lechner, G. Inverse Scattering and Local Observable Algebras in Integrable Quantum Field Theories. Commun. Math. Phys. 354, 913–956 (2017). https://doi.org/10.1007/s00220-017-2891-0

Download citation