Skip to main content
SpringerLink
Log in
Book cover

Advances in Algebraic Quantum Field Theory pp 397–448Cite as

Algebraic Constructive Quantum Field Theory: Integrable Models and Deformation Techniques

  • Chapter
  • First Online:

Part of the book series: Mathematical Physics Studies ((MPST))

Abstract

Several related operator-algebraic constructions for quantum field theory models on Minkowski spacetime are reviewed. The common theme of these constructions is that of a Borchers triple, capturing the structure of observables localized in a Rindler wedge. After reviewing the abstract setting, we discuss in this framework (i) the construction of free field theories from standard pairs, (ii) the inverse scattering construction of integrable QFT models on two-dimensional Minkowski space, and (iii) the warped convolution deformation of QFT models in arbitrary dimension, inspired from non-commutative Minkowski space.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    A term coined by S.J. Summers, see also his online article http://people.clas.ufl.edu/sjs/constructive-quantum-field-theory/ for a review.

  2. 2.

    For large parts of this review, we will rely on Tomita-Takesaki modular theory, see for example [35] for an introduction and [24] for an overview of applications to QFT.

  3. 3.

    Although we will mostly be working with Minkowski space here, it should be noted that similar families of regions can also be defined in other situations: On the one-dimensional line, the half lines \((a,\infty )\) and \((-\infty ,a)\), \(a\in {\mathbb {R}}\), have the same properties as the wedges in Minkowski space (see also the discussion in Sect. 10.3.5). Also the family of all intervals on a circle, of prominent importance in chiral conformal field theory (Chap. 8), shares many properties with the family of wedges on Minkowski space, as it is the orbit of a reference region (e.g., the upper semi circle) under a symmetry group (the Möbius group PSL\((2,{\mathbb {R}})\)), see for example [108].

    Furthermore, on certain curved spacetimes, such as de Sitter space [25, 36], anti de Sitter space [45, 93], and more general curved spacetimes [51, 58], families of regions with properties analogous to Minkowski space wedges exist.

  4. 4.

    This implies that \(W_R\) is globally hyperbolic, and can be regarded as a spacetime in its own right.

  5. 5.

    Note, for example, that for any \(n\in \mathbb {N}\), there exists a family of n wedges \(W_1,\ldots ,W_n\subset {\mathbb {R}}^4\) such that \(W_i\cap W_j=\emptyset \) for \(i\ne j\) [135].

  6. 6.

    Note that we deviate here slightly from the definition in [53, Definition 4.1], where the term “causal Borchers triple” has been used. Also note that in [82], there is a related but different definition of the term “Borchers triple”. We will always stick to the definition given here.

  7. 7.

    Here and in many places in the following text, we will make use of the Tomita-Takesaki modular theory of von Neumann algebras with cyclic separating vector, see for example [35] for an introduction.

  8. 8.

    This statement is stronger than the one of Theorem 10.2.6, which does not yield equality of the modular data \(J_W,\varDelta _W^{it}\) with the Lorentz transformations \(U(j_W)\), \(U(\varLambda _W(t))\). However, in the context of a local net satisfying further assumptions, including asymptotic completeness, Mund proved that the Bisognano-Wichmann property does follow from Borchers’ theorem [114].

  9. 9.

    This is even the case for the projection \(Q=1\).

  10. 10.

    In case \(U_1\) does not contain the trivial representation, as is adequate for a single particle representation, the algebras corresponding to spacelike cones are also known to be factors of type III\(_1\) [36, 71].

  11. 11.

    That is, a von Neumann algebra isomorphic to \({\mathcal B}(\tilde{\mathcal {H}})\) for some Hilbert space \(\tilde{\mathcal {H}}\).

  12. 12.

    The continuity assumption can be relaxed, cf. [104].

  13. 13.

    Picking this particular Hilbert space is a matter of choice, see [3, 98] for other possibilities.

  14. 14.

    It is in fact p-nuclear for any \(p>0\), i.e. its singular values decay faster than any inverse power.

  15. 15.

    Note that the stronger results claimed in [97] contain an incorrect estimate [99], spotted by S. Alazzawi. At the time of writing, the result stated here is the strongest one with a complete proof.

  16. 16.

    That this is a non-trivial issue can be seen (in the setting of Wightman QFT) at the example of a family of complicated Wightman functions [121] which were only later realized to be equivalent to generalized free fields [122].

  17. 17.

    However, boundedness of A is not directly reflected in its coefficients \(f^{[A]}_{n,m}\) because the expansion (10.77) involves the unbounded operators \(z_S^\#\).

  18. 18.

    See, however, [38, 40, 66] for results on scattering involving massless particles.

  19. 19.

    As we saw in Theorem 10.2.5, wedge algebras are always type III\(_1\) factors, and are in fact expected to be isomorphic to the unique hyperfinite type III\(_1\) factor in generic cases. Thus “deforming \({\mathcal M}\)” does not mean deforming the (fixed) internal algebraic structure of \({\mathcal M}\), as in other algebraic deformation procedures [75]. Rather, “deforming \({\mathcal M}\)” means deforming/changing the position of \({\mathcal M}\) inside \({\mathcal B}(\mathcal {H})\), i.e. deforming the subfactor \({\mathcal M}\subset {\mathcal B}(\mathcal {H})\).

  20. 20.

    However, it is entirely possible to “deform” Borchers triples by keeping the wedge algebra fixed while changing U and \(\varOmega \).

  21. 21.

    The Weyl-Moyal product (see, for example [77]) features prominently in deformation quantization [136], where it describes the transition from classical mechanics to quantum mechanics. In that setting, one considers suitable number-valued functions f, g on the simple classical phase space \({\mathbb {R}}^d\) (d even) and equips them with the non-commutative product

    $$(f\star g)(y)=(2\pi )^{-d}\int _{{\mathbb {R}}^d}dp\int _{{\mathbb {R}}^d}dx\,e^{-ipx}f(y+\hbar \theta p)g(y+x),$$

    where the antisymmetric matrix \(\theta \) is given by the Poisson bracket.

  22. 22.

    See also [118] for another recent closely related approach, drawing from [90].

  23. 23.

    This is related to the fact that Q is skew-symmetric and for \(d>4\), the edge of \(W_R\) is fixed by the subgroup SO\((d-2)\) of rotations in the edge.

  24. 24.

    There is also a sharp difference between the undeformed and deformed case in the thermal equilibrium states. In the deformed \((Q\ne 0\)) situation, the thermal representation leads to a decoupling of the noncommutativity parameters \(\varLambda Q\varLambda ^{-1}\) related to different wedges [101].

References

  1. Abdalla, E., Abdalla, C., Rothe, K.D.: Non-perturbative Methods in 2-Dimensional Quantum Field Theory. World Scientific, Singapore (1991)

    Google Scholar 

  2. Åks, S.: Proof that scattering implies production in quantum field theory. J. Math. Phys. 6, 516–532 (1965)

    Article  ADS  MathSciNet  Google Scholar 

  3. Alazzawi, S.: Deformations of fermionic quantum field theories and integrable models. Lett. Math. Phys. 103(1), 37–58 (2013)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Alazzawi, S.: Deformations of quantum field theories and the construction of interacting models. Ph.D. thesis, University of Vienna, 2014. arXiv:1503.00897

  5. Andersson, A.: Operator deformations in quantum measurement theory. Lett. Math. Phys. 104(4), 415–430 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Araki, H.: A lattice of von Neumann algebras associated with the quantum theory of a free bose field. J. Math. Phys. 4(11), 1343–1362 (1963)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Araki, H.: Von Neumann algebras of local observables for free scalar field. J. Math. Phys. 5(1), 1–13 (1964)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Arinshtein, A.E., Fateev, V.A., Zamolodchikov, A.B.: Quantum S-matrix of the (1+1)-dimensional Toda chain. Phys. Lett. B 87, 389–392 (1979)

    Article  ADS  Google Scholar 

  9. Babujian, H.M., Karowski, M.: Towards the construction of Wightman functions of integrable quantum field theories. Int. J. Mod. Phys. A19S2, 34–49 (2004)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. Babujian, H.M., Foerster, A., Karowski, M.: SU(N) and O(N) off-shell nested Bethe ansatz and exact form factors. J. Phys. A: Math. Theor. 41(27), 275202 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Balog, J., Hauer, T.: Polynomial form-factors in the O(3) nonlinear sigma model. Phys. Lett. B 337, 115–121 (1994)

    Article  ADS  Google Scholar 

  13. Barata, J., Jäkel, C., Mund, J.: The \({\cal P}(\varphi )_2\) Model on the de Sitter Space. Preprint, arXiv:1311.2905 (2013)

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

    Google Scholar 

  15. Baumgärtel, H., Jurke, M., Lledó, F.: On free nets over Minkowski space. Rept. Math. Phys. 35(1), 101–127 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Baumgärtel, H., Jurke, M., Lledo, F.: Twisted duality of the CAR algebra. J. Math. Phys. 43, 4158–4179 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Berg, B., Karowski, M., Weisz, P.: Construction of Green’s functions from an exact S-matrix. Phys. Rev. D 19(8), 2477–2479 (1979)

    Article  ADS  Google Scholar 

  18. Bischoff, M.: Construction of models in low-dimensional quantum field theory using operator algebraic methods. Ph.D. thesis, Rome (2012)

    Google Scholar 

  19. Bischoff, M., Tanimoto, Y.: Construction of wedge-local nets of observables through Longo-Witten endomorphisms. II. Comm. Math. Phys. 317(3), 667–695 (2013)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Borchers, H.-J.: On revolutionizing quantum field theory with Tomita’s modular theory. J. Math. Phys. 41, 3604–3673 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Borchers, H.-J., Buchholz, D.: Global properties of vacuum states in de Sitter space. Ann. Poincare Phys. Theor. A70, 23–40 (1999)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Bostelmann, H.: Operator product expansions as a consequence of phase space properties. J. Math. Phys. 46, 082304 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Bostelmann, H.: Phase space properties and the short distance structure in quantum field theory. J. Math. Phys. 46, 052301 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. Bostelmann, H., Cadamuro, D.: An operator expansion for integrable quantum field theories. J. Phys. A: Math. Theor. 46, 095401 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Bostelmann, H., Cadamuro, D.: Characterization of local observables in integrable quantum field theories. Comm. Math. Phys. (2014) (To appear)

    Google Scholar 

  31. Bostelmann, H., Cadamuro, D.: Negative energy densities in integrable quantum field theories at one-particle level. Preprint, arXiv:1502.01714 (2015)

  32. Bostelmann, H., D’Antoni, C., Morsella, G.: Scaling algebras and pointlike fields: a nonperturbative approach to renormalization. Comm. Math. Phys. 285, 763–798 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Bostelmann, H., Lechner, G., Morsella, G.: Scaling limits of integrable quantum field theories. Rev. Math. Phys. 23(10), 1115–1156 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Bostelmann, H., Cadamuro, D., Fewster, C.: Quantum energy inequality for the massive Ising model. Phys. Rev. D 88(2), 025019 (2013)

    Article  ADS  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  37. Buchholz, D.: Product states for local algebras. Comm. Math. Phys. 36, 287–304 (1974)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. Buchholz, D.: Collision theory for massless fermions. Comm. Math. Phys. 42, 269 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  39. Buchholz, D.: Collision theory for waves in two dimensions and a characterization of models with trivial S-matrix. Comm. Math. Phys. 45, 1–8 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  40. Buchholz, D.: Collision theory for massless bosons. Comm. Math. Phys. 52, 147 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  41. Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states. Comm. Math. Phys. 84, 1 (1982)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. Buchholz, D., Junglas, P.: On the existence of equilibrium states in local quantum field theory. Comm. Math. Phys. 121, 255–270 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  44. Buchholz, D., Porrmann, M.: How small is the phase space in quantum field theory? Ann. Poincare Phys. Theor. 52, 237 (1990)

    MathSciNet  MATH  Google Scholar 

  45. Buchholz, D., Summers, S.J.: Stable quantum systems in anti-de Sitter space: causality, independence and spectral properties. J. Math. Phys. 45, 4810–4831 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  46. Buchholz, D., Summers, S.J.: Warped convolutions: a novel tool in the construction of quantum field theories. In: Seiler, E., Sibold, K. (eds.) Quantum Field Theory and Beyond: Essays in Honor of Wolfhart Zimmermann, pp. 107–121. World Scientific, Singapore (2008)

    Google Scholar 

  47. Buchholz, D., Verch, R.: Scaling algebras and renormalization group in algebraic quantum field theory. II: instructive examples. Rev. Math. Phys. 10, 775–800 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  48. Buchholz, D., Wichmann, E.H.: Causal independence and the energy level density of states in local quantum field theory. Comm. Math. Phys. 106, 321 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. Buchholz, D., D’Antoni, C., Longo, R.: Nuclear maps and modular structures 2: applications to quantum field theory. Comm. Math. Phys. 129, 115 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  50. Buchholz, D., D’Antoni, C., Longo, R.: Nuclear maps and modular structures. I. General properties. J. Funct. Anal. 88, 233–250 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  51. Buchholz, D., Mund, J., Summers, S.J.: Transplantation of local nets and geometric modular action on Robertson-walker space-times. Fields Inst. Commun. 30, 65–81 (2001)

    MathSciNet  MATH  Google Scholar 

  52. Buchholz, D., D’Antoni, C., Longo, R.: Nuclearity and thermal states in conformal field theory. Comm. Math. Phys. 270, 267–293 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  53. Buchholz, D., Lechner, G., Summers, S.J.: Warped convolutions, Rieffel deformations and the construction of quantum field theories. Comm. Math. Phys. 304, 95–123 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  54. Cadamuro, D.: A characterization theorem for local operators in factorizing scattering models. Ph.D. thesis, Göttingen University, 2012

    Google Scholar 

  55. Cadamuro, D., Tanimoto, Y.: Wedge-local fields in integrable models with bound states. Preprint, arXiv:1502.01313 (2015)

  56. D’Antoni, C., Longo, R.: Interpolation by type I factors and the flip automorphism. J. Funct. Anal. 51, 361 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  57. D’Antoni, C., Morsella, G., Verch, R.: Scaling algebras for charged fields and short-distance analysis for localizable and topological charges. Ann. Henri Poincare 5, 809–870 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  58. Dappiaggi, C., Lechner, G., Morfa-Morales, E.: Deformations of quantum field theories on spacetimes with Killing fields. Comm. Math. Phys. 305(1), 99–130 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  60. Doplicher, S., Fredenhagen, K., Roberts, J.E.: The Quantum structure of space-time at the Planck scale and quantum fields. Comm. Math. Phys. 172, 187–220 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  61. Doplicher, S., Haag, R., Roberts, J.E.: Fields, observables and gauge transformations. I. Comm. Math. Phys. 13, 1–23 (1969)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  62. Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics. II. Comm. Math. Phys. 35, 49–85 (1974)

    Article  MathSciNet  ADS  Google Scholar 

  63. Dorey, P.: Exact S-matrices. In: Horváth, Z., Palla, L. (eds.) Conformal Field Theories and Integrable Models. Lecture Notes in Physics, vol. 498, pp. 85–125. Springer, New York (1997)

    Google Scholar 

  64. Driessler, W.: Comments on lightlike translations and applications in relativistic quantum field theory. Comm. Math. Phys. 44, 133–141 (1975)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  65. Dütsch, M., Fredenhagen, K.: Algebraic quantum field theory, perturbation theory, and the loop expansion. Comm. Math. Phys. 219, 5–30 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  66. Dybalski, W.: Haag-Ruelle scattering theory in presence of massless particles. Lett. Math. Phys. 72, 27–38 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  67. Dybalski, W., Tanimoto, Y.: Asymptotic completeness in a class of massless relativistic quantum field theories. Comm. Math. Phys. 305, 427–440 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  68. Eckmann, J.-P., Osterwalder, K.: An application of Tomita’s theory of modular Hilbert algebras: duality for free Bose fields. J. Funct. Anal. 13(1), 1–12 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  69. Faddeev, L.D.: Quantum completely integrable models in field theory, volume 1 of Mathematical Physics Reviews, pp. 107–155. In: Novikov, S.P. (ed.) Mathematical Physics Reviews, vol. 1, pp. 107–155 (1984)

    Google Scholar 

  70. Fassarella, L., Schroer, B.: Wigner particle theory and local quantum physics. J. Phys. A 35, 9123–9164 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  71. Figliolini, F., Guido, D.: On the type of second quantization factors. J. Oper. Theory 31(2), 229–252 (1994)

    MathSciNet  MATH  Google Scholar 

  72. Florig, M.: On Borchers’ theorem. Lett. Math. Phys. 46, 289–293 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  73. Foit, J.J.: Abstract twisted duality for quantum free Fermi fields. Publ. Res. Inst. Math. Sci. Kyoto 19, 729–741 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  74. Fring, A., Mussardo, G., Simonetti, P.: Form-factors of the elementary field in the Bullough-Dodd model. Phys. Lett. B 307, 83–90 (1993)

    Article  MathSciNet  ADS  Google Scholar 

  75. Gerstenhaber, M.: On the deformation of rings and algebras. Ann. Math. 79, 59–103 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  76. Glimm, J., Jaffe, A.: Quantum Physics. A Functional Integral Point of View, 535p. Springer, New York, (1987)

    Google Scholar 

  77. Gracia-Bondia, J.M., Varilly, J.C.: Algebras of distributions suitable for phase space quantum mechanics. I. J. Math. Phys. 29, 869–879 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  78. Grosse, H., Lechner, G.: Wedge-local quantum fields and noncommutative Minkowski space. JHEP 11, 012 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  79. Grosse, H., Lechner, G.: Noncommutative deformations of wightman quantum field theories. JHEP 09, 131 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  80. Grosse, H., Lechner, G., Ludwig, T., Verch, R.: Wick rotation for quantum field theories on degenerate Moyal space (-time). J. Math. Phys. 54, 022307 (2013)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  81. Guido, D., Longo, R.: An algebraic spin and statistics theorem. Comm. Math. Phys. 172(3), 517 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  82. Guido, D., Longo, R., Wiesbrock, H.W.: Extensions of conformal nets and superselection structures. Comm. Math. Phys. 192, 217–244 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  83. Haag, R.: Quantum field theories with composite particles and asymptotic conditions. Phys. Rev. 112, 669–73 (1958)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  84. Haag, R.: Local Quantum Physics—Fields, Particles, Algebras, 2nd edn. Springer, Berlin (1996)

    Google Scholar 

  85. Haagerup, U.: Conne’s bicentralizer problem and uniqueness of the injective factor of type III1. Acta Math. 158(1), 95 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  86. Hepp, K.: On the connection between Wightman and LSZ quantum field theory. Comm. Math. Phys. 1, 95–111 (1965)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  87. Iagolnitzer, D.: Scattering in Quantum Field Theories. Princeton University Press, Princeton (1993)

    Google Scholar 

  88. Jost, R.: The General Theory of Quantized Fields. American Mathematical Society, Providence (1965)

    Google Scholar 

  89. Karowski, M., Thun, H.J., Truong, T.T., Weisz, P.H.: On the uniqueness of a purely elastic S-matrix in (1+1) dimensions. Phys. Lett. B 67(3), 321–322 (1977)

    Article  ADS  Google Scholar 

  90. Kasprzak, P.: Rieffel Deformation via crossed products. J. Funct. Anal. 257(5), 1288–1332 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  91. Kawahigashi, Y., Longo, R.: Classification of local conformal nets: case c \(<\) 1. Ann. Math. 160, 493–522 (2004)

    Google Scholar 

  92. Köhler, C.: Ph.D. thesis, University of Vienna (2015) (To appear)

    Google Scholar 

  93. Lauridsen-Ribeiro, P.: Structural and dynamical aspects of the AdS/CFT correspondence: a rigorous approach. Ph.D. thesis, Sao Paulo, 2007

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  95. Lechner, G.: On the existence of local observables in theories with a factorizing S-matrix. J. Phys. A 38, 3045–3056 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  96. Lechner, G.: On the construction of quantum field theories with factorizing S-matrices. Ph.D. thesis, University of Göttingen, 2006

    Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  98. Lechner, G.: Deformations of quantum field theories and integrable models. Comm. Math. Phys. 312(1), 265–302 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  99. Lechner, G.: Erratum (2015)

    Google Scholar 

  100. Lechner, G., Longo, R.: Localization in nets of standard spaces. Comm. Math. Phys. 336, 27–61 (2015)

    Google Scholar 

  101. Lechner, G., Schlemmer, J.: Thermal equilibrium states for quantum fields on non-commutative spacetimes. arXiv:1503.01639 (2015)

  102. Lechner, G., Schützenhofer, C.: Towards an operator-algebraic construction of integrable global gauge theories. Ann. Henri Poincaré 15(4), 645–678 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  103. Lechner, G., Waldmann, S.: Strict deformation quantization of locally convex algebras and modules. Preprint, arXiv:1109.5950 (2011)

  104. Lechner, G., Schlemmer, J., Tanimoto, Y.: On the equivalence of two deformation schemes in quantum field theory. Lett. Math. Phys. 103(4), 421–437 (2013)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  105. Leylands, P., Roberts, J.E., Testard, D.: Duality for Quantum Free Fields. Preprint, 1978

    Google Scholar 

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

    Google Scholar 

  107. Longo, R.: Notes on algebraic invariants for noncommutative dynamical systems. Comm. Math. Phys. 69, 195–207 (1979)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  108. Longo, R.: Lectures on conformal Nets—Part 1. In: Von Neumann algebras in Sibiu, Theta (2008)

    Google Scholar 

  109. Longo, R., Witten, E.: An algebraic construction of boundary quantum field theory. Comm. Math. Phys. 303(1), 213–232 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  110. Martin, A.: Can one continue the scattering amplitude through the elastic cut? (1969)

    Google Scholar 

  111. Morfa-Morales, E.: Deformations of quantum field theories on curved spacetimes. Ph.D. thesis, University of Vienna, 2012

    Google Scholar 

  112. Much, A.: Quantum mechanical effects from deformation theory. J. Math. Phys. 55, 022302 (2014)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  113. Müger, M.: Superselection structure of massive quantum field theories in 1+1 dimensions. Rev. Math. Phys. 10, 1147–1170 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  114. Mund, J.: The Bisognano-Wichmann theorem for massive theories. Ann. Henri Poincare 2, 907–926 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  115. Mund, J.: An algebraic Jost-Schroer theorem for massive theories. Comm. Math. Phys. 315, 445–464 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  116. Mund, J., Schroer, B., Yngvason, J.: String-localized quantum fields and modular localization. Comm. Math. Phys. 268, 621–672 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  117. Mussardo, G.: Off critical statistical models: factorized scattering theories and bootstrap program. Phys. Rept. 218, 215–379 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  118. Neshveyev, S.: Smooth crossed products of Rieffel’s deformations. Lett. Math. Phys. 104(3), 361–371 (2014)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  119. Niedermaier, M.R.: A derivation of the cyclic form factor equation. Comm. Math. Phys. 196, 411–428 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  120. Plaschke, M.: Wedge local deformations of charged fields leading to anyonic commutation relations. Lett. Math. Phys. 103(5), 507–532 (2013)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  121. Read, C.J.: Quantum field theories in all dimensions. Comm. Math. Phys. 177(3), 631 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  122. Rehren, K.-H.: Comments on a recent solution to Wightman’s axioms. Comm. Math. Phys. 178, 453–466 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  123. Rieffel, M.A.: Deformation Quantization for Actions of \(R^d\). Memoirs of the American Mathematical Society, vol. 106. American Mathematical Society, Providence (1992)

    Google Scholar 

  124. Ruelle, D.: On the asymptotic condition in quantum field theory. Helv. Phys. Acta 35, 147–163 (1962)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  126. Schroer, B.: Modular wedge localization and the d=1+1 formfactor program. Ann. Phys. 275, 190–223 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  127. Smirnov, F.A.: Form Factors in Completely Integrable Models of Quantum Field Theory. World Scientific, Singapore (1992)

    Book  MATH  Google Scholar 

  128. Smirnov, F.A.: A New set of exact form-factors. Int. J. Mod. Phys. A 9, 5121–5144 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  129. Soloviev, M.A.: On the failure of microcausality in noncommutative field theories. Phys. Rev. D 77, 125013 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  130. Streater, R.F., Wightman, A.: PCT, Spin and Statistics, and All That. Benjamin-Cummings, Reading (1964)

    MATH  Google Scholar 

  131. Summers, S.J.: Normal product states for fermions and twisted duality for CCR- and CAR-type algebras with application to the Yukawa2 quantum field model. Comm. Math. Phys. 86, 111–141 (1982)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  132. Summers, S.J.: On the independence of local algebras in quantum field theory. Rev. Math. Phys. 2, 201–247 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  133. Summers, S.J.: A Perspective on Constructive Quantum Field Theory. arXiv:1203.3991 (2012)

  134. Tanimoto, Y.: Construction of wedge-local nets of observables through Longo-Witten endomorphisms. Comm. Math. Phys. 314(2), 443–469 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  135. Thomas, L.J., Wichmann, E.H.: On the causal structure of Minkowski space-time. J. Math. Phys. 38, 5044–5086 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  136. Waldmann, S.: Poisson-Geometrie und Deformationsquantisierung. Springer, New York (2007)

    Google Scholar 

  137. Weinberg, S.: The Quantum Theory of Fields I—Foundations. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  138. Yngvason, J.: Zero-mass infinite spin representations of the Poincaré group and quantum field theory. Comm. Math. Phys. 18, 195–203 (1970)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

Download references

Acknowledgments

The publications reviewed in this article include joint work with my colleagues S. Alazzawi, H. Bostelmann, D. Buchholz, C. Dappiaggi, H. Grosse, R. Longo, T. Ludwig, E. Morfa-Morales, G. Morsella, J. Schlemmer, C. Schützenhofer, S.J. Summers, Y. Tanimoto, R. Verch, and S. Waldmann. I wish to thank them all for fruitful and enjoyable collaborations.

Author information

Authors and Affiliations

  1. School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, Wales, UK

    Gandalf Lechner

Authors
  1. Gandalf Lechner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gandalf Lechner .

Editor information

Editors and Affiliations

  1. University of Trento, Trento, Italy

    Romeo Brunetti

  2. University of Pavia, Pavia, Italy

    Claudio Dappiaggi

  3. University of Hamburg, Hamburg, Germany

    Klaus Fredenhagen

  4. University of Vienna, Vienna, Austria

    Jakob Yngvason

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Lechner, G. (2015). Algebraic Constructive Quantum Field Theory: Integrable Models and Deformation Techniques. In: Brunetti, R., Dappiaggi, C., Fredenhagen, K., Yngvason, J. (eds) Advances in Algebraic Quantum Field Theory. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-21353-8_10

Download citation

Publish with us

Policies and ethics

Search

Navigation