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

SIGMA 10 (2014), 085, 45 pages      arXiv:1407.2124      https://doi.org/10.3842/SIGMA.2014.085

### The Ongoing Impact of Modular Localization on Particle Theory

Bert Schroer a, b
a) CBPF, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil
b) Institut für Theoretische Physik, FU-Berlin, Arnimallee 14, 14195 Berlin, Germany

Received July 05, 2013, in final form July 28, 2014; Published online August 13, 2014

Abstract
Modular localization is the concise conceptual formulation of causal localization in the setting of local quantum physics. Unlike QM it does not refer to individual operators but rather to ensembles of observables which share the same localization region, as a result it explains the probabilistic aspects of QFT in terms of the impure KMS nature arising from the local restriction of the pure vacuum. Whereas it played no important role in the perturbation theory of low spin particles, it becomes indispensible for interactions which involve higher spin $s\geq1$ fields, where is leads to the replacement of the operator (BRST) gauge theory setting in Krein space by a new formulation in terms of stringlocal fields in Hilbert space. The main purpose of this paper is to present new results which lead to a rethinking of important issues of the Standard Model concerning massive gauge theories and the Higgs mechanism. We place these new findings into the broader context of ongoing conceptual changes within QFT which already led to new nonperturbative constructions of models of integrable QFTs. It is also pointed out that modular localization does not support ideas coming from string theory, as extra dimensions and Kaluza-Klein dimensional reductions outside quasiclassical approximations. Apart from hologarphic projections on null-surfaces, holograhic relations between QFT in different spacetime dimensions violate the causal completeness property, this includes in particular the Maldacena conjecture. Last not least, modular localization sheds light onto unsolved problems from QFT's distant past since it reveals that the Einstein-Jordan conundrum is really an early harbinger of the Unruh effect.

Key words: modular localization; string-localization; integrable models.

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References

1. Aste A., Scharf G., Dütsch M., On gauge invariance and spontaneous symmetry breaking, J. Phys. A: Math. Gen. 30 (1997), 5785-5792, hep-th/9705216.
2. Babujian H., Fring A., Karowski M., Zapletal A., Exact form factors in integrable quantum field theories: the sine-Gordon model, Nuclear Phys. B 538 (1999), 535-586, hep-th/9805185.
3. Belavin A.A., Polyakov A.M., Zamolodchikov A.B., Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys. B 241 (1984), 333-380.
4. Bogolyubov N.N., Logunov A., Oksak A.I., Todorov I.T., General principles of quantum field theory, Mathematical Physics and Applied Mathematics, Vol. 10, Kluwer Academic Publishers, Dordrecht, 1990.
5. Borchers H.-J., Field operators as $C^{\infty }$ functions in spacelike directions, Nuovo Cimento 33 (1964), 1600-1613.
6. Borchers H.-J., On revolutionizing quantum field theory with Tomita's modular theory, J. Math. Phys. 41 (2000), 3604-3673.
7. Borchers H.-J., Buchholz D., Schroer B., Polarization-free generators and the $S$-matrix, Comm. Math. Phys. 219 (2001), 125-140, hep-th/0003243.
8. Bros J., Epstein H., Glaser V., A proof of the crossing property for two-particle amplitudes in general quantum field theory, Comm. Math. Phys. 1 (1965), 240-264.
9. Bros J., Mund J., Braid group statistics implies scattering in three-dimensional local quantum physics, Comm. Math. Phys. 315 (2012), 465-488, arXiv:1112.5785.
10. Brower R.C., Spectrum-generating algebra and no-ghost theorem for the dual model, Phys. Rev. D 6 (1972), 1655-1662.
11. Brunetti R., Guido D., Longo R., Modular localization and Wigner particles, Rev. Math. Phys. 14 (2002), 759-785, math-ph/0203021.
12. Buchholz D., Haag-Ruelle approximation of collision states, Comm. Math. Phys. 36 (1974), 243-253.
13. Buchholz D., New light on infrared problems: sectors, statistics, spectrum and all that, arXiv:1301.2516.
14. Buchholz D., D'Antoni C., Longo R., Nuclear maps and modular structures. I. General properties, J. Funct. Anal. 88 (1990), 233-250.
15. Buchholz D., Fredenhagen K., Locality and the structure of particle states, Comm. Math. Phys. 84 (1982), 1-54.
16. Buchholz D., Mack G., Todorov I., The current algebra on the circle as a germ of local field theories, Nuclear Phys. B Proc. Suppl. 5B (1988), 20-56.
17. Buchholz D., Solveen C., Unruh effect and the concept of temperature, Classical Quantum Gravity 30 (2013), 085011, 9 pages, arXiv:1212.2409.
18. Buchholz D., Summers S.J., Scattering in relativistic quantum field theory: fundamental concepts and tools, math-ph/0509047.
19. Buchholz D., Wichmann E.H., Causal independence and the energy-level density of states in local quantum field theory, Comm. Math. Phys. 106 (1986), 321-344.
20. Di Vecchia P., The birth of string theory, in String theory and fundamental interactions, Lecture Notes in Phys., Vol. 737, Springer, Berlin, 2008, 59-118, arXiv:0704.0101.
21. Dong C., Xu F., Conformal nets associated with lattices and their orbifolds, Adv. Math. 206 (2006), 279-306, math.OA/0411499.
22. Duncan A., Janssen M., Pascual Jordan's resolution of the conundrum of the wave-particle duality of light, Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Modern Phys. 39 (2008), 634-666, arXiv:0709.3812.
23. Dütsch M., Gracia-Bondía J.M., Scheck F., Várilly J.C., Quantum gauge models without (classical) Higgs mechanism, Eur. Phys. J. C 69 (2010), 599-621, arXiv:1001.0932.
24. Epstein H., Glaser V., The role of locality in perturbation theory, Ann. Inst. H. Poincaré A 19 (1973), 211-295.
25. Ezawa H., Swieca J.A., Spontaneous breakdown of symmetries and zero-mass states, Comm. Math. Phys. 5 (1967), 330-336.
26. Fassarella L., Schroer B., Wigner particle theory and local quantum physics, J. Phys. A: Math. Gen. 35 (2002), 9123-9164, hep-th/0112168.
27. Haag R., Local quantum physics. Fields, particles, algebras, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, Berlin, 1996.
28. Haag R., Schroer B., Postulates of quantum field theory, J. Math. Phys. 3 (1962), 248-256.
29. Haag R., Swieca J.A., When does a quantum field theory describe particles?, Comm. Math. Phys. 1 (1965), 308-320.
30. Hollands S., Wald R.M., Quantum field theory is not merely quantum mechanics applied to low energy effective degrees of freedom, Gen. Relativity Gravitation 36 (2004), 2595-2603, gr-qc/0405082.
31. Jaffe A.M., High-energy behavior in quantum field theory. I. Strictly localizable fields, Phys. Rev. 158 (1967), 1454-1461.
32. Jost R., TCP-Invarianz der Streumatrix und interpolierende Felder, Helvetica Phys. Acta 36 (1963), 77-82.
33. Kähler R., Wiesbrock H.-W., Modular theory and the reconstruction of four-dimensional quantum field theories, J. Math. Phys. 42 (2001), 74-86.
34. Kawahigashi Y., Longo R., Local conformal nets arising from framed vertex operator algebras, Adv. Math. 206 (2006), 729-751, math.OA/0407263.
35. Kay B.S., Ortíz L., Brick walls and AdS/CFT, Gen. Relativity Gravitation 46 (2014), 1727, 55 pages, arXiv:1111.6429.
36. Lechner G., An existence proof for interacting quantum field theories with a factorizing $S$-matrix, Comm. Math. Phys. 227 (2008), 821-860, math-ph/0601022.
37. Lechner G., Deformations of quantum field theories and integrable models, Comm. Math. Phys. 312 (2012), 265-302, arXiv:1104.1948.
38. Lowe D.A., Causal properties of free string field theory, Phys. Lett. B 326 (1994), 223-230, hep-th/9312107.
39. Lowenstein J.H., Swieca J.A., Quantum electrodynamics in two dimensions, Ann. Physics 68 (1971), 172-195.
40. Lüscher M., Mack G., Global conformal invariance in quantum field theory, Comm. Math. Phys. 41 (1975), 203-234.
41. Mack G., $D$-independent representation of conformal field theories in $D$ dimensions via transformation to auxiliary dual resonance models. Scalar amplitudes, arXiv:0907.2407.
42. Mack G., $D$-dimensional conformal field theories with anomalous dimensions as dual resonance models, 2009, arXiv:0909.1024.
43. Majorana E., Teoria relativistica di particelle con momentum internisico arbitrario, Nuovo Cimento 9 (1932), 335-344.
44. Maldacena J., The illusion of gravity, Sci. Amer. 293 (2005), no. 5, 56-63.
45. Mandelstam S., Determination of the pion-nucleon scattering amplitude from dispersion relations and unitarity. General theory, Phys. Rev. 112 (1958), 1344-1360.
46. Mandelstam S., Analytic properties of transition amplitudes in perturbation theory, Phys. Rev. 115 (1959), 1741-1751.
47. Mandelstam S., Feynman rules for electromagnetic and Yang-Mills fields from the gauge-independent field-theoretic formalism, Phys. Rev. 175 (1968), 1580-1603.
48. Martinec E., The light cone in string theory, Classical Quantum Gravity 10 (1993), L187-L192, hep-th/9304037.
49. Mund J., An algebraic Jost-Schroer theorem for massive theories, Comm. Math. Phys. 315 (2012), 445-464, arXiv:1012.1454.
50. Mund J., String-localized massive vector bosons without ghosts and indefinite metric: the example of massive QED, in preparation.
51. Mund J., Schroer B., Renormalization theory of string-localized self-coupled massive vectormesons in Hilbert space, in preparation.
52. Mund J., Schroer B., Yngvason J., String-localized quantum fields and modular localization, Comm. Math. Phys. 268 (2006), 621-672, math-ph/0511042.
53. Plaschke M., Yngvason J., Massless, string localized quantum fields for any helicity, J. Math. Phys. 53 (2012), 042301, 15 pages, arXiv:1111.5164.
54. Polchinski J., String theory. I. An introduction to the bosonic string, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1998.
55. Rehren K.-H., Local quantum observables in the anti-de-Sitter - conformal QFT correspondence, Phys. Lett. B 493 (2000), 383-388, hep-th/0003120.
56. Scharf G., Quantum gauge theories. A true ghost story, John Wiley & Sons, New York, 2001.
57. Schroer B., Modular localization and the bootstrap-formfactor program, Nuclear Phys. B 499 (1997), 547-568, hep-th/9702145.
58. Schroer B., Modular wedge localization and the $d=1+1$ formfactor program, Ann. Physics 275 (1999), 190-223, hep-th/9712124.
59. Schroer B., Jorge A. Swieca's contributions to quantum field theory in the 60s and 70s and their relevance in present research, Eur. Phys. J. H 35 (2010), 53-88, arXiv:0712.0371.
60. Schroer B., Localization and the interface between quantum mechanics, quantum field theory and quantum gravity I: the two antagonistic localizations and their asymptotic compatibility, Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Modern Phys. 41 (2010), 104-127, arXiv:0912.2874.
61. Schroer B., Unexplored regions in QFT and the conceptual foundations of the Standard Model, arXiv:1006.3543.
62. Schroer B., An alternative to the gauge theoretic setting, Found. Phys. 41 (2011), 1543-1568, arXiv:1012.0013.
63. Schroer B., Modular localization and the foundational origin of integrability, Found. Phys. 43 (2013), 329-372, arXiv:1109.1212.
64. Schroer B., Causality and dispersion relations and the role of the $S$-matrix in the ongoing research, Found. Phys. 42 (2012), 1481-1522, arXiv:1107.1374.
65. Schroer B., The Einstein-Jordan conundrum and its relation to ongoing foundational research in local quantum physics, Eur. Phys. J. H 38 (2013), 137-173, arXiv:1101.0569.
66. Schroer B., Dark matter and Wigner's third positive-energy representation class, arXiv:1306.3876.
67. Schroer B., A Hilbert space setting for interacting higher spin fields and the Higgs issue, arXiv:1407.0365.
68. Schroer B., Swieca J.A., Conformal transformations for quantized fields, Phys. Rev. D 10 (1974), 480-485.
69. Schwinger J., Trieste lectures, IAEA, Vienna, 1963.
70. Sewell G., Quantum fields on manifolds: PCT and gravitationally induced thermal states, Ann. Physics 141 (1982), 201-224.
71. Staskiewicz C.P., Die lokale Struktur abelscher Stromalgebren auf dem Kreis, Ph.D. Thesis, Freie Universität Berlin, 1995.
72. Streater R.F., Wightman A.S., PCT, spin and statistics, and all that, W. A. Benjamin, Inc., New York - Amsterdam, 1964.
73. Swieca J.A., Charge screening and mass spectrum, Phys. Rev. D 13 (1976), 312-314.
74. Unruh W.G., Notes on black-hole evaporation, Phys. Rev. D 14 (1976), 870-892.
75. Veneziano G., Construction of a crossing-simmetric, Regge-behaved amplitude for linearly rising trajectories, Nuovo Cimento A 57 (1968), 190-197.
76. Weinberg S., The quantum theory of fields. I. Foundations, Cambridge University Press, Cambridge, 2005.
77. Yennie D., Frautschi S., Suura H., The infrared divergence phenomena and high-energy processes, Ann. Physics 13 (1961), 379-452.
78. Yngvason J., Zero-mass infinite spin representations of the Poincaré group and quantum field theory, Comm. Math. Phys. 18 (1970), 195-203.