×

Consistent descriptions of quantum fields. (English) Zbl 1384.81065

Summary: Study of explicit generalized functions identifies a technical generalization of the Wightman functional analytic axioms that admits realizations of quantum fields with interaction. Scalar field examples are discussed.

MSC:

81T05 Axiomatic quantum field theory; operator algebras
81T08 Constructive quantum field theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Wightman, A. S., Hilbert’s Sixth Problem: Mathematical Treatment of the Axioms of Physics, Mathematical Development Arising from Hilbert Problems, (Browder, F. E., Symposia in Pure Mathematics 28 (1976), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 147 · Zbl 0339.46051
[2] Wightman, A. S., Quantum field theory in terms of vacuum expectation values, Phys. Rev., 101, 860 (1956) · Zbl 0074.22902
[3] Streater, R. F.; Wightman, A. S., PCT, Spin and Statistics and All That (1964), W. A. Benjamin: W. A. Benjamin Reading, MA · Zbl 0135.44305
[4] Bogolubov, N. N.; Logunov, A. A.; Todorov, I. T., (Fulling, Stephen; Popova, Ludmilla, Introduction to Axiomatic Quantum Field Theory (1975), W. A. Benjamin: W. A. Benjamin Reading, MA) · Zbl 1114.81300
[5] Baumgärtel, H.; Wollenberg, M., A class of nontrivial weakly local massive wightman fields with interpolating properties, Commun. Math. Phys., 94, 331 (1984) · Zbl 0578.47007
[6] Lechner, G., Deformations of quantum field theories and integrable models, Commun. Math. Phys., 312, 265 (2012) · Zbl 1243.81107
[7] Albeverio, S.; Gottschalk, H.; Wu, J.-L., Convoluted generalized white noise, Schwinger functions and their analytic continuation to Wightman functions, Rev. Math. Phys., 8, 763 (1996) · Zbl 0870.60038
[8] Jaffe, A., Constructive quantum field theory, (Fokas, A. S., Mathematical Physics 2000 (2000), Imperial College Press: Imperial College Press London) · Zbl 1074.81543
[9] Johnson, G. E., Introduction to quantum field theory exhibiting interaction (Feb. 2015), arXiv:math-ph/1502.07727
[10] Johnson, G. E., Algebras without involution and quantum field theories (March 2012), arXiv:math-ph/1203.2705
[11] Weinberg, S., The Quantum Theory of Fields, Volume I, Foundations (1995), Cambridge University Press: Cambridge University Press New York, NY · Zbl 0959.81002
[12] Johnson, G. E., Classical approximations of relativistic quantum physics (April 2016), arXiv:quant-ph/1604.07654
[13] Johnson, G. E., Fields and Quantum Mechanics (December 2013), arXiv:math-ph/1312.2608
[14] Borchers, H. J., On the structure of the algebra of field operators, Nuovo Cimento, 24, 214 (1962) · Zbl 0129.42205
[15] Gel’fand, I. M.; Shilov, G. E., (Friedman, M. D.; Feinstein, A.; Peltzer, C. P., Generalized Functions, 2 (1968), Academic Press: Academic Press New York, NY) · Zbl 0159.18301
[16] Wightman, A. S.; Gårding, L., Fields as operator-valued distributions in relativistic quantum theory, Arkiv för Fysik, 28, 129 (1965) · Zbl 0138.45401
[17] Schwabl, F., (Hilton, R.; Lahee, A., Advanced Quantum Mechanics (1999), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0997.81505
[18] Dieudonné, J., Treatise on Analysis, Volume II, (MacDonald, I. G., 10-II in Pure and Applied Mathematics, A Series of Monographs and Textbooks (1970), Academic Press: Academic Press New York, NY)
[19] Newton, T. D.; Wigner, E. P., Localized states for elementary systems, Rev. Modern Phys., 21, 400 (1949) · Zbl 0036.26704
[20] Yngvason, J., Localization and Entanglement in Relativistic Quantum Physics (Jan. 2014), arXiv:quant-ph/1401.2652
[21] Brunetti, R.; Guido, D.; Longo, R., Modular localization and Wigner particles, Rev. Math. Phys., 14, 759 (2002) · Zbl 1033.81063
[22] von Neumann, J., Mathematical Foundations of Quantum Mechanics (1955), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0064.21503
[23] Johnson, G. E., Massless Particles in QFT from Algebras without Involution (May 2012), arXiv:math-ph/1205.4323
[24] Ruelle, D., On the asymptotic condition in quantum field theory, Helv. Phys. Acta., 35, 147 (1962) · Zbl 0158.45702
[25] Steinmann, O., Structure of the two-point function, J. Math. Phys., 4, 583 (1963) · Zbl 0128.22101
[26] Reeh, H.; Schlieder, S., Bemerkungen zur Unitäräquivalenz von Lorentzinvarianten Feldern, Nuovo Cimento, 22, 1051 (1961) · Zbl 0101.22402
[27] Segal, I. E.; Goodman, R. W., Anti-locality of certain Lorentz-invariant operators, J. Math. Mech., 14, 629 (1965) · Zbl 0151.44201
[28] Masuda, K., Anti-locality of the one-half power of elliptic differential operators, Publ. RIMS, Kyoto Univ., 8, 207 (1972) · Zbl 0245.35026
[29] Hegerfeldt, G. C., Prime field decompositions and infinitely divisible states on Borchers’ tensor algebra, Commun. Math. Phys., 45, 137 (1975) · Zbl 0315.46065
[30] Yngvason, J., The Role of Type III Factors on Quantum Field Theory, Rep. Math. Phys., 55, 1, 135 (2005) · Zbl 1140.81427
[31] Schrödinger, E., Der stetige Übergang von der Mikro-zur Makromechanik, Die Naturwissenschaften, 14, Issue 28, 664 (1926) · JFM 52.0967.01
[32] Ehrenfest, P., Bemerkung über die angenäherte Gültigkeit der klassichen Mechanik innerhalb der Quanatenmechanik, Z. Physik, 45, 455 (1927) · JFM 53.0843.01
[33] Messiah, A., Quantum Mechanics, I (1968), John Wiley and Sons: John Wiley and Sons New York, NY
[34] DeWitt, B. S.; Everett, H.; Graham, N.; Wheeler, J. A., (DeWitt, B. S.; Graham, N., The Many-worlds Interpretation of Quantum Mechanics (1973), Princeton University Press: Princeton University Press Princeton, NJ)
[35] Pusey, M. F.; Barrett, J.; Rudolph, T., On the reality of the quantum state, Nat. Phys., 8, 475 (2012)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.