Abstract
The crossing property is perhaps the most subtle aspect of the particle-field relation. Although it is not difficult to state its content in terms of certain analytic properties relating different matrixelements of the S-matrix or formfactors, its relation to the localization- and positive energy spectral principles requires a level of insight into the inner workings of QFT which goes beyond anything which can be found in typical textbooks on QFT. This paper presents a recent account based on new ideas derived from “modular localization” including a mathematic appendix on this subject. Its main achievement is the proof of the crossing property from a two-algebra generalization of the KMS condition.
The main content is an in-depth criticism of the dual model and its string theoretic extension. The conceptual flaws of these models are closely related to misunderstandings of crossing. The correct interpretation of string theory is that of a dynamic infinite component one particle space where “dynamic” means that, unlike a mere collection of independent free fields, the formalism contains also operators which communicate between the different irreducible Poincaré representations and set the mass/spin spectrum. Whereas in pre-string times there were unsuccessful attempts to achieve this in analogy to the O(4,2) hydrogen spectrum by the use of higher noncompact groups, the superstring in d=9+1, which uses instead (bosonic/fermionic) oscillators obtained from multicomponent chiral currents, is the only known solution of the dynamical infinite component pointlike field (or pointlike generated wave function) project.
Similar content being viewed by others
References
Heisenberg, W.: Z. Naturforschung 1, 608 (1946)
Bros, J., Epstein, H., Glaser, V.: Commun. Math. Phys. 1, 240 (1965)
Epstein, H., Glaser, V., Martin, A.: Commun. Math. Phys. 13, 257 (1969)
Martin, A.: Scattering Theory, Unitarity, Analyticity and Crossing. Springer, Berlin (1969)
Bros, J.: Phys. Rep. 134 (1986)
Karowski, M., Thun, H.-J., Truoung, T.T., Weiss, P.: Phys. Rev. Lett. B 67, 321 (1977)
Babujian, H., Förster, A., Karowski, M.: Nucl. Phys. B 825, 396 (2010)
Di Vecchia, P.: arXiv:0704.0101
Jost, R.J.: Helvetica Phys. Acta 36, 77 (1963)
Heisenberg, W.: Verh. Sächs. Akad. 86, 317 (1934)
Furry, W.H., Oppenheimer, J.R.: Phys. Rev. 45, 245 (1934)
Borchers, H.J., Buchholz, D., Schroer, B.: Commun. Math. Phys. 219, 125 (2001)
Streater, R.F., Wightman, A.S.: PCT, Spin and Statistics and All That. Benjamin, New York (1964)
Haag, R.: Local Quantum Physics, 2nd edn. Springer, Berlin (1996)
Wald, R.E.: arXiv:gr-qc/0608018
Epstein, H., Glaser, V.: Ann. Inst. H. Poincare A XIX, 211 (1973)
Duetsch, M., Fredenhagen, K.: Commun. Math. Phys. 203, 71 (1999). hep-th/9807078
Coester, F., Polyzou, W.N.: Phys. Rev. D 26, 1348 (1982) and references therein
Schroer, B.: arXiv:0912.2874
Schroer, B.: arXiv:0912.2886
Brunetti, R., Guido, D., Longo, R.: Rev. Math. Phys. 14, 759 (2002)
Schroer, B.: Nucl. Phys. B 499, 447 (1997)
Lechner, G.: Commun. Math. Phys. 227, 821 (2008). arXiv:math-ph/0601022
Buchholz, D., Mack, G., Todorov, I.: Nucl. Phys. B, Proc. Suppl. 5B, 20 (1988)
Källén, G., Wightman, A.S.: Mat. Fys. Skrifter Kongl. Dansk. Vid. Selsk. 1, 6 (1958)
Bogoliubov, N.N., Logunov, A., Oksak, A.I., Todorov, I.T.: General Principles of Quantum Field Theory. Kluwer Academic, Dordrecht (1990)
Fubini, S., Veneziano, G.: Ann. Phys. 63, 12 (1971)
Del Giudice, E., Di Vecchia, P., Fubini, S.: Ann. Phys. 70, 378 (1972)
Staszkiewicz, C.P.: Die lokale Struktur abelscher Stromalgebren auf dem Kreis. Freie Universitaet, Thesis, Berlin (1995)
Doplicher, S., Roberts, J.E.: Commun. Math. Phys. 131, 51 (1990)
Longo, R., Kawahigashi, Y.: Adv. Math. 206, 729 (2006) and references therein
Mack, G.: arXiv:0909.1024
Mack, G.: arXiv:0907.2407
Mund, J., Schroer, B., Yngvason, J.: CMP 268, 621 (2006). math-ph/0511042
Polchinski, J.: String Theory. Cambridge University Press, Cambridge (1998)
Schroer, B.: arXiv:1006.3543
Schroer, B.: Pascual Jordan’s legacy and the ongoing research in quantum field theory. In preparation
Buchholz, D., Fredenhagen, K.: Commun. Math. Phys. 84, 1 (1982)
Nambu, Y.: Lectures at the Copenhagen Symposium (1970, unpublished)
Goto, T.: Progr. Theor. Phys. 46, 1560 (1971)
Brower, R.C.: Phys. Rev. D 6, 1655 (1972)
Goddard, P., Thorn, C.B.: Phys. Lett. B 40, 235 (1972)
Weinberg, S.: The Quantum Theory of Fields I. Cambridge University Press, Cambridge (1995)
Dimock, J.: Ann. H. Poincaré 3, 613 (2002). math-ph/0102027
Martinec, E.: Class. Quantum Gravity 10, 187 (1993)
Lowe, D.A.: Phys. Lett. B 326, 223 (1994)
Scherk, J., Schwarz, J.: Nucl. Phys. B 81, 118 (1974)
Clifton, R., Halvorson, H.: Br. J. Philos. Sci. 52, 417 (2001)
Arageorgis, J.E., Rutsche, L.: Philos. Sci. 70 (2003)
Fraser, D.: SHPMP 19, 841 (2008)
Clifton, R., Halvorson, H.: Stud. Hist. Philos. Mod. Phys. 32, 1 (2001). arXiv:quant-ph/0001107
Summers, S.J.: arXiv:0802.1854
Araki, H.: Publ. RIMS, Kyoto Univ. Ser. A 4, 361 (1968)
Schroer, B.: arXiv:0905.4435
Aks, S.: J. Math. Phys. 6, 516 (1965)
Schroer, B.: Ann. Phys. 307, 421 (2003). arXiv:hep-th/0106066
Newton, T.D., Wigner, E.P.: Rev. Mod. Phys. 21, 400 (1949)
Zamolodchikov, A.B., Zamolodchikov, A.: AOP 120, 253 (1979)
Schroer, B.: Ann. Phys. 321, 435 (2006) and references to previous publications of the author
Babujian, H., Foerster, A., Karowski, M.: Nucl. Phys. B 736, 169 (2006)
Babujian, H., Karowski, M.: Int. J. Mod. Phys. A 1952, 34 (2004) and references therein to the beginnings of the bootstrap-formfactor program
Schroer, B., Truong, T.T., Weiss, P.: Phys. Lett. B 63, 422 (1976)
Born, M.: Z. Phys. 38, 803 (1926)
Lowenstein, J.H., Schroer, B.: Phys. Rev. D 7, 1929 (1973)
Gomes, M., Lowenstein, J.H.: Nucl. Phys. B 45, 252 (1972)
Mund, J., Schroer, B.: An extended KMS property and the analytic crossing identity. In preparation
Rehren, K.-H.: hep-th/0411086
Duetsch, M., Rehren, K.-H.: Lett. Math. Phys. 62, 171 (2002)
Maldacena, J.A.: Adv. Theor. Math. Phys. 2, 231 (1998)
Haag, R., Swieca, J.A.: Commun. Math. Phys. 1, 308 (1965)
Buchholz, D., Wichmann, E.: Commun. Math. Phys. 106, 321 (1986)
Schroer, B.: arXiv:0712.0371
Karowski, M.-, Weisz, P.: Nucl. Phys. B 139, 445 (1978)
Zapata Marin, O.: arXiv:0905.1439, see also: Spinning the superweb, essays on the history of string theory, http://www.spinningthesuperweb.blogspot.com/
Smolin, L.: The trouble with physics: the rise of string theory, the fall of a science, and what comes next. September 2006
Woit, P.: Not Even Wrong, the Failure of String Theory and the Continuing Challenge to Unify the Laws of Physics. Jonathan Cape, London (2006)
Hedrich, R.: physics/0610168
Fredenhagen, K., Rehren, K.-H., Seiler, E.: Springer Lecture Notes Phys. 721, 61 (2007). arXiv:hep-th/0603155
Fassarella, L., Schroer, B.: J. Phys. A 35, 9123 (2002)
Mund, J.: Commun. Math. Phys. 286, 1159 (2009)
Brunetti, R., Fredenhagen, K., Verch, R.: Commun. Math. Phys. 237, 31 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Ivan Todorov on the occasion of his 75th birthday.
Rights and permissions
About this article
Cite this article
Schroer, B. A Critical Look at 50 Years Particle Theory from the Perspective of the Crossing Property. Found Phys 40, 1800–1857 (2010). https://doi.org/10.1007/s10701-010-9492-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-010-9492-5