Abstract
This paper presents a general framework for a refined spectral analysis of a group of isometries acting on a Banach space, which extends the spectral theory of Arveson. The concept of a continuous Arveson spectrum is introduced and the corresponding spectral subspace is defined. The absolutely continuous and singular-continuous parts of this spectrum are specified. Conditions are given, in terms of the transposed action of the group of isometries, which guarantee that the pure-point and continuous subspaces span the entire Banach space. In the case of a unitarily implemented group of automorphisms, acting on a C*-algebra, relations between the continuous spectrum of the automorphisms and the spectrum of the implementing group of unitaries are found. The group of spacetime translation automorphisms in quantum field theory is analyzed in detail. In particular, it is shown that the structure of its continuous spectrum is relevant to the problem of existence of (infra-)particles in a given theory.
Similar content being viewed by others
References
Araki H.: Mathematical Theory of Quantum Fields. Oxford University Press, Oxford (1999)
Araki H., Haag R.: Collision cross sections in terms of local observables. Commun. Math. Phys. 4, 77–91 (1967)
Arendt W., Batty C.J.K.: Almost periodic solutions of first and second order Cauchy problems. J. Diff. Eqs. 137, 363–383 (1997)
Arveson W.: On groups of automorphisms of operator algebras. J. Funct. Anal. 15, 217–243 (1974)
Arveson, W.: The harmonic analysis of automorphism groups. In: Operator Algebras and Applications, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math. 38, Providence, RI: Amer. Math. Soc., 1982, pp. 199–269
Baskakov A.G.: Spectral criteria for almost periodicity of solutions of functional equations. Math. Notes 24, 606–612 (1978)
Borchers H.J., Haag R., Schroer B.: The vacuum state in quantum field theory. Nuovo Cimento 29, 148–162 (1963)
Bostelmann H.: Phase space properties and the short distance structure in quantum field theory. J. Math. Phys. 46, 052301–052318 (2005)
Buchholz D.: The physical state space of quantum electrodynamics. Commun. Math. Phys. 85, 49–71 (1982)
Buchholz D.: Gauss’ law and the infraparticle problem. Phys. Lett. B 174, 331–334 (1986)
Buchholz D.: Harmonic analysis of local operators. Commun. Math. Phys. 129, 631–641 (1990)
Buchholz, D.: On the manifestations of particles. In: Mathematical Physics Towards the 21st Century. Proceedings Beer-Sheva 1993, Sen, R.N., Gersten, A. Eds., Ben-Gurion University of the Negev Press 1994, pp. 177–202
Buchholz D., Fredenhagen K.: Locality and the structure of particle states. Commun. Math. Phys. 84, 1–54 (1982)
Buchholz D., Porrmann M., Stein U.: Dirac versus Wigner: Towards a universal particle concept in quantum field theory. Phys. Lett. B 267, 377–381 (1991)
Buchholz D., Wanzenberg R.: The realm of the vacuum. Commun. Math. Phys. 143, 577–589 (1992)
Chen T., Fröhlich J., Pizzo A.: Infraparticle scattering states in non-relativistic QED: I. The Bloch-Nordsieck paradigm. Commun. Math. Phys. 294, 761–825 (2010)
Dereziński J.: Asymptotic completeness of long-range N-body quantum systems. Ann. of Math. 138, 427–476 (1993)
Dereziński J., Gerard C.: Spectral and scattering theory of spatially cut-off \({P(\phi)_2}\) Hamiltonians. Commun. Math. Phys. 213, 39–125 (2000)
Dybalski W.: Haag-Ruelle scattering theory in presence of massless particles. Lett. Math. Phys. 72, 27–38 (2005)
Dybalski W.: A sharpened nuclearity condition and the uniqueness of the vacuum in QFT. Commun. Math. Phys. 283, 523–542 (2008)
Dybalski, W.: Spectral theory of automorphism groups and particle structures in quantum field theory. PhD thesis, Universität Göttingen (2008). Preprint: http://webdoc.sub.gwdg.de/diss/2009/dybalski/, 2009
Dybalski W.: Coincidence arrangements of local observables and uniqueness of the vacuum in QFT. J. Phys. A 42, 365201–365223 (2009)
Enss V.: Asymptotic completeness for quantum mechanical potential scattering. Commun. Math. Phys. 61, 285–291 (1978)
Evans D.: On the spectrum of a one parameter strongly continuous representation. Math. Scand. 39, 80–82 (1976)
Exel R.: Unconditional integrability for dual actions. Bol. Soc. Brasil. Mat. (N.S.) 30, 99–124 (1999)
Exel R.: Morita-Rieffel equivalence and spectral theory for integrable automorphism groups of C*-algebras. J. Funct. Anal. 172, 404–465 (2000)
Fröhlich J., Morchio G., Strocchi F.: Infrared problem and spontaneous breaking of the Lorentz group in QED. Phys. Lett. B 89, 61–64 (1979)
Fröhlich J., Griesemer M., Schlein B.: Asymptotic completeness for Compton scattering. Commun. Math. Phys. 252, 415–476 (2004)
Fredenhagen K., Hertel J.: Local algebras of observables and pointlike localized fields. Commun. Math. Phys. 80, 555–561 (1981)
Graf G.M.: Asymptotic completeness for N-body short-range quantum systems: a new proof. Commun. Math. Phys. 132, 73–101 (1990)
Haag R.: Local Quantum Physics. Springer, Berlin-Heidelbreg-New York (1996)
Herdegen A.: Infrared problem and spatially local observables in electrodynamics. Ann. Henri Poincaré 9, 373–401 (2008)
Huang S.-Z.: Completeness of eigenvectors of group representations of operators whose Arveson spectrum is scattered. Proc. Amer. Math. Soc. 127, 1473–1482 (1999)
Jarchow, H.: Locally Convex Spaces. Stuttgart: B. G. Teubner, 1981
Johannsen, K.: Teilchenaspekte im Schroermodell. Diplomarbeit, Universität Hamburg, 1991
Jorgensen P.E.T.: Spectral theory of one-parameter groups of isometries. J. Math. Anal. Appl. 168, 131–146 (1992)
Jorgensen P.E.T.: Spectral theory for infinitesimal generators of one-parameter groups of isometries: The mini-max principle and compact perturbations. J. Math. Anal. Appl. 90, 347–370 (1982)
Lechner G.: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277, 821–860 (2008)
Longo, R.: Some aspects of C*-dynamics. In: Algèbres d’opérateurs et leur applications en physique mathèmatique. Colloques Internationaux du C.N.R.S. 274, Paris: CNRS, 1979, pp. 261–273
Meyer R.: Generalized fixed point algebras and square-integrable group actions. J. Funct. Anal. 186, 167–195 (2001)
Pizzo A.: Scattering of an infraparticle: the one particle sector in Nelson’s massless models. Ann. Henri Poincaré 6, 553–606 (2005)
Pedersen G.K.: C*-Algebras and Their Automorphism Groups. Academic Press, London-New York (1979)
Porrmann M.: Particle weights and their disintegration I. Commun. Math. Phys. 248, 269–304 (2004)
Porrmann M.: Particle weights and their disintegration II. Commun. Math. Phys. 248, 305–333 (2004)
Reed M., Simon B.: Methods of Modern Mathematical Physics. Part I: Functional Analysis. Academic Press, London-New York (1972)
Rejzner, K.: Asymptotic Algebra of Fields in Quantum Electrodynamics. Master’s thesis, University of Cracow, 2009
Rieffel, M.A.: Proper actions of groups on C*-algebras. In: Mappings of Operator Algebras (Philadelphia, PA,1988), Boston, MA: Birkhäuser Boston, 1990, pp. 141–182
Rudin W.: Functional Analysis. McGraw-Hill, New York (1977)
Schroer B.: Infrateilchen in der Quantenfeldtheorie. Fortschr. Phys. 11, 1–32 (1963)
Sigal I.M., Soffer A.: The N-particle scattering problem: asymptotic completeness for short-range systems. Ann. of Math. 126, 35–108 (1987)
Spohn H.: Dynamics of Charged Particles and Their Radiation Field. Cambridge University Press, Cambridge (2004)
Wigner E.P.: On unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40, 149–204 (1939)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
Rights and permissions
About this article
Cite this article
Dybalski, W. Continuous Spectrum of Automorphism Groups and the Infraparticle Problem. Commun. Math. Phys. 300, 273–299 (2010). https://doi.org/10.1007/s00220-010-1091-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-010-1091-y