No CrossRef data available.
Article contents
SOME C*-ALGEBRAS ASSOCIATED TO QUANTUM GAUGE THEORIES
Part of:
Quantum field theory; related classical field theories
Categories with structure
Selfadjoint operator algebras
Published online by Cambridge University Press: 18 May 2011
Abstract
Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
Algebras associated with quantum electrodynamics and other gauge theories share some mathematical features with T-duality. Exploiting this different perspective and some category theory, the full algebra of fermions and bosons can be regarded as a braided Clifford algebra over a braided commutative boson algebra, sharing much of the structure of ordinary Clifford algebras.
Keywords
- Type
- Research Article
- Information
- Copyright
- Copyright © Australian Mathematical Publishing Association Inc. 2011
References
[1]Abramsky, S. and Coecke, B., ‘A categorical semantics of quantum protocols’, Proceedings of the 19th IEEE Conference on Logic in Computer Science (LiCS’04) (IEEE Computer Society Press, Los Alamitos, CA, 2004).Google Scholar
[2]Ashtekar, A., Lectures on Non-perturbative Canonical Gravity (World Scientific, Singapore, 1991).CrossRefGoogle Scholar
[3]Ashtekar, A. and Lewandowsi, J., ‘Representation theory of analytic holonomy C*-algebras’, in: Knots and Quantum Gravity, Oxford Lecture Series in Mathematics and its Applications, 1 (ed. Baez, J.) (Oxford University Press, New York, 1994).Google Scholar
[4]Beggs, E. and Majid, S., ‘Bar categories and star operations’, Algebr. Represent. Theory 12 (2009), 103–152.CrossRefGoogle Scholar
[5]Bouwknegt, P., Hannabuss, K. C. and Mathai, V., ‘Nonassociative tori and applications toT-duality’, Comm. Math. Phys. 264 (2006), 41–69.CrossRefGoogle Scholar
[7]Carey, A. L., ‘Inner automorphisms of hyperfinite factors and Bogoliubov transformations’, Ann. Inst. H. Poincaré 40 (1984), 141–149.Google Scholar
[8]Carey, A. L., Gaffney, J. M. and Hurst, C. A., ‘A C*-algebra formulation of the quantization of the electromagnetic field’, J. Math. Phys. 18 (1977), 629–640.CrossRefGoogle Scholar
[9]Carey, A. L. and Grundling, H., ‘On the problem of the amenability of the gauge group’, Lett. Math. Phys. 68 (2004), 113–120.CrossRefGoogle Scholar
[10]Carey, A. L., Grundling, H., Hurst, C. A. and Langmann, E., ‘Realizing 3-cocycles as obstructions’, J. Math. Phys. 36 (1995), 2605–2620.CrossRefGoogle Scholar
[11]Carey, A. L., Grundling, H., Raeburn, I. and Sutherland, C., ‘Group actions on C*-algebras, 3-cocycles and quantum field theory’, Comm. Math. Phys. 168 (1995), 389–416.CrossRefGoogle Scholar
[12]Connes, A. and Kreimer, D., ‘Hopf algebras, renormalization and noncommutative geometry’, Comm. Math. Phys. 199 (1998), 203–242.CrossRefGoogle Scholar
[13]Dirac, P. A. M., Principles of Quantum Mechanics, 4th edn (Clarendon Press, Oxford, 1958).Google Scholar
[14]Dyson, F., ‘The radiation theories of Tomonaga, Schwinger, and Feynman’, Phys. Rev. 75 (1949), 486–502.CrossRefGoogle Scholar
[15]Epstein, H. and Glaser, V., ‘The role of locality in perturbation theory’, Ann. Inst. H. Poincaré Sect. A (N.S.) 19 (1973), 211–295.Google Scholar
[16]Fredenhagen, K., Rehren, K.-H. and Seiler, E., Quantum Field Theory: Where We Are. Approaches to Fundamental Physics, Lecture Notes in Physics, 721 (Springer, Berlin, 2007), pp. 61–87.Google Scholar
[17]Gracia-Bondia, J. M., Varilly, J. C. and Figueroa, H., Elements of Noncommutative Geometry (Birkhäuser, Boston, 2001).CrossRefGoogle Scholar
[18]Grundling, H. and Neeb, H., ‘Full regularity for a C *-algebra of the canonical commutation relations’, Rev. Math. Phys. 21 (2009), 587–613.CrossRefGoogle Scholar
[19]Kreimer, D., Knots and Feynman Diagrams, Cambridge Lecture Notes in Physics, 13 (Cambridge University Press, Cambridge, 2000).CrossRefGoogle Scholar
[20]Mathai, V. and Rosenberg, J., ‘T-duality for torus bundles with H-fluxes via noncommutative geometry’, Comm. Math. Phys. 253 (2005), 705–721.CrossRefGoogle Scholar
[21]Plymen, R. and Robinson, P. L., Spinors in Hilbert Space, Cambridge Tracts in Mathematics, 114 (Cambridge University Press, Cambridge, 1994).Google Scholar
[22]Rieffel, M. A., ‘Induced representations of C*-algebras’, Adv. Math. 13 (1974), 176–257.CrossRefGoogle Scholar
[23]Robinson, P. L., ‘Modular theory and Bogoliubov autmomorphisms of Clifford algebras’, J. Lond. Math. Soc. (2) 49 (1994), 463–476.CrossRefGoogle Scholar
[25]Schweber, S. S., QED and the Men Who Made it (Princeton University Press, Princeton, NJ, 1994).CrossRefGoogle Scholar
[26]Schwinger (ed.), J., Selected Papers on Quantum Electrodynamics (Dover Publications, New York, 1958).Google Scholar
[27]Selinger, P., ‘Dagger compact closed categories and completely positive maps’, Proceedings of the 3rd International Workshop on Quantum Programming Languages, Chicago, June 30–July 1, 2005.Google Scholar
[28]Shale, D. and Stinespring, W. F., ‘States of the Clifford algebra’, Ann. of Math. (2) 80 (1964), 365–381.CrossRefGoogle Scholar
[29]Spera, M. and Wurzbacher, T., ‘Determinants, Pfaffians and quasi-free representations of the CAR algebra’, Rev. Math. Phys. 10 (1998), 705–721.CrossRefGoogle Scholar
[30]Takai, H., ‘On a duality for crossed products of C *-algebras’, J. Funct. Anal. 19 (1975), 25–39.CrossRefGoogle Scholar
[31]Wurzbacher, T., ‘Fermionic second quantization and the geometry of the restricted Grassmannian’, in: Infinite-Dimensional Kaehler Manifolds, Birkhäuser Series on DMV-Seminars, 31 (eds. Huckleberry, A. and Wurzbacher, T.) (Birkhäuser, Basel, 2001).Google Scholar
You have
Access