Derivation-based noncommutative field theories on \(AF\) algebras. (English) Zbl 07822285
Summary: In this paper, we start the investigation of a new natural approach to “unifying” noncommutative gauge field theories (NCGFT) based on approximately finite-dimensional (\(AF\)) \(C^\ast\)-algebras. The defining inductive sequence of an \(AF\) \(C^\ast\)-algebra is lifted to
enable the construction of a sequence of NCGFT of Yang-Mills-Higgs types. This paper focuses on derivation-based noncommutative field theories. A mathematical study of the ingredients involved in the construction of a NCGFT is given in the framework of \(AF\) \(C^\ast\)-algebras: derivation-based differential calculus, modules, connections, metrics and Hodge \(\star\)-operators, and Lagrangians. Some physical applications concerning mass spectra generated by Spontaneous Symmetry Breaking Mechanisms (SSBM) are proposed using numerical computations for specific situations.
MSC:
| 81R60 | Noncommutative geometry in quantum theory |
| 46L89 | Other “noncommutative” mathematics based on \(C^*\)-algebra theory |
| 16W25 | Derivations, actions of Lie algebras |
| 81T75 | Noncommutative geometry methods in quantum field theory |
| 46L35 | Classifications of \(C^*\)-algebras |
| 81R40 | Symmetry breaking in quantum theory |
Keywords:
gauge field theories; noncommutative geometry; \(AF\) algebras; derivation-based noncommutative geometry; symmetry breakingSoftware:
MathematicaReferences:
| [1] | Connes, A.Lott, J.Particle models and noncommutative geometryNuclear Phys. B Proc. Suppl.18B19902947 · Zbl 0957.46516 |
| [2] | Dubois-Violette, M.Kerner, R.Madore, J.Noncommutative differential geometry and new models of gauge theoryJ. Math. Phys.311990323 · Zbl 0704.53082 |
| [3] | Dubois-Violette, M.Kerner, R.Madore, J.Noncommutative differential geometry of matrix algebrasJ. Math. Phys.311990316 · Zbl 0704.53081 |
| [4] | van Suijlekom, W. D.Noncommutative Geometry and Particle PhysicsSpringerNetherlands2015 · Zbl 1305.81007 |
| [5] | Fournel, C.Lazzarini, S.Masson, T.Formulation of gauge theories on transitive Lie algebroidsJ. Geom. Phys.642013174191 · Zbl 1268.58002 |
| [6] | Lazzarini, S.Masson, T.Connections on Lie algebroids and on derivation-based non-commutative geometryJ. Geom. Phys.622012387402 · Zbl 1260.58002 |
| [7] | Blackadar, B.Operator Algebras, Theory of \(C^\ast\) Springer-Verlag2006 · Zbl 1092.46003 |
| [8] | Davidson, K. R.\( C^\ast\) AMS1996 |
| [9] | Rørdam, M.Larsen, F.Laustsen, N. J.An Introduction to \(K C^\ast\) Cambridge University Press2000 · Zbl 0967.19001 |
| [10] | Masson, T.Gauge theories in noncommutative geometryFFP11 Symp. ProceedingsAIP20127398 |
| [11] | Dubois-Violette, M.Dérivations et calcul differentiel non commutatifC.R. Acad. Sci. Paris, Série I3071988403408 · Zbl 0661.17012 |
| [12] | Masson, T.Submanifolds and quotient manifolds in noncommutative geometryJ. Math. Phys.375199624842497 · Zbl 0877.58006 |
| [13] | Dubois-Violette, M.Masson, T.\(SU(n)\) J. Geom. Phys.251,21998104 · Zbl 0933.58007 |
| [14] | Masson, T.On the noncommutative geometry of the endomorphism algebra of a vector bundleJ. Geom. Phys.142199931 · Zbl 0940.46046 |
| [15] | Dubois-Violette, M.Michor, P. W.Dérivations et calcul differentiel non commutatif IIC.R. Acad. Sci. Paris, Série I3191994927931 · Zbl 0829.16028 |
| [16] | Dubois-Violette, M.Michor, P. W.Connections on central bimodules in noncommutative differential geometryJ. Geom. Phys.201996218232 · Zbl 0867.53023 |
| [17] | Dubois-Violette, M.Michor, P. W.More on the Frölicher-Nijenhuis bracket in non commutative differential geometryJ. Pure Appl. Algebra1211997107135 · Zbl 0889.58011 |
| [18] | Cagnache, E.Masson, T.Wallet, J.-C.Noncommutative Yang-Mills-Higgs actions from derivation-based differential calculusJ. Noncommut. Geom.5120113967 · Zbl 1226.81279 |
| [19] | Dubois-Violette, M.Lectures on graded differential algebras and noncommutative geometryNoncommutative Differential Geometry and Its Applications to PhysicsMaeda, Y.Moriyoshi, H.Kluwer Academic Publishers2001245306 · Zbl 1038.58004 |
| [20] | Masson, T.Noncommutative generalization of \(SU(n)\) J. Phys.: Conf. Ser.1032008012003 |
| [21] | Masson, T.Examples of derivation-based differential calculi related to noncommutative gauge theoriesInt. J. Geom. Methods Mod. Phys.58200813151336 · Zbl 1165.81384 |
| [22] | T. Masson, Géométrie non commutative et applications à la théorie des champs, Thèse de doctorat, Université Paris XI (1995), Thèse soutenue le 13 décembre 1995 |
| [23] | Chamseddine, A. H.Connes, A.Marcolli, M.Gravity and the standard model with neutrino mixingAdv. Theor. Math. Phys.1120079911089 · Zbl 1140.81022 |
| [24] | François, J.Lazzarini, S.Masson, T.Gauge field theories: various mathematical approachesMathematical Structures of the UniverseEckstein, M.Heller, M.Szybka, S. J.Kraków, Poland2014177225Copernicus Center Press |
| [25] | Wolfram Research, Inc., Mathematica, Version 12.2, Champaign, IL, 2020, https://www.wolfram.com/mathematica |
| [26] | Bertlmann, R.Anomalies in Quantum Field TheoryOxford Science Publications1996 · Zbl 1223.81003 |
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.
