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Quantum Isometries of the finite noncommutative geometry of the Standard Model  [PDF]
Jyotishman Bhowmick,Francesco D'Andrea,Ludwik Dabrowski
Mathematics , 2010, DOI: 10.1007/s00220-011-1301-2
Abstract: We compute the quantum isometry group of the finite noncommutative geometry F describing the internal degrees of freedom in the Standard Model of particle physics. We show that this provides genuine quantum symmetries of the spectral triple corresponding to M x F where M is a compact spin manifold. We also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries.
A brief introduction to the noncommutative geometry description of particle physics standard model  [PDF]
Pierre Martinetti
Physics , 2003,
Abstract: Notes from a course given at Oujda university, Morocco, october 2002 - march 2003 within the support of a fellowship from the Agence Universitaire de la Francophonie. These notes present a brief introduction to Connes' non commutative geometry description of the standard model of particle physics. The notion of distance is emphasized, especially the possible interpretation of the Higgs field as the component of a discrete internal dimension. These notes are in french and are taken from the author's phD thesis.
Non-Commutative Geometry, Non-Associative Geometry and the Standard Model of Particle Physics  [PDF]
Latham Boyle,Shane Farnsworth
Physics , 2014, DOI: 10.1088/1367-2630/16/12/123027
Abstract: Connes' notion of non-commutative geometry (NCG) generalizes Riemannian geometry and yields a striking reinterepretation of the standard model of particle physics, coupled to Einstein gravity. We suggest a simple reformulation with two key mathematical advantages: (i) it unifies many of the traditional NCG axioms into a single one; and (ii) it immediately generalizes from non-commutative to non-associative geometry. Remarkably, it also resolves a long-standing problem plaguing the NCG construction of the standard model, by precisely eliminating from the action the collection of 7 unwanted terms that previously had to be removed by an extra, non-geometric, assumption. With this problem solved, the NCG algorithm for constructing the standard model action is tighter and more explanatory than the traditional one based on effective field theory.
A Lorentzian version of the non-commutative geometry of the standard model of particle physics  [PDF]
John W. Barrett
Mathematics , 2006, DOI: 10.1063/1.2408400
Abstract: A formulation of the non-commutative geometry for the standard model of particle physics with a Lorentzian signature metric is presented. The elimination of the fermion doubling in the Lorentzian case is achieved by a modification of Connes' internal space geometry so that it has signature 6 (mod 8) rather than 0. The fermionic part of the Connes-Chamseddine spectral action can be formulated, and it is shown that it allows an extension with right-handed neutrinos and the correct mass terms for the see-saw mechanism of neutrino mass generation.
Rethinking Connes' approach to the standard model of particle physics via non-commutative geometry  [PDF]
Shane Farnsworth,Latham Boyle
Physics , 2014, DOI: 10.1088/1367-2630/17/2/023021
Abstract: Connes' non-commutative geometry (NCG) is a generalization of Riemannian geometry that is particularly apt for expressing the standard model of particle physics coupled to Einstein gravity. In a previous paper, we suggested a reformulation of this framework that is: (i) simpler and more unified in its axioms, and (ii) allows the Lagrangian for the standard model of particle physics (coupled to Einstein gravity) to be specified in a way that is tighter and more explanatory than the traditional algorithm based on effective field theory. Here we explain how this same reformulation yields a new perspective on the symmetries of a given NCG. Applying this perspective to the NCG traditionally used to describe the standard model we find, instead, an extension of the standard model by an extra $U(1)_{B-L}$ gauge symmetry, and a single extra complex scalar field $\sigma$, which is a singlet under $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$, but has $B-L=2$. This field has cosmological implications, and offers a new solution to the discrepancy between the observed Higgs mass and the NCG prediction.
Parameter restrictions in a non-commutative geometry model do not survive standard quantum corrections  [PDF]
E. Alvarez,J. M. Gracia-Bondia,C. P. Martin
Physics , 1993, DOI: 10.1016/0370-2693(93)91137-C
Abstract: We have investigated the standard one-loop quantum corrections for a particularly simple non-commutative geometry model containing fermions interacting with a unique abelian gauge field and a unique scalar through Yukawa couplings. In this model there are certain relations among the different coupling constants quite similar to the ones appearing in the Connes-Lott version of the standard model. We find that it is not possible to implement those relations in a renormalization-group invariant way.
Quantum theory, gravity, and the standard model of particle physics : using the hints of today to build the final theory of tomorrow  [PDF]
T. P. Singh
Physics , 2010,
Abstract: When a mountaineer is ascending one of the great peaks of the Himalayas she knows that an entirely new vista awaits her at the top, whose ramifications will be known only after she gets there. Her immediate goal though, is to tackle the obstacles on the way up, and reach the summit. In a similar vein, one of the immediate goals of contemporary theoretical physics is to build a quantum, unified description of general relativity and the standard model of particle physics. Once that peak has been reached, a new (yet unknown) vista will open up. In this essay I propose a novel approach towards this goal. One must address and resolve a fundamental unsolved problem in the presently known formulation of quantum theory : the unsatisfactory presence of an external classical time in the formulation. Solving this problem takes us to the very edge of theoretical physics as we know it today!
60 years of Broken Symmetries in Quantum Physics (From the Bogoliubov Theory of Superfluidity to the Standard Model)  [PDF]
D. V. Shirkov
Physics , 2009, DOI: 10.1142/S0217732309001017
Abstract: A retrospective historical overview of the phenomenon of spontaneous symmetry breaking (SSB) in quantum theory, the issue that has been implemented in particle physics in the form of the Higgs mechanism. The main items are: -- The Bogoliubov's microscopical theory of superfluidity (1946); -- The BCS-Bogoliubov theory of superconductivity (1957); -- Superconductivity as a superfluidity of Cooper pairs (Bogoliubov - 1958); -- Transfer of the SSB into the QFT models (early 60s); -- The Higgs model triumph in the electro-weak theory (early 80s). The role of the Higgs mechanism and its status in the current Standard Model is also touched upon.
A Wave Equation including Leptons and Quarks for the Standard Model of Quantum Physics in Clifford Algebra  [PDF]
Claude Daviau, Jacques Bertrand
Journal of Modern Physics (JMP) , 2014, DOI: 10.4236/jmp.2014.518210

A wave equation with mass term is studied for all fermionic particles and antiparticles of the first generation: electron and its neutrino, positron and antineutrino, quarks u and d with three states of color and antiquarks \"\" and \"\". This wave equation is form invariant under the \"\" group generalizing the relativistic invariance. It is gauge invariant under the U(1)×SU(2)×SU(3) group of the standard model of quantum physics. The wave is a function of space and time with value in the Clifford algebra Cl1,5. Then many features of the standard model, charge conjugation, color, left waves, and Lagrangian formalism, are obtained in the frame of the first quantization.

The Standard Model of Particle Physics  [PDF]
Tom W. B. Kibble
Physics , 2014,
Abstract: This is a historical account from my personal perspective of the development over the last few decades of the standard model of particle physics. The model is based on gauge theories, of which the first was quantum electrodynamics, describing the interactions of electrons with light. This was later incorporated into the electroweak theory, describing electromagnetic and weak nuclear interactions. The standard model also includes quantum chromodynamics, the theory of the strong nuclear interactions. The final capstone of the model was the Higgs particle discovered in 2012 at CERN. But the model is very far from being the last word; there are still many gaps in our understanding.
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