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Register a new userOn anomalies and noncommutative geometry
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Edwin Langmann
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I discuss examples where basic structures from Connes noncommutative geometry naturally arise in quantum field theory. The discussion is based on recent work, partly collaboration with J. Mickelsson.
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Consistent Yang--Mills anomalies $intom_{2n-k}^{k-1}$ ($ninN$, $ k=1,2, ldots ,2n$) as described collectively by Zuminos descent equations $deltaom_{2n-k}^{k-1}+ddom_{2n-k-1}^{k}=0$ starting with the Chern character $Ch_{2n}=ddom_{2n-1}^{0}$ of a principal $SU(N)$ bundle over a $2n$ dimensional manifold are considered (i.e. $intom_{2n-k}^{k-1}$ are the Chern--Simons terms ($k=1$), axial anomalies ($k=2$), Schwinger terms ($k=3$) etc. in $(2n-k)$ dimensions). A generalization in the spirit of Connes noncommutative geometry using a minimum of data is found. For an arbitrary graded differential algebra $CC=bigoplus_{k=0}^infty CC^{(k)}$ with exterior differentiation $dd$, form valued functions $Ch_{2n}: CC^{(1)}to CC^{(2n)}$ and $om_{2n-k}^{k-1}: underbrace{CC^{(0)}timescdots times CC^{(0)}}_{mbox{{small $(k-1)$ times}}} times CC^{(1)}to CC^{(2n-k)}$ are constructed which are connected by generalized descent equations $deltaom_{2n-k}^{k-1}+ddom_{2n-k-1}^{k}=(cdots)$. Here $Ch_{2n}= (F_A)^n$ where $F_A=dd(A)+A^2$ for $AinCC^{(1)}$, and $(cdots)$ is not zero but a sum of graded commutators which vanish under integrations (traces). The problem of constructing Yang--Mills anomalies on a given graded differential algebra is thereby reduced to finding an interesting integration $int$ on it. Examples for graded differential algebras with such integrations are given and thereby noncommutative generalizations of Yang--Mills anomalies are found.
We study some consequences of noncommutativity to homogeneous cosmologies by introducing a deformation of the commutation relation between the minisuperspace variables. The investigation is carried out for the Kantowski-Sachs model by means of a comparative study of the universe evolution in four different scenarios: the classical commutative, classical noncommutative, quantum commutative, and quantum noncommutative. The comparison is rendered transparent by the use of the Bohmian formalism of quantum trajectories. As a result of our analysis, we found that noncommutativity can modify significantly the universe evolution, but cannot alter its singular behavior in the classical context. Quantum effects, on the other hand, can originate non-singular periodic universes in both commutative and noncommutative cases. The quantum noncommutative model is shown to present interesting properties, as the capability to give rise to non-trivial dynamics in situations where its commutative counterpart is necessarily static.
We review the noncommutative approach to the standard model. We start with the introduction if the mathematical concepts necessary for the definition of noncommutative spaces, and manifold in particular. This defines the framework of spectral geometry. This is applied to the standard model of particle interaction, discussing the fermionic and bosonic spectral action. The issues relating to the calculation of the mass of the Higgs are discussed, as well as the role of neutrinos and Wick rotations. Finally, we present the possibility of solving the problem of the Higgs mass by considering a pregeometric grand symmetry.
I review results from recent investigations of anomalies in fermion--Yang Mills systems in which basic notions from noncommutative geometry (NCG) where found to naturally appear. The general theme is that derivations of anomalies from quantum field theory lead to objects which have a natural interpretation as generalization of de Rham forms to NCG, and that this allows a geometric interpretation of anomaly derivations which is useful e.g. for making these calculations efficient. This paper is intended as selfcontained introduction to this line of ideas, including a review of some basic facts about anomalies. I first explain the notions from NCG needed and then discuss several different anomaly calculations: Schwinger terms in 1+1 and 3+1 dimensional current algebras, Chern--Simons terms from effective fermion actions in arbitrary odd dimensions. I also discuss the descent equations which summarize much of the geometric structure of anomalies, and I describe that these have a natural generalization to NCG which summarize the corresponding structures on the level of quantum field theory. Contribution to Proceedings of workshop `New Ideas in the Theory of Fundamental Interactions, Szczyrk, Poland 1995; to appear in Acta Physica Polonica B.
We calculate conformal anomalies in noncommutative gauge theories by using the path integral method (Fujikawas method). Along with the axial anomalies and chiral gauge anomalies, conformal anomalies take the form of the straightforward Moyal deformation in the corresponding conformal anomalies in ordinary gauge theories. However, the Moyal star product leads to the difference in the coefficient of the conformal anomalies between noncommutative gauge theories and ordinary gauge theories. The $beta$ (Callan-Symanzik) functions which are evaluated from the coefficient of the conformal anomalies coincide with the result of perturbative analysis.
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