Structure of Fluctuation Terms in the Trace Dynamics Ward Identity

We give a detailed analysis of the anti-self-adjoint operator contribution to the fluctuation terms in the trace dynamics Ward identity. This clarifies the origin of the apparent inconsistency between two forms of this identity discussed in Chapter 6 of our recent book on emergent quantum theory.Comment: TeX; 14 pages. Dedicated to Rafael Sorkin on the occasion of his 60th birthda

Poincar\'e Supersymmetry Representations Over Trace Class Noncommutative Graded Operator Algebras

We show that rigid supersymmetry theories in four dimensions can be extended to give supersymmetric trace (or generalized quantum) dynamics theories, in which the supersymmetry algebra is represented by the generalized Poisson bracket of trace supercharges, constructed from fields that form a trace class noncommutative graded operator algebra. In particular, supersymmetry theories can be turned into supersymmetric matrix models this way. We demonstrate our results by detailed component field calculations for the Wess-Zumino and the supersymmetric Yang-Mills models (the latter with axial gauge fixing), and then show that they are also implied by a simple and general superspace argument.Comment: plaintex, 23 Page

Anomalies to All Orders

I give an account of my involvement with the chiral anomaly, and with the nonrenormalization theorem for the chiral anomaly and the all orders calculation of the trace anomaly, as well as related work by others. I then briefly discuss implications of these results for more recent developments in anomalies in supersymmetric theories.Comment: 35 pages, latex; To appear in Fifty Years of Yang-Mills Theory, G. 't Hooft editor, to be published by World Scientific. Final version; references adde

SU(n)SU(n) symmetry breaking by rank three and rank two antisymmetric tensor scalars

We study $SU(n)$ symmetry breaking by rank three and rank two antisymmetric tensor fields. Using tensor analysis, we derive branching rules for the adjoint and antisymmetric tensor representations, and explain why for general $SU(n)$ one finds the same $U(1)$ generator mismatch that we noted earlier in special cases. We then compute the masses of the various scalar fields in the branching expansion, in terms of parameters of the general renormalizable potential for the antisymmetric tensor fields.Comment: Latex, 11 pages; v2 has a minor revision above Eq. (30

SU(8) family unification with boson-fermion balance

We formulate an $SU(8)$ family unification model motivated by requiring that the theory should incorporate the graviton, gravitinos, and the fermions and gauge fields of the standard model, with boson--fermion balance. Gauge field $SU(8)$ anomalies cancel between the gravitinos and spin $\frac {1}{2}$ fermions. The 56 of scalars breaks $SU(8)$ to $SU(3)_{family} \times SU(5)\times U(1)/Z_5$, with the fermion representation content needed for "flipped" $SU(5)$ with three families, and with residual scalars in the $10$ and $\overline{10}$ representations that break flipped $SU(5)$ to the standard model. Dynamical symmetry breaking can account for the generation of $5$ representation scalars needed to break the electroweak group. Yukawa couplings of the 56 scalars to the fermions are forbidden by chiral and gauge symmetries, so in the first stage of $SU(8)$ breaking fermions remain massless. In the limit of vanishing gauge coupling, there are $N=1$ and $N=8$ supersymmetries relating the scalars to the fermions, which restrict the form of scalar self-couplings and should improve the convergence of perturbation theory, if not making the theory finite and "calculable". In an Appendix we give an analysis of symmetry breaking by a Higgs component, such as the $(1,1)(-15)$ of the $SU(8)$ 56 under $SU(8) \supset SU(3) \times SU(5) \times U(1)$, which has nonzero $U(1)$ generator.Comment: Latex, 20 pages. To appear in a World Scientific volume celebrating the 50th anniversary of the quark model, H. Fritzsch and M. Gell-Mann, eds., and also in International Journal of Modern Physics A, Vol. 29 (2014) 1450130 (18 pages