One-dimensional stable probability density functions for rational index 0<α≤2

Fox’s H-function provide a unified and elegant framework to tackle several physical phenomena. We solve the space fractional diffusion equation on the real line equipped with a delta distribution initial condition and identify the corresponding H-function by studying the small x expansion of the solution. The asymptotic expansions near zero and infinity are expressed, for rational values of the index α, in terms of a finite series of generalized hypergeometric functions. In x-space, the α=1 stable law is also derived by solving the anomalous diffusion equation with an appropriately chosen infinitesimal generator for time translations. We propose a new classification scheme of stable laws according to which a stable law is now characterized by a generating probability density function. Knowing this elementary probability density function and bearing in mind the infinitely divisible property we can reconstruct the corresponding stable law. Finally, using the asymptotic behavior of H-function in terms of hypergeometric functions we can compute closed expressions for the probability density functions depending on their parameters α β c τ. Known cases are then reproduced and new probability density functions are presented

QOperAv, a Code Generator for Generating Quantum Circuits for Evaluating Certain Quantum Operator Averages

This paper introduces QOperAv v1.5, a Java application available for free. (Source code included in the distribution.) QOperAv is a "code generator" for generating quantum circuits. The quantum circuits generated by QOperAv can be used to evaluate with polynomial efficiency the average of $f(A)$ for some simple (that is, computable with polynomial efficiency) function $f$ and a Hermitian operator $A$, provided that we know how to compile $\exp(iA)$ with polynomial efficiency. QOperAv implements an algorithm described in earlier papers, that combines various standard techniques such as quantum phase estimation and quantum multiplexors.Comment: 5 pages, 5 files(1 .tex, 1 .sty, 1 .pdf, 1 .txt, 1 .xxx)Source code in QOperAv1-5.tx

Unified Theory of Fundamental Interactions

Based on local gauge invariance, four different kinds of fundamental interactions in Nature are unified in a theory which has $SU(3)_c \otimes SU(2)_L \otimes U(1) \otimes_s Gravitational Gauge Group$ gauge symmetry. In this approach, gravitational field, like electromagnetic field, intermediate gauge field and gluon field, is represented by gauge potential. Four kinds of fundamental interactions are formulated in the similar manner, and therefore can be unified in a direct or semi-direct product group. The model discussed in this paper can be regarded as extension of the standard model to gravitational interactions. The model discussed in this paper is a renormalizable quantum model, so it can be used to study quantum effects of gravitational interactions.Comment: 23 pages, no figur

Complex Spinors and Unified Theories

In this paper delivered by Murray Gell-Mann at the Stony Brook Supergravity Workshop in 1979, several paths to unification are discussed, from N=8 supergravity to $SU_5$, $SO_{10}$, and $E_6$. Generalizations of $SO_{10}$ to spinor representations of larger groups are introduced. A natural mechanism for generating tiny neutrino masses is proposed in the context of $SO_{10}$, and finally, focus on $SU_3$ rather than $SU_2$ or $SO_3$ family symmetries is noted.Comment: Retro-preprint of 1979 paper. Originally published in Supergravity, P. van Nieuwenhuizen and D.Z. Freedman, eds, North Holland Publishing Co, 197

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