arXiv subjects

Lionel Bréhamet

Publications and source records attributed to Lionel Bréhamet.

6 recordsLinked to original sources

arXiv2016

The Ponzano-Regge asymptotic of the supersymmetric 6jS symbols

We adapt the Gurau's proof (2008) about the asymptotic limit of Ponzano-Regge formula to supersymmetric 6jS symbols according to their intrinsic parities alpha, beta, gamma. The behaviour at a large scaling shows significant differences depending on these parities. The decay is slowed and the angles of the oscillating parts in cosine are generally shifted or more altered.Our results should be relevant in 3-D Quantum Supergravity and Spin Foam models. PACS: 03.65.Fd Algebraic methods PACS: 04.60.-m Quantum gravity PACS: 11.30.Pb Supersymmetry MSC: 81Q60; 83D05.

math-ph↗

arXiv2015

Regge symmetry and partition of Wigner 3-j or super 3-jS symbols: unknown properties

For each generic $(3\mbox{-}j)$ the column parities, $2(j \pm m)$, define $3$ intrinsic parities: $α, β,γ$. In algebra $so(3)$ only $(3\mbox{-}j)\_α$ exists whereas super-algebra $osp(1|2)$ admits $3$ kinds of super-symbols $(3\mbox{-}j)^{S}\_α, (3\mbox{-}j)^{S}\_β, (3\mbox{-}j)^{S}\_γ$. Instead of 4 for $\{6\mbox{-}j\}$ symbols, Regge symmetry this time produces 5 partitions ${\tt S}\_{\boldsymbol{\wr}}(0), {\tt S}\_{\boldsymbol{\wr}}(1), {\tt S}\_{\boldsymbol{\wr}}(2), {\tt S}\_{\boldsymbol{\wr}}(4), {\tt S}\_{\boldsymbol{\wr}}(5)$, with ${\tt S}\_{\boldsymbol{\wr}}(3) = \emptyset$. Valid for $(3\mbox{-}j)\_α, (3\mbox{-}j)^{S}\_{α,γ}$ they reduce to 2 for $(3\mbox{-}j)^{S}\_β$ with ${\tt S}\_{\boldsymbol{\wr}}(0), {\tt S}\_{\boldsymbol{\wr}}(1)$. Unexpectedly a symbol $(3\mbox{-}j)^{S}\_β$ and its 'Regge-transformed' may be opposite in sign. In terms of integer parts and super-triangle $\triangle^{S}$ a formula similar to that of a $(3\mbox{-}j)$ is obtained for the $(3\mbox{-}j)^{S}$. Some forbidden $(3\mbox{-}j)^{S}\_β$ require an analytic prolongation, consistent with Regge $β$-partitions.

math-ph↗

arXiv2015

Regge symmetry of 6-j or super 6-jS symbols: a re-analysis with partition properties

It shown that the five Regge transformations act as a spectrometric splitter on any 6-j symbol. Four unknown partitions are brought out: S4(0), S4(1), S(2) and S4(5). They are stable subsets, with well defined parameters depending only on triangles and quadrangles. These findings are easily generalized to super 6-jS symbols, properly labelled by their own parity alpha, beta, gamma. Super Regge symmetry is reduced only for beta where S4(2), S4(5) vanish. In addition, all tools for computing exact values of any 6-jS are provided.

math-ph↗

arXiv2014

Remarks on the Egyptian 2/D table in favor of a global approach (D composite number)

Egyptian decompositions of 2/D as a sum of two unit fractions are studied by means of certain divisors of D, namely r and s. Our analysis does not concern the method to find r and s, but just why the scribes have chosen a solution instead of another. A comprehensive approach, unconventional, is developed which implies having an overview of all pre-calculated alternatives. Difference s-r is the basis of the classification to be examined before taking the right decisions, step by step, if difficulties are encountered. An adequate adjustment of s-r limits the number of all alternatives to 57, what is very few. A four-component generator (2/3, 2/5, 2/7, 2/11) operates as a (hidden) mother-table. Adding few logical rules of common sense is enough to explain the reasons of the Egyptian choices. Even the singular case 2/95, which can not be decomposed into two fractions but only into three, reveals a doubly justified explanation.

math.HO↗

arXiv2014

Remarks on the Egyptian 2/D table in favor of a global approach (D prime number)

For h=3 or 4, Egyptian decompositions into h unit fractions, like 2/D = 1/D1 + ... +1/Dh, were given by using (h-1) divisors (di) of D1. This ancient modus operandi, well recognized today, provides Di=DD1/di for i greater than 1. Decompositions selected (depending on di) have generally been studied by modern researchers through the intrinsic features of di itself. An unconventional method is presented here without considering the di properties but just the differences d(h-1)-dh. In contrast to widespread ideas about the last denominator like 'Dh smaller than 1000', it is more appropriate to adopt a global boundary of the form 'Dh smaller or equal to 10D', where 10 comes from the Egyptian decimal system. Singular case 2/53 (with 15 instead of 10) is explained. The number of preliminary alternatives before the final decisions is found tobe so low (71) for h=3 or 4 that a detailed overview is possible. A simple additive method of trials, independent of any context, can be carried out, namely 2n+1= d2+ ... + dh. Clearly the decisions fit with a minimal value of the differences d(h-1)-dh, independently of any di values.

math.HO↗

arXiv2008

Concept of a veritable osp(1$|$2) super-triangle sum rule with 6-$j^S$ symbols from intrinsic operator techniques: an open problem

Efficiency of intrinsic operator techniques (using only products and ranks of tensor operators) is first evidenced by condensed proofs of already known $\bigtriangledown$-triangle sum rules of su(2)/su$_q$(2). {\em A new compact} su$_q$(2)-{\em expression} is found, using a $q$-series $Φ$, with $Φ(n)_{| q=1}=1$. This success comes from an ultimate identification process over monomials like $(c_0)^p$. For osp(1$|$2), analogous principles of calculation are transposed, involving a second parameter $d_0$. Ultimate identification process then must be done over binomials like ${(c_{0}+{d_{0}}^{2})}^{Ω-m} ({d_{0}}^{2})^{m}$. {\em Unknown} polynomials ${\cal P}$ are introduced as well as their expansion coefficients, $x$, over the binomials. It is clearly shown that a hypothetical super-triangle sum rule requires super-triangles $\bigtriangleup^{S}$, instead of $\bigtriangledown$ for su(2)/su$_q$(2). Coefficients $x$ are integers ({\em conjecture 1}). Massive unknown advances are done for intermediate steps of calculation. Among other, are proved {\em two theorems} on tensor operators, "zero" by construction. However, the ultimate identification seems to lead to a dead end, due to analytical apparent complexities. Up today, except for a few of coefficients $x$, no general formula is really available.

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