scholarly journals GROUP THEORY FACTORS FOR FEYNMAN DIAGRAMS

1999 ◽  
Vol 14 (01) ◽  
pp. 41-96 ◽  
Author(s):  
T. VAN RITBERGEN ◽  
A. N. SCHELLEKENS ◽  
J. A. M. VERMASEREN
Keyword(s):  
Group Theory ◽  
Feynman Diagrams ◽  
Computer Algorithms ◽  
Coxeter Number ◽  
Exceptional Algebras ◽  
Dual Coxeter Number ◽  
Invariant Tensors

We present algorithms for the group independent reduction of group theory factors of Feynman diagrams. We also give formulas and values for a large number of group invariants in which the group theory factors are expressed. This includes formulas for various contractions of symmetric invariant tensors, formulas and algorithms for the computation of characters and generalized Dynkin indices and trace identities. Tables of all Dynkin indices for all exceptional algebras are presented, as well as all trace identities to order equal to the dual Coxeter number. Further results are available through efficient computer algorithms.

scholarly journals Coloured Generalised Young Diagrams for Affine Weyl-Coxeter Groups

10.37236/931 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
R. C. King ◽  
T. A. Welsh
Keyword(s):  
Lie Algebras ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Slight Variation ◽  
Affine Lie Algebras ◽  
Young Diagrams ◽  
Bijective Correspondence ◽  
Coxeter Number ◽  
Finite Dimensional ◽  
Dual Coxeter Number

Coloured generalised Young diagrams $T(w)$ are introduced that are in bijective correspondence with the elements $w$ of the Weyl-Coxeter group $W$ of $\mathfrak{g}$, where $\mathfrak{g}$ is any one of the classical affine Lie algebras $\mathfrak{g}=A^{(1)}_\ell$, $B^{(1)}_\ell$, $C^{(1)}_\ell$, $D^{(1)}_\ell$, $A^{(2)}_{2\ell}$, $A^{(2)}_{2\ell-1}$ or $D^{(2)}_{\ell+1}$. These diagrams are coloured by means of periodic coloured grids, one for each $\mathfrak{g}$, which enable $T(w)$ to be constructed from any expression $w=s_{i_1}s_{i_2}\cdots s_{i_t}$ in terms of generators $s_k$ of $W$, and any (reduced) expression for $w$ to be obtained from $T(w)$. The diagram $T(w)$ is especially useful because $w(\Lambda)-\Lambda$ may be readily obtained from $T(w)$ for all $\Lambda$ in the weight space of $\mathfrak{g}$. With $\overline{\mathfrak{g}}$ a certain maximal finite dimensional simple Lie subalgebra of $\mathfrak{g}$, we examine the set $W_s$ of minimal right coset representatives of $\overline{W}$ in $W$, where $\overline{W}$ is the Weyl-Coxeter group of $\overline{\mathfrak{g}}$. For $w\in W_s$, we show that $T(w)$ has the shape of a partition (or a slight variation thereof) whose $r$-core takes a particularly simple form, where $r$ or $r/2$ is the dual Coxeter number of $\mathfrak{g}$. Indeed, it is shown that $W_s$ is in bijection with such partitions.

THE CRITICAL REPRESENTATIONS OF AFFINE LIE ALGEBRAS

1986 ◽  
Vol 01 (10) ◽  
pp. 557-564 ◽  
Author(s):  
D. ALTSCHÜLER
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Central Charge ◽  
Affine Lie Algebras ◽  
Affine Algebra ◽  
Open Problems ◽  
Simple Lie Algebra ◽  
Coxeter Number ◽  
Dual Coxeter Number ◽  
Current Algebras

A critical representation of an affine algebra Ĝ is a representation with central charge k=−g, g being the dual Coxeter number of the underlying simple Lie algebra G. These representations arise naturally in the study of conformal current algebras and BRS cohomology. The author shows how to construct them explicitly in a number of cases, and some intriguing open problems are mentioned.

MASS OPERATOR OF THE INTERACTING ELECTRONS IN FORMALISM OF PERMUTATION GROUP THEORY

2010 ◽  
Vol 24 (19) ◽  
pp. 3735-3748
Author(s):  
CORNELIA TOVSTYUK
Keyword(s):  
Perturbation Theory ◽  
Group Theory ◽  
Fourth Order ◽  
Mass Operator ◽  
Feynman Diagrams ◽  
Natural Order ◽  
Young Diagrams ◽  
The Third ◽  
Interacting Electrons

Double permutation (DP) method is developed here for designing Feynman diagrams for mass operator (MO) of interacting electrons in high orders of perturbation theory (PT). The derived expression allows finding the Young diagrams for the class of permutations corresponding to disconnect Feynman diagrams. The classification of DPs, carried out before, allows to identify the permutations corresponding to disconnected, singly connected (improper) diagrams and to derive expressions for intolerant cycles of permutations. Ordering the nonprimed digits in natural order in the cycles of DP, we avoid the permutations, corresponding to the Feynman diagrams of the same topology because of other numbering of nodes. Thus, the numbers of considered permutations is sufficiently reduced: (from 24 to 6 and from 720 to 42) in the second and the third orders of PT. All 414 expressions (diagrams) for MO in the fourth order of PT were designed using this method. The use of group theory allows us to conclude that no more Feynman diagrams can be designed. The developed method can be used as algorithm for Feynman diagrams designing for MO of interacting electrons (one sort fermions) in higher orders of PT.

Analysis of 3-d molecular distribution with the digital imaging microscope

1988 ◽  
Vol 46 ◽  
pp. 40-41
Author(s):  
W.A. Carrington ◽  
F.S. Fay ◽  
K.E. Fogarty ◽  
L. Lifshitz
Keyword(s):  
Image Restoration ◽  
Digital Imaging ◽  
Fluorescent Dyes ◽  
Optical Axis ◽  
Computer Hardware ◽  
Computer Algorithms ◽  
Low Contrast ◽  
Specific Proteins ◽  
Effective Use ◽  
Single Living Cells

Advances in digital imaging microscopy and in the synthesis of fluorescent dyes allow the determination of 3D distribution of specific proteins, ions, GNA or DNA in single living cells. Effective use of this technology requires a combination of optical and computer hardware and software for image restoration, feature extraction and computer graphics.The digital imaging microscope consists of a conventional epifluorescence microscope with computer controlled focus, excitation and emission wavelength and duration of excitation. Images are recorded with a cooled (-80°C) CCD. 3D images are obtained as a series of optical sections at .25 - .5 μm intervals.A conventional microscope has substantial blurring along its optical axis. Out of focus contributions to a single optical section cause low contrast and flare; details are poorly resolved along the optical axis. We have developed new computer algorithms for reversing these distortions. These image restoration techniques and scanning confocal microscopes yield significantly better images; the results from the two are comparable.

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