GRMS or Graphical Representation of Model Spaces: Vol. 1 BasicsThe purpose of these notes is to give some simple tools and pictures to physicists and ' chemists working on the many-body problem. Abstract thinking and seeing have much in common - we say "I see" meaning "I understand" , for example. Most of us prefer to have a picture of an abstract object. The remarkable popularity of the Feynman diagrams, and other diagrammatic approaches to many-body problem derived thereof, may be partially due to this preference. Yet, paradoxically, the concept of a linear space, as fundamental to quantum physics as it is, has never been cast in a graphical form. We know that is a high-order contribution to a two-particle scattering process (this one invented by Cvitanovic(1984)) corresponding to a complicated matrix element. The lines in such diagrams are labeled by indices of single-particle states. When things get complicated at this level it should be good to take a global view from the perspective of the whole many-particle space. But how to visualize the space of all many-particle states ? Methods of such visualization or graphical representation of the ,spaces of interest to physicists and chemists are the main topic of this work. |
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GRMS or Graphical Representation of Model Spaces: Vol. 1 Basics Wlodzislaw Duch Limited preview - 2012 |
Common terms and phrases
antisymmetric basis branching diagram Clebsch-Gordan coefficients coefficients computed corresponding coupled cycles cyclic permutation D-part d(SL Diophantine equations doubly occupied orbital Duch and Karwowski elementary loops example fagot graph formulas four-segment loops four-slope graph full space Gelfand gives graph of Fig graph theory graphical representation graphical rules group theory intermediate Karwowski 1985 labels loop value M₁ many-particle matrix elements model space multiplication non-fagot graphs non-zero elements number of particles number of paths obtained one-body segment one-particle orbital orderings point group symmetries positions positions k represented reversed lexical ordering S-diagram Ŝ2-adapted segment values SFT graph Shavitt shift operators shown in Fig singlet singly occupied arcs singly occupied orbitals spin eigenfunctions spin functions spin orbitals spin path subgraph subspaces symmetric group symmetry Sz-adapted Table tableaux three-slope graph transposition two-particle operators two-segment loops unitary group vertex vertices