Shiva diagrams for composite-boson many-body effects : How they work

摘要

The purpose of this paper is to show how the diagrammatic expansion infermion exchanges of scalar products of $N$-composite-boson (``coboson'')states can be obtained in a practical way. The hard algebra on which thisexpansion is based, will be given in an independent publication. Due to the composite nature of the particles, the scalar products of$N$-coboson states do not reduce to a set of Kronecker symbols, as forelementary bosons, but contain subtle exchange terms between two or morecobosons. These terms originate from Pauli exclusion between the fermioniccomponents of the particles. While our many-body theory for composite bosonsleads to write these scalar products as complicated sums of products of ``Pauliscatterings'' between \emph{two} cobosons, they in fact correspond to fermionexchanges between any number P of quantum particles, with $2 \leq P\leq N$.These $P$-body exchanges are nicely represented by the so-called ``Shivadiagrams'', which are topologically different from Feynman diagrams, due to theintrinsic many-body nature of Pauli exclusion from which they originate. TheseShiva diagrams in fact constitute the novel part of our composite-excitonmany-body theory which was up to now missing to get its full diagrammaticrepresentation. Using them, we can now ``see'' through diagrams the physics ofany quantity in which enters $N$ interacting excitons -- or more generally $N$composite bosons --, with fermion exchanges included in an \emph{exact} -- andtransparent -- way.