Zeons, lattices of partitions, and free probability

Abstract

Central to the theory of free probability is the notion of summing multiplicative functionals on the lattice of non-crossing partitions. In this paper, a graph-theoretic perspective of partitions is investigated in which independent sets in graphs correspond to non-crossing partitions. By associating particular graphs with elements of “zeon” algebras (commutative subalgebras of fermion algebras), multiplicative functions can be summed over segments of lattices of partitions by employing methods of “zeon-Berezin” operator calculus. In particular, properties of the algebra are used to “sieve out” the appropriate segments and sub-lattices. The work concludes with an application to joint moments of quantum random variables.