Wick polynomials in noncommutative probability: a group-theoretical approach

Wick polynomials and Wick products are studied in the context of noncommutative probability theory. It is shown that free, Boolean, and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf-algebraic approach to cumulants and Wick products in classical probability theory.

The chordal Loewner equation and monotone probability theory

In Ref. 5, O. Bauer interpreted the chordal Loewner equation in terms of noncommutative probability theory. We follow this perspective and identify the chordal Loewner equations as the non-autonomous versions of evolution equations for semigroups in monotone and anti-monotone probability theory. We also look at the corresponding equation for free probability theory.

THE FIVE INDEPENDENCES AS NATURAL PRODUCTS

Let [Formula: see text] be the class of all algebraic probability spaces. A "natural product" is, by definition, a map [Formula: see text] which is required to satisfy all the canonical axioms of Ben Ghorbal and Schürmann for "universal product" except for the commutativity axiom. We show that there exist only five natural products, namely tensor product, free product, Boolean product, monotone product and anti-monotone product. This means that, in a sense, there exist only five universal notions of stochastic independence in noncommutative probability theory.

COLLECTIVE POTENTIAL FOR LARGE-N HAMILTONIAN MATRIX MODELS AND FREE FISHER INFORMATION

We formulate the planar "large N limit" of matrix models with a continuously infinite number of matrices directly in terms of U(N) invariant variables. Noncommutative probability theory is found to be a good language to describe this formulation. The change of variables from matrix elements to invariants induces an extra term in the Hamiltonian, which is crucial in determining the ground state. We find that this collective potential has a natural meaning in terms of noncommutative probability theory: it is the "free Fisher information" discovered by Voiculescu. This formulation allows us to find a variational principle for the classical theory described by such large N limits. We then use the variational principle to study models more complex than the one describing the quantum mechanics of a single Hermitian matrix (i.e. go beyond the so-called D = 1 barrier). We carry out approximate variational calculations for a few models and find excellent agreement with known results where such comparisons are possible. We also discover a lower bound for the ground state by using the noncommutative analog of the Cramer–Rao inequality.