Free Fock Space Research Articles

Combinatorial aspects of weighted free Poisson random variables

This paper will be devoted to the study of weighted (deformed) free Poisson random variables from the viewpoint of orthogonal polynomials and statistics of non-crossing partitions. A family of weighted (deformed) free Poisson random variables will be defined in a sense by the sum of weighted (deformed) free creation, annihilation, scalar, and intermediate operators with certain parameters on a weighted (deformed) free Fock space together with the vacuum expectation. We shall provide a combinatorial moment formula of non-commutative Poisson random variables. This formula gives us a very nice combinatorial interpretation to two parameters of weights. One can see that the deformation treated in this paper interpolates free and boolean Poisson random variables, their distributions and moments, and yields some conditionally free Poisson distribution by taking limit of the parameter.

Quantum Finance Models

If a Riemann sum limit type quantum integral I is defined on a subspace of a filtrated *-algebra B acting on a separable Hilbert space H , we show that one can construct a quantum finance model, representing a financial market of finitely many basic assets whose numeraire normalized values are {(Xi (t))t0}di=1 as a filtrated von Neumann algebra A of bounded operators on H . A fundamental asset pricing theorem states that the market is arbitrage free if and only if there exists a faithful normal state rho in the predual A of A such that each (Xi (t)t0 is a rho-martingale. This allows one to assign fair prices to European contingent claims written on these basic assets in the market.A quantum finance model of this kind is established where I is the quantum stochastic integrals defined by the canonical unbounded quantum noises acting on a Boson Fock space. Two additional quantum finance models are sketched in this note. In the first model, the integration I is defined through the standard bounded quantum noises acting on a free Fock space. The quantum Brownian motions on two Fock spaces are tensor independent and free independent, free in the sense, respectively. The second model in this note is an abstraction of the above two when the quantum noises are Brownian motions. Utilizing the quantum integral theory on non-commutative Lp-space, we focus on a multi-factor free Brownian motion quantum finance model.

Quantum Group as Semi-infinite Cohomology

We obtain the quantum group SL q (2) as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges \({c+\bar{c}=26}\). Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the generators of SL q (2) and verify their relations. We also discuss the possible extensions of our construction and its connection to the Liouville model and minimal string theory.

Open BPS wall crossing and M-theory

Consider the degeneracies of BPS bound states of one D6-brane wrapping Calabi–Yau X with D0-branes and D2-branes. When we include D4-branes wrapping Lagrangian cycles in addition, D2-branes can end on them. These give rise to new bound states in the d = 2 , N = ( 2 , 2 ) theory of the D4-branes. We call these “open” BPS states, in contrast to closed BPS states that arise from D-branes without boundaries. Lifting this to M-theory, we show that the generating function is captured by free Fock space spanned by M2-brane particles ending on M5-branes wrapping the Lagrangian. This implies that the open BPS bound states are counted by the square of the open topological string partition function on X, reduced to the corresponding chamber. Our results give new predictions for open BPS invariants and their wall crossing phenomena when we change the open and closed string moduli. We relate our results to the work of Cecotti and Vafa on wall crossing in the two-dimensional N = ( 2 , 2 ) theories. The findings from the crystal melting model for the open BPS invariants proposed recently fit well with the M-theory predictions.

NONCOMMUTATIVE BROWNIAN MOTIONS ASSOCIATED WITH KESTEN DISTRIBUTIONS AND RELATED POISSON PROCESSES

We introduce and study a noncommutative two-parameter family of noncommutative Brownian motions in the free Fock space. They are associated with Kesten laws and give a continuous interpolation between Brownian motions in free probability and monotone probability. The combinatorics of our model is based on ordered non-crossing partitions, in which to each such partition P we assign the weight w(P) = pe(P)qe'(P), where e(P) and e'(P) are, respectively, the numbers of disorders and orders in P related to the natural partial order on the set of blocks of P implemented by the relation of being inner or outer. In particular, we obtain a simple relation between Delaney's numbers (related to inner blocks in non-crossing partitions) and generalized Euler's numbers (related to orders and disorders in ordered non-crossing partitions). An important feature of our interpolation is that the mixed moments of the corresponding creation and annihilation processes also reproduce their monotone and free counterparts, which does not take place in other interpolations. The same combinatorics is used to construct an interpolation between free and monotone Poisson processes.

On the nonrelativistic Lee model

In this work we present two rigorous results on the nonrelativistic Lee model following a method proposed by Rajeev in an unpublished article (S. G. Rajeev, hep-th 9902025). Thus this short paper should be considered as a commentary on Rajeev. In the unpublished paper of Rajeev, the renormalization of the Hamiltonian is accomplished at the level of resolvents. We first establish that the renormalized resolvent of the interacting Hamiltonian indeed defines a unique closed densely defined operator acting on the free Fock space of bosons. Next we give a justification in the mean field approximation that the ground state energy is bounded from below and the system has a good thermodynamic limit by elaborating along the original arguments of Rajeev. Our arguments in two dimensions do not yield better bounds, but this could be due to the inadequacy of the method used. In both cases though the ground state energy is not significantly altered to give a nontrivial ground state energy per particle.

DIFFUSION PROCESSES WITH RESPECT TO FREE BROWNIAN MOTION

We prove the existence, uniqueness and Markov property for SDE of diffusion type in the context of the stochastic analysis on the free Fock space introduced in Ref. 1.

Stochastic calculus with respect to free Brownian motion and analysis on Wigner space

We define stochastic integrals with respect to free Brownian motion, and show that they satisfy Burkholder-Gundy type inequalities in operator norm. We prove also a version of Ito's predictable representation theorem, as well as product form and functional form of Ito's formula. Finally we develop stochastic analysis on the free Fock space, in analogy with stochastic analysis on the Wiener space.

On the interacting Free Fock space and the deformed Wigner law

The Fock space is a basic structure for the quantum field theory and quantum stochastic calculus. In all the cases, a Fock space can be described as a direct sum of a sequence of some Hilbert spaces, i.e. a Fock space has the form of , where, is the complex field and is a given Hilbert space.

THE MASTER FIELD FOR LARGE-N MATRIX MODELS AND QUANTUM GROUPS

In recent works by Singer, Douglas, Gopakumar and Gross an application of results of Voiculescu from noncommutative probability theory to constructions of the master field for large-N matrix field theories have been suggested. It turns out that this master field becomes the Boltzmann field in the free Fock space. In this note we consider interrelations between the master field and quantum semigroups. We define the master field algebra and observe that it is isomorphic to the algebra of functions on the quantum semigroup SU q(2) for q=0. The master field becomes a central element of the quantum group bialgebra. The quantum Haar measure on the SU q(2) for any q gives the Wigner semicircle distribution for the master field. Coherent states on SU q(2) become coherent states in the master field theory.