We prove that the free Fock space F(R + ; C), which is very commonly used in Free Probability Theory, is the continuous free product of copies of the space C 2. We describe an explicit embedding and approximation of this continuous free product structure by means of a discrete-time approximation: the free toy Fock space, a countable free product of copies of C 2. We show that the basic creation...

We prove that the free Fock space F(R+;C), which is very commonly used in Free Probability Theory, is the continuous free product of copies of the space C2. We describe an explicit embeding and approximation of this continuous free product structure by means of a discrete-time approximation: the free toy Fock space, a countable free product of copies of C2. We show that the basic creation, anni...

Using ab initio unrestricted Hartree−Fock, restricted open-shell Hartree−Fock, and fourth-order perturbation theory methods, 13 minima on the Si2CH4+ potential energy surface have been calculated including ion−molecule complexes, Si−H and C−H insertion products, and other Si2CH4+ isomers. The transition states connecting those isomers have been located and the minimum energy paths have been tra...

In this study, the proton is taken as an ensemble of quark-gluon Fock states. Using the principle of detailed balance, the probabilities of finding every Fock states of the proton are obtained without any parameter. A new origin of the light flavor sea quark asymmetry, i.e., ū 6= d̄, is given as a pure statistical effect. It is found that d̄− ū ≈ 0.124, which is in surprisingly agreement with the...

The method of constrained Hamiltonian systems can be used to reduce Fock modules. It is applied to the Virasoro algebra, where a possibly new realization is found.

Given any representation of an arbitrary Lie algebra g over a field K of characteristic 0, we construct representations of a central extension of g on bosonic and fermionic Fock space. The method gives an explicit formula for a (sometimes trivial) 2-cocycle in H(g;K). We illustrate these techniques with several concrete examples. MSC: 17B10, 17B56, 17B65, 17B68.

Recent progress in mathematical theory of random processes provides us with non-Fock product systems (continuous tensor products of Hilbert spaces) used here for constructing a toy model for fermions. Some state vectors describe infinitely many particles in a finite region; the particles accumulate to a point. Electric charge can be assigned to the particles, the total charge being zero. Time d...

We define the i-restriction and i-induction functors on the category O of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.

We prepare a general set-up of random point fields and introduce a densely defined operator T on the Fock space. Then, we will give a simple construction of the so-called Fock isomorphism associated with Poisson measures, an explicit formula between correlation functions and local density functions, and some applications to fermion(determinantal) and other processes. Furthermore, we will give t...