Narrow Results

We define a deformation of free creations (and annihilations), given by operators on the full Fock space, acting nontrivially only between the vacuum subspace ℂΩ and the twofold tensor product . Then we study the distribution of the deformed free gaussian operators, with the deformation containing also a real parameter d. The recurrence formula for moments is shown, and the Cauchy transform of the distribution measure is computed. This yields the description of the measure: absolutely continuous part and the atomic part. The existence of atoms depends on the parameter d. The special case d =1 is studied with all details, with the formula for moments is given as values of the hypergeometric series. Finally we show the formula for computing the mixed moments of the deformed operators.

As a general study for uniqueness of integrands in quantum martingale representation, we present a necessary and sufficient condition for uniqueness of integrands in a quantum stochastic integral. Also, several equivalent conditions to the necessary and sufficient condition are studied.

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) = p^e(P)q^e'(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.

We discuss the local structure of the net of von Neumann algebras generated by a representation of a local gauge group . Our discussion is independent of the singularity of spectral measures, which has been discussed by many authors since the pioneering work of Gelfand–Graev–Veršic. We show that, for type (S) operators U_A,b, second quantized operators with some twists, the commutativity only with those U(ψ) is sufficient for the triviality of them, where ψ belongs to an arbitrary (small) neighborhood of constant function 1. Some properties of 1-cocycles for the representation V : ψ ↦ Ad_ψ are also discussed.

An interaction system of a fermionic quantum field is considered. The state space is defined by a tensor product space of a fermion Fock space and a Hilbert space. It is assumed that the total Hamiltonian is a self-adjoint operator on the state space and bounded from below. Then it is proven that a subset of real numbers is the essential spectrum of the total Hamiltonian. It is applied to a Yukawa interaction system, which is a system of a Dirac field coupled to a Klein–Gordon, and the HVZ theorem is obtained.

Semi-classical asymptotic expansions up to any finite orders of the partition function of an abstract Bose field model is considered.

In this paper, we present analogues of the Poisson limit distribution for the noncommutative bm-independence, which is associated with several positive symmetric cones. We construct related discrete Fock spaces with creation, annihilation and conservation operators, and prove Poisson type limit theorems for them. Properties of the positive cones, in particular the volume characteristic property they enjoy, and the combinatorics of labeled noncrossing partitions, play crucial role in these considerations.

Quantum $N$-toroidal algebras are generalizations of quantum affine algebras and quantum toroidal algebras. In this paper, we construct a level-one vertex representation of the quantum $N$-toroidal algebra for type $C$. In particular, we also obtain a level-one module of the quantum toroidal algebra for type $C$ as a special case.

The theory of logical gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence α is identified with a quantum information quantity, represented by a density operator of a Hilbert space, whose dimension depends on the logical complexity of α. At the same time, the logical connectives of the language are interpreted as operations defined in terms of quantum logical gates. Standard quantum computational models can be described as special cases of Fock space models, where the meaning of any sentence is localized in a precise sector of a Fock space ℱ. From an intuitive point of view, the increasing number of particles described in the different sectors of ℱ can be interpreted as increasing information.

The adaptive dynamics is known as a new mathematics to treat with a complex phenomena, for example, chaos, quantum algorithm and psychological phenomena. In this paper, we briefly review the notion of the adaptive dynamics, and explain the definition of the generalized Turing machine (GTM) and recognition process represented by the Fock space. Moreover, we show that there exists the quantum channel which is described by the GKSL master equation to achieve the Chaos Amplifier used in [M. Ohya and I. V. Volovich, J. Opt. B5(6) (2003) 639., M. Ohya and I. V. Volovich, Rep. Math. Phys.52(1) (2003) 25.]

Let M be an approximately finite codimensional quasi-invariant subspace of the Fock space. This paper gives a formula to calculate the codimension of such spaces and uses this formula to study the structure of quasi-invariant subspaces of the Fock space. Especially, as one of applications, it is showed that the analogue of Beurling's theorem is not true for the Fock space in the case of n ≥ 2.

Quantum $N$-toroidal algebras are generalization of the quantum toroidal algebras. In this paper, we construct two types of level-one vertex representations of the quantum $N$-toroidal algebras in type $B$, which also realize the quantum toroidal algebras of type $B$ as special cases.

In a recent paper, using a splitting formula for the multi-dimensional Heisenberg group, we derived a formula for the vacuum characteristic function (Fourier transform) of quantum random variables defined as self-adjoint sums of Fock space operators satisfying the multidimensional Heisenberg Lie algebra commutation relations. In this paper we use that formula to compute the characteristic function of quantum random variables defined as suitably truncated sums of the Virasoro algebra generators. By relating the structure of the Virasoro fields to the quadratic quantization program and using techniques developed in that context we prove that the vacuum distributions of the truncated Virasoro fields are products of independent, but not identically distributed, shifted Gamma-random variables.

In this paper, we classify the family of entire functions $\{\rho ,\sigma \}$ such that the induced weighted composition map $A_{\rho ,\sigma }$, is a conjugation namely $J_{b,\alpha }$, where $b$ is an arbitrary complex number and $\alpha$ is a real number such that $0<\alpha <1$. In the last section, we characterize the $J_{b,\alpha }$-symmetric maximal differential operators on the Fock space ${𝔽}_{\alpha }^2$.

In this paper we construct and study an integral of operator-valued functions with respect to Hilbert space-valued measures generated by a resolution of identity. Our integral generalizes the Itô stochastic integral with respect to normal martingales and the Itô integral on a Fock space.

We explain the representation theoretic treatment of Fock spaces by the Kyoto school, and their use in the categorification of basic modules of the Kac-Moody Lie algebras of affine Lie type by cyclotomic quiver Hecke algebras. As the (affine) quiver Hecke algebras are also interesting objects for study, I briefly review known results on the representation theory of those algebras of finite Lie type in the final section. This survey is aimed at non-experts.

The paper gives an overview of our previous results concerning heat kernel measures in infinite dimensions. We give a history of the subject first, and then describe the construction of heat kernel measure for a class of infinite-dimensional groups. The main tool we use is the theory of stochastic differential equations in infinite dimensions. We provide examples of groups to which our results can be applied. The case of finite-dimensional matrix groups is included as a particular case.

An operator on a Fock space is considered as a non-linear and non-commutative function of annihilation and creation operators at points . The derivatives with respect to a_t and , called respectively the annihilation- and creation-derivatives, are formulated within the framework of quantum white noise theory. We prove the differentiability of an admissible white noise operator and give explicit formulae for the derivatives in terms of integral kernel operators. The qwn-derivative is a non-commutative counterpart of the Gross derivative in stochastic analysis.

An integral representation of Hilbert–Schmidt operators on Boson Fock space is proved with explicit forms of integrands which is a quantum version of (classical) Clark–Haussmann–Ocone formula.

We define two analytical Goodwillie's theorem, either by using an Hida type Fock space or by using an holomorphic type Fock space.