Full Fock Space Research Articles

K-Theoretic Duality for Shifts of Finite Type

C*-algebras generalizing Cuntz-Krieger algebras can be associated to hyperbolic homeomorphisms of compact metric spaces. They satisfy a non-commutative form of Spanier-Whitehead duality with respect to K-theory. We prove this for the case of subshifts of finite type. The special feature of the present situation is that the constructions are all done on the full Fock space and are very explicit, while the general theorem requires much more abstract machinery.

Non-commutative disc algebras and their representations

It is shown that the smallest closed subalgebra A l g ( I K , V 1 , … , V n ) ⊂ B ( K ) ( n = 2 , 3 , … , ∞ ) \begin{equation*}Alg(I_{ \mathcal {K}} ,V_{1},\dots ,V_{n})\subset \mathcal {B} (\mathcal {K}) \qquad (n=2,3,\dots ,\infty )\end{equation*} generated by any sequence V 1 , … , V n V_{1},\dots , V_{n} of isometries on a Hilbert space K \mathcal {K} such that V 1 V 1 ∗ + ⋯ + V n V n ∗ ≤ I K V_{1}V_{1}^{*}+\cdots +V_{n}V_{n}^{*}\le I_{\mathcal {K}} is completely isometrically isomorphic to the non-commutative “disc” algebra A n \mathcal {A} _{n} introduced in Math. Scand. 68 (1991), 292–304. We also prove that for n ≠ m n\ne m the Banach algebras A n \mathcal {A} _{n} and A m \mathcal {A} _{m} are not isomorphic. In particular, we give an example of two non-isomorphic Banach algebras which are completely isometrically embedded in each other. The completely bounded (contractive) representations of the “disc” algebras A n ( n = 2 , 3 , … , ∞ ) \mathcal {A} _{n} (n=2,3,\dots ,\infty ) on a Hilbert space are characterized. In particular, we prove that a sequence of operators A 1 , A 2 , … A_{1},A_{2},\dots is simultaneously similar to a contractive sequence T 1 , T 2 , … T_{1},T_{2},\dots (i.e., T 1 T 1 ∗ + ⋯ + T n T n ∗ ≤ I T_{1}T_{1}^{*}+\cdots +T_{n}T_{n}^{*} \le I ) if and only if it is completely polynomially bounded. The first cohomology group of A n \mathcal {A} _{n} with coefficients in C \mathbb {C} is calculated, showing, in particular, that the disc algebras are not amenable. Similar results are proved for the non-commutative Hardy algebras F n ∞ F_{n}^{\infty } introduced in Math. Scand. 68 (1991), 292–304. The right joint spectrum of the left creation operators on the full Fock space is also determined.

Off-shell fields and pauli-villars regularization

We analyze the correspondence between a five-dimensional U(1)gauge invariant theory and four-dimensional scalar QED, where the fifth dimension (τ)is an invariant parameter of evolution of the manifestly covariant one-particle sector as well as for the full Fock space. The correspondence is represented by the limit in which the width of the photon mass distribution Δs tends to zero and large τ correlations occur. In the limiting procedure, calculation of a twopoint diagram shows that the PauliVillars regularization is intrinsically related to these longrange correlations.

Factorization and reflexivity on Fock spaces

The framework of the paper is that of the full Fock space $$\mathcal{F}^2 (\mathcal{H}_n )$$ and the Banach algebraF ∞ which can be viewed as non-commutative analogues of the Hardy spacesH 2 andH ∞ respectively. An inner-outer factorization for any element in $$\mathcal{F}^2 (\mathcal{H}_n )$$ as well as characterization of invertible elements inF ∞ are obtained. We also give a complete characterization of invariant subspaces for the left creation operatorsS 1 ,..., S n of $$\mathcal{F}^2 (\mathcal{H}_n )$$ . This enables us to show that every weakly (strongly) closed unital subalgebra of {φ(S 1 ,..., S n ) ∶ φ∈F ∞} is reflexive, extending in this way the classical result of Sarason [S]. Some properties of inner and outer functions and many examples are also considered.

Free E0-Semigroups

AbstractGiven a strongly continuous semigroup of isometries ∪ acting on a Hilbert space ℋ, we construct an E0-semigroup α∪, the free E0-semigroup over ∪, acting on the algebra of all bounded linear operators on full Fock space over ℋ. We show how the semigroup αU⊗V can be regarded as the free product of α∪ and αV. In the case where U is pure of multiplicity n, the semigroup au, called the Free flow of rank n, is shown to be completely spatial with Arveson index +∞. We conclude that each of the free flows is cocycle conjugate to the CAR/CCR flow of rank +∞.

Functional integrals for Hubbard operators and projection methods for strong interaction

We discuss problems of functional-integral formalisms in a constrained fermionic Fock space. A functional integral is set up for the Hubbard model using generalized coherent states which lie either in the constrained or in the full Fock space. The projection for the latter states is implemented through a reduction of the charge fluctuations which induce transitions between the constrained and full space. The lagrangian is expressed in terms of two complex fields representing spin and charge excitations, and one Grassmann field signifying hole excitations. Here, the charge excitations denote transitions between states with empty and doubly occupied sites. The projection method is inspired by the observation that the local interaction in the model resembles a magnetic field in the space of charge fluctuations. Hence the projection is understood as an infinite magnetic field in a spin path integral.

On the Fock representation of the q-commutation relations.

The q-commutation relations in the title are those that have recently received much attention, and that for -1<q<1 provide an interpolation between Bosonic and Fermionic statistics, passing through free statistics at q=0. We look at the C*-algebra R^q generated by the representation of these relations on the twisted Fock space of Bozejko-Speicher, and we find a canonical unitary U_q from the twisted Fock space to usual full Fock space, such that U_q R^q (U_q)^* contains the Cuntz algebra R^0, and such that we have equality for |q|<0.44.

Stochastic integration on the Cuntz algebra O∞

We develop a theory of non-commutative stochastic integration with respect to the creation and annihilation process on the full Fock space over L 2( R ). Our theory largely parallels the theories of non-commutative stochastic Itô integration on Boson and Fermion Fock space as developed by R. Hudson and K. R. Parthasarathy. It provides the first example of a non-commutative stochastic calculus which does not depend on the quantum mechanical commutation or anticommutation relations, but it is based on the theory of reduced free products of C ∗-algebras by D. Voiculescu. This theory shows that the creation and annihilation processes on the full Fock space over L 2( R ), which generate the Cuntz algebra O ∞, can be interpreted as a generalized Brownian motion. We should stress the fact that in contrast to the other theories of stochastic integration our integrals converge in the C ∗-norm on O ∞, i.e., uniformly rather than in some state-dependent strong operator topology or (non-commutative) L 2-norm.

Stochastic Integration on the Full Fock Space with the Help of a Kernel Calculus

We develop a stochastic integration theory with respect to creation, annihilation and gauge operators on the full Fock space. This is done by using a kernel representation for a large class of bounded operators on the full Fock space. It is shown that the kernels form a Banach algebra. Having established the definition of processes and stochastic integrals we go on to prove an Ito formula and use this for examining stochastic evolutions and constructing dilations of special completely positive semigroups. Explicit solutions of the corresponding stochastic differential equations are given.

A new example of ‘independence’ and ‘white noise’

We examine the notion of ‘free independence’ according to Voiculescu. This form of independence is used for defining ‘free white noise’ or ‘process with stationary and freely independent increments’. We prove a general limit theorem giving the combinatorics of infinitely freely divisible states and thus of free white noises with the help of ‘admissible’ partitions. We realize the free analogues of the Wiener process and of the Poisson process as processes on the full Fock space ofL 2 (—).

Analyzing multiparticle reactions. II. Exactly solvable production models.

A relativistic Lippmann-Schwinger equation for production reactions is developed. To obtain such equations a relativistic generalization of the Schroedinger and Lippmann-Schwinger equations is given, which, when applied to a Fock space appropriate for particles of mass {ital m} and zero spin, describes production reactions. Separable potentials are used for the production channels, which then incorporate the correct inelasticity in the elastic channel. Exact solutions to the relativistic Lippmann-Schwinger equations are given for potentials that are separable on all subspaces of the full Fock space.

Quantum dynamics of a massless relativistic string

Abstract We develop the classical and quantum mechanics of a massless relativistic string, the light string, which is characterized by an action proportional to the area of the world sheet swept out by the string in space time. We show that, classically, there are only D -2 dynamically independent components among the D functions x μ ( σ , τ ) which represent the world sheet ( D is the dimension of space time). Quantizing only these independent components, we find that the angular momentum operators suggested by the correspondence principle generate O( D -1,1) only when the first excited state is a photon, i.e., a spin-one massless state, and when D = 26. By allowing additional degrees of freedom in the quantum mechanics, we are able to quantize the string in a Lorentz covariant manner for any value of D and any mass for the first excited state. In this latter scheme the full Fock space contains negative norm states. However, when D ⩽ 26 for a massless first excited state and D ⩽ 25 for a real massive first excited state, the physical states span a positive subspace of the Fock space. The excitation spectrum of the light string coincides with the space of physical states in the dual resonance model for unit intercept of the leading Regge trajectory. We point out the connection of this work to previous studies of the physical states in dual models.