Abstract. We examine, for −1 < q < 1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) Xt = at + a ∗ t – where the at fulfill the q-commutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a q-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possess a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret [FB].

We prove the analogue of the classical Burkholder-Gundy inequalites for noncommutative martingales. As applications we give a characterization for an Ito-Clifford integral to be an Lp-martingale via its integrand, and then extend the Ito-Clifford integral theory in L2, developed by Barnett, Streater and Wilde, to Lp for all 1 < p < ∞. We include an appendix on the non-commutative analogue of the classical Fefferman duality between H¹ and BMO.

: Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of non-commutative integration are established. On leave from School of Math., Georgia Institute of Technology, Atlanta, GA 30332 Work supported by U.S. National Science Foundation grant no. PHY90--19433--A01. 1 I. INTRODUCTION Observables pertaining to the configuration of a quantum system with n degrees of freedom are operators Q 1 ; Q 2 ; : : : ; Q n which, depending on the system, may or may not commute. Our main concern is with the case in which the configuration variables are amplitudes of certain field modes. For boson fields, these configuration observables do commute, and the state space H can be taken as the space of all complex square integrable functions on their joint spectrum. This is t...

ABSTRACT. We prove an analog of Janson’s strong hypercontractivity inequality in a class of non-commutative “holomorphic ” algebras. Our setting is the q-Gaussian algebras Γq associated to the q-Fock spaces of Bozejko, Kümmerer and Speicher, for q ∈ [−1, 1]. We construct subalgebras Hq ⊂ Γq, a q-Segal-Bargmann transform, and prove Janson’s strong hypercontractivity L 2 (Hq) → L r (Hq) for r an even integer. 1.

Within the framework of nonrelativisitic quantum electrodynamics we consider a single nucleus and N electrons coupled to the radiation field. Since the total momentum P is conserved, the Hamiltonian H admits a fiber decomposition with respect to P with fiber Hamiltonian H(P). A stable atom, resp. ion, means that the fiber Hamiltonian H(P) has an eigenvalue at the bottom of its spectrum. We establish the existence of a ground state for H(P) under (i) an explicit bound on P, (ii) a binding condition, and (iii) an energy inequality. The binding condition is proven to hold for a heavy nucleus and the energy inequality for spinless electrons.

We study asymptotic behavior of the spectrum of a Schrödinger type operator L V on the Wiener space as # ##. Here L

Dedicated to Prof. Huzihiro Araki on his 60 th birthday Abstract: Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of non-commutative integration are established.