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].

Abstract. The notions of reflection, symmetry, and positivity from quantum field theory are shown to induce a duality operation for a general class of unitary representations of Lie groups. The semisimple Lie groups which have this c-duality are identified and they are placed in the context of Harish-Chandra’s legacy for the unitary representations program. Our paper begins with a discussion of path space measures, which is the setting where reflection positivity (Osterwalder-Schrader) was first identified as a useful tool of analysis. Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe. —Jacques Hadamard

Sample paths properties of certain stochastic processes connected to models of Euclidean Quantum Field Theory are studied. In particular, the Holder continuity of paths of the coordinate processes and trace processes is proven. The results are obtained by an application of classical probabilistic criteria together with basic estimates proven in Constructive Quantum Field Theory. 1 Introduction The sample paths space properties of the Euclidean Field Theory models which have been constructed ([Si74, GJ81, AFHKL86, BaeSeZh97] and references therein), were studied in the past quite intensively see i.e. [Can74, CL73, ReRo74, AHK77, Car77, BeGN80, Ha79]. However, most efforts have been devoted to the case of the Nelson free field [Nel73a, Nel73b]. In the case of d = 1, detailed results on sample path properties can be found in [RoSi76]. For the case of space-dimension d 2, the only author who studied with some generality the Holder continuity of sample paths of the interacting models se...

The trace properties of the sample paths of su#ciently regular generalized random #elds are studied. In particular, nice localisation properties are shown in the case of hyperplanes. Using techniques of Euclidean quantum #eld theory a constructive description of the conditional expectation values with respect to some Gibbs measures describing Euclidean quantum #eld theory models and the #-algebras localised in halfspaces is given. In particular the Global Markov property with respect to hyperplanes follows from these constructions in an explicit way. Key words Quantum #eld theory, Global Markov Property, Gibbsian perturbation of the free #eld AMS Classi#cation 60G60; 35Q99; 60H15 1 Introduction 1.1 Generalities Generalized random #elds play an important role in physics. Of particular interest are the #elds which are homogeneous #stationary# with respect to the action of the Euclidean group. Quantum #eld theory applications of generalized random #elds also require that the Markov ...

We study scalar quantum field theory on a compact manifold. The free theory is defined in terms of functional integrals. For positive mass it is shown to have the Markov property in the sense of Nelson. This property is used to establish a reflection positivity result when the manifold has a reflection symmetry. In dimension d=2 we use the Markov property to establish a sewing operation for manifolds with boundary circles. Also in d=2 the Markov property is proved for interacting fields.

Symmetries on contact manifolds and almost S-manifolds Candidata: Giulia DILEO

Abstract. For the P(φ)

Abstract. We explore a framework for complex classical fields, appropriate for describing quantum field theories. Our fields are linear transformations on a Hilbert space, so they are more general than random variables for a probability measure. Our method generalizes Osterwalder and Schrader’s construction of Euclidean fields. We allow complex-valued classical fields in the case of quantum field theories that describe neutral particles. From an analytic point-of-view, the key to using our method is reflection positivity. We investigate conditions on the Fourier representation of the fields to ensure that reflection positivity holds. We also show how reflection positivity is preserved by various

Abstract. We prove general reflection positivity results for both scalar fields and Dirac fields on a Riemannian manifold, and comment on applications to quantum field theory. As another application, we prove the inequality CD ≤ CN between Dirichlet and Neumann covariance operators on a manifold with a reflection. 1.

A stochastic process with self-interaction as a model of quantum field theory is studied. We consider an Ornstein-Uhlenbeck stochastic process x(t) with interaction of the form x (α) (t) 4, where α indicates the fractional derivative. Using Bogoliubov’s R−operation we investigate ultraviolet divergencies for the various parameters α. Ultraviolet properties of this one-dimensional model in the case α = 3/4 are similar to those in the ϕ4 4 theory but there are extra counterterms. It is shown that the model is two-loops renormalizable. For 5/8 ≤ α < 3/4 the model has a finite number of divergent Feynman diagrams. In the case α = 2/3 the model is similar to the ϕ4 3 theory. If 0 ≤ α < 5/8 then the model does not have ultraviolet divergencies at all. Finally if α> 3/4 then the model is nonrenormalizable. This model can be used for a non-perturbative study of ultraviolet divergencies in quantum field theory and also in theory of phase transitions. 1