Raphael Høegh-Krohn

Hypercontractive semigroups and two dimensional self-coupled Bose fields

We present an abstract perturbation theory for operators of the form H0 + V obeying four properties: (1) H0 is a positive self-adjoint operator on L2(M, μ) with μ a probability measure so that e−tH0 is a contraction on L1 for each t > 0; (2) e−TH0 is a bounded map of L2 to L4 for some T; (3) V ϵ Lp(M, μ) for some p > 2; (4) e−tV ϵ L1 for all t > 0. We then show that spatially cutoff Bose fields in two-dimensional space-time fit into this framework. Finally, we discuss some details of two-dimensional Bose fields in the abstract including coupling constant analyticity in the spatially cutoff case.

The Wightman Axioms and the Mass Gap for Strong Interactions of Exponential Type in Two-Dimensional Space-Time

Abstract We construct boson models in two space-time dimensions which satisfy all the Wightman axioms with mass gap. The interactions are exponential and no restriction on the size of the coupling constant is made. The Schwinger functions for the space cut-off interaction are shown to be non-negative and to decrease monotonically to their unique, non zero, infinite volume limit, as the space cut-off is removed. The correspondent Wightman functions satisfy all the Wightman axioms. The mass gap of the space cut-off Hamiltonian is non decreasing as the space cut-off is removed and the Hamiltonian for the infinite volume limit has mass gap at least as large as the bare mass. The infinite volume Schwinger functions and the mass gap depend monotonically on the coupling constant and the bare mass. The coupling of the first power of the field to the first excited state is proven and an equation of motion for the interacting field is derived.

Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics I.*)

We give a theory of oscillatory integrals in infinitely many dimensions which extends, for a class of phase functions, the finite dimensional theory. In particular we extend the method of stationary phase, the theory of Lagrange immersions and the corresponding asymptotic expansions to the infinite dimensional case. A particular application of the theory to the Feyman path integrals defined in previous work by the authors yields asymptotic expansions to all orders of quantum mechanical quantities in powers of Plancks constant.

Uniqueness and the Global Markov Property for Euclidean Fields. The Case of Trigonometric Interactions

We prove the global Markov property for the Euclidean measure given by weak trigonometric interactions. To obtain this result we first prove a uniqueness theorem concerning the set of regular Gibbs measures corresponding to a given interaction.

A class of explicitly soluble, local, many‐center Hamiltonians for one‐particle quantum mechanics in two and three dimensions. I

We derive an explicit formula for the resolvent of a class of one‐particle, many‐center, local Hamiltonians. This formula gives, in particular, a full description of a model molecule given by point interactions at n arbitrarily placed fixed centers in three dimensions. It also gives a three−dimensional analog of the Kronig–Penney model.

A class of exactly solvable three-body quantum mechanical problems and the universal low energy behavior

Abstract Three-body systems with two-body point interactions are studied. These systems are the universal low energy limits of three-body problems with short-range two-body forces. Hence if there are infinitely many spherically symmetric three-body bound states with energies E n then lim n →∞ E n / E n +1 = e 2 λσ , where σ is explicitly computed.

Frobenius Theory for Positive Maps of von Neumann Algebras

Frobenius theory about the cyclic structure of eigenvalues of irreducible non negative matrices is extended to the case of positive linear maps of von Neumann algebras. Semigroups of such maps and ergodic properties are also considered.

Relativistic quantum statistical mechanics in two-dimensional space--time

We construct for a boson field in two-dimensional space-time with polynomial or exponential interactions and without cut-offs, the positive temperature state or the Gibbs state at temperature 1/β. We prove that at positive temperatures i.e. β<∞, there is no phase transitions and the thermodynamic limit exists and is unique for all interactions. It turns out that the Schwinger functions for the Gibbs state at temperature 1/β is after interchange of space and time equal to the Schwinger functions for the vacuum or temperature zero state for the field in a periodic box of length β, and the lowest eigenvalue for the energy of the field in a periodic box is simply related to the pressure in the Gibbs state at temperature 1/β.

Uniqueness of the physical vacuum and the Wightman functions in the infinite volume limit for some non polynomial interactions

We consider quantum field theoretical models inn dimensional space-time given by interaction densities which are bounded functions of an ultraviolet cut-off boson field. Using methods of euclidean Markov field theory and of classical statistical mechanics, we construct the infinite volume imaginary and real time Wightman functions as limits of the corresponding quantities for the space cut-off models. In the physical Hilbert space, the space-time translations are represented by strongly continuous unitary groups and the generator of time translationsH is positive and has a unique, simple lowest eigenvalue zero, with eigenvector Ω, which is the unique state invariant under space-time translations. The imaginary time Wightman functions and the infinite volume vacuum energy density are given as analytic functions of the coupling constant. The Wightman functions have cluster properties also with respect to space translations.

Classification and construction of Quasisimple Lie algebras

Abstract We study a class of (possibly infinite-dimensional) Lie algebras, called the Quasisimple Lie algebras (QSLAs), and generalizing semisimple and affine Kac-Moody Lie algebras. They are characterized by the existence of a finite-dimensional Cartan subalgebra, a non-degenerate symmetric ad-invariant Killing form, and nilpotent rootspaces attached to non-isotropic roots. We are then able to derive a classification theorem for the possible irreducible elliptic quasisimple root systems; moreover, we construct explicit realizations of some of them as (untwisted and twisted) current algebras, generalizing the affine loop algebras.