We establish necessary and sufficient conditions for Euclidean Green's functions to define a unique Wightman field theory. Contents1 Introduction 83 2. Test Functions and Distributions 85 3. The Axioms, Main Theorems 87 4. TheoremE->R 90 4.1. Construction of the Wightman Distributions 90 4.

In this article we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with n channels. The corresponding on-shell S-matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy E > 0 is explicitly given in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff's law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory. Finally we provide a composition rule by which the on-shell S-matrix of a graph is factorizable in terms of the S-matrices of its subgraphs. All proofs only use known facts from the theory of self-adjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitian symplectic forms.

We give new (necessary and) sufficient conditions for Euclidean Green's functions to have analytic continuations to a relativistic field theory. These results extend and correct a previous paper. Table of ContentsI. Introduction 281 II. Notations 283 III. The Equivalence Theorem Revisited 285 IV. The Main Result: Another Reconstruction Theorem 287 IV. 1 Linear Growth Condition and Statement of Results 287 IV.2 Proof of Theorem £'->#' 288 V. The Analytic Continuation . 289 V.I Real Analyticity 291 V.2 Towards the Real World 293 VI. The Temperedness Estimate 297 VI.1 From Distributions to Functions 297 VI.2 Continuing the Estimates 301 VII. Appendix 303 References 305 * Supported in part by the National Science Foundation under Grant MPS73-05037 A01. Alfred P. Sloan Foundation Fellow. 1 For verification of this assertion the reader should consult the 1973 Erice Lectures on Constructive Quantum Field Theory [19], where also references and historical accounts can be found.

Abstract. We consider analogs of the Lipschitz-Killing curvatures of smooth Riemannian manifolds for piecewise flat spaces. In the special case of scalar curvature, the definition is due to T. Regge considerations in this spirit date back to J. Steiner. We show that if a piecewise flat space approximates a smooth space in a suitable sense, then the corresponding curvatures are close in the sense of measures.

Covariant solutions of the Dirac (and Klein‐Gordon) equation in a homogeneous classical electromagnetic field are constructed. This is done using the symmetry group of the equation, the Maxwell group. These covariant solutions are obtained starting from solutions in the frame where the electromagnetic field is described by a magnetic field pointing in the 3‐direction and then using the theory of induced representations.