Nuclear Physics B (Prec. Suppl.) 4 (1988) 639-643 639 North-Holland, Amsterdam SCATTERING THEORY FOR EUCLIDEAN LATTICE FIELDS JoSo C. A. Barata "and Klaus Fredenhagen t II. Institut flir Theoretische Physik, Universit~t Hamburg Luruper Chausse 149, D-2000 Hamburg S0, FRG. A Haag-Ruelle scattering theory for Euclidean lattice fields is developped. The main motivation for the investigation of Eu- clidean Lattice Field Theories (ELFT) is the aim to un- derstand particle physics. It is, however, not obvious at all that ELFT directly describe particles. Usually for the relation to particle physics the continuum limit is used. Continuum Euclidean field theories lead (under rather general circumstances) to quantum field theories on Minkowski space which (again under quite general conditions) describe the behavior of particles. The per- tinent quantities in the continuum limit are then ap- proximated on the lattice. This procedure is of course not unique. A direct particle interpretation of ELFT would be highly desirable. It would lead to an unambiguous def- inition of physical quantities like cross sections. It also would justify the interpretation of ELFT with a triv- ial continuum limit as effective theories for particles. In any case ELFT would become a respectable theory which shares some structural properties with continuum quantum field theory and which is better accessible by analytical and numerical methods. Let us briefly review the status of the particle in- terpretation of continuum quantum field theory. The basic structural properties which are exploited are local commutativity of space like separated observables as an implementation of Einstein causality, and the spectrum condition. A rather satisfactory analysis can be made in theories without massless particles. In theories with *Supported by DAAD. Heisenberg fellow. physical massless particles problems with infraparticles occur which are presently not under full control 1 In purely massive theories one first has to find the single particle states. If they belong to the vacuum sec- tor, i.e. if the particles carry no charge, there exist so- called almost local operators which create the particle states from the vacuum. Using them one constructs the outgoing and incoming multiparticle states via methods of the Haag-Ruelle scattering theory 2. If the single par- ticle state is not in the vacuum sector, i.e. if the particle is charged, one first has to apply the theory of super- selection sectors (see 3 for a review) to construct the group of global gauge transformations and the charged fields. Then one can again apply the methods of the Haag-Ruelle scattering theory for the construction of all scattering states. There are also some unsolved problems. Besides the already mentioned problem of infraparticles the main open problem is the asymptotic completeness, i.e. the question whether each state is an incoming and an out- going scattering state. (For recent progress in this prob- lem see 1, 4). The main obstruction for performing a correspond- ing analysis in ELFT is the absence of local commu- tativity for spacelike separations in the corresponding real time quantum theory. Consider e.g. the theory of a scalar Euclidean field ~(¢), x = (x°,x_) E ~d+l, d :> 1. Let (.) be a translation invariant and reflection positive state. Then, using the transfer matrix formal- ism, one finds a Hilbert space ~, a vector ~ E ?t, 0920-5632/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)