Space and time from translation symmetry

We show that the notions of space and time in algebraic quantum field theory arise from translation symmetry if we assume asymptotic commutativity. We argue that this construction can be applied to string theory. The goal of the present paper is to understand better the foundations of string theory. I will start with the question ”What is quantum field theory?” I hope to explain that in quantum field theory space and time can be considered as secondary notions arising from the existence of commutative symmetry group of the theory. The explanation will be based on the paper [1]. This paper generalizes Haag -Ruelle theory [3] [4 ] of scattering in algebraic quantum field theory 1to the case when the assumption of locality or of commutativity of observables in domains separated by a space-like interval is replaced by the assumption of asymptotic commutativity. 2 (See, for example, [2] for a review of Haag-Ruelle theory.) Haag-Ruelle constructions as well as the results of [1] can be applied only if all particles are massive. However, D. Buchholz and his collaborators developed scattering theory also in the case when there exist massless particles (see [5], [6],[7]). One can show that in this case also one can get rid of the assumption of locality replacing it by the condition of asymptotic commutativity in a form much weaker than in [1]. It seems that the condition of asymptotic commutativity is sufficient to guarantee macroscopic causality. It seems that the explanation of origin of space-time from translation symmetry can be applied also to string theory. At the end of the paper we will discuss the application of our ideas to string field theory. The string theory is non-local, however, people understand that still it should be local in some sense (see, for example, [17]). Our statement is that asymptotic commutativity replaces locality in string theory. We argue that that the problems of string theory can be rigorously formulated in the framework of algebraic quantum field theory. It was suggested that more appropriate name for axiomatic quantum field theory in its modern form is algebraic quantum field theory [18]. I agree with this suggestion. In the case of local theory this condition is satisfied for quasilocal observables.