Abstract. We propose a generalization of the Haag-Kastler axioms for local observables to Lorentzian manifolds. The framework is intended to resolve ambiguities in the construction of quantum field theories on manifolds. As an example we study linear scalar fields for globally hyperbolic manifolds. AxiomsQuantum field theories are usually defined on Minkowski space-time, but it seems desirable to generalize to arbitrary Lorentzian manifolds. This is so not only to accommodate physical systems that require a manifold model for spacetime, but also as a means of gaining perspective on the general structure of quantum field theories. General references for quantization on manifolds are De Witt [3] and Isham [12].The problem can be posed as finding field operators which satisfy given field equations. However, on a general manifold there is no natural choice for the Hilbert space on which the operators act, and, in particular, there is no vacuum state to be used as a reference point. This suggests formulating the problem in terms of the algebraic structure of the field operators, and leaving the specification of states as a secondary step.One algebraic approach has been developed by Isham [12], Kay [13], and Hajicek [t0]. They associate with each Cauchy surface S the C* algebra d s generated by the canonical commutation relations (CCR) over functions on S. The field equations then determine isomorphisms ~4 s ~ ~¢~ which give the dynamics. This type of approach seems quite satisfactory for linear problems, but one can anticipate troubles in extending it to nonlinear problems: things are probably too singular to allow a definition of the algebras d s.In this paper we propose another algebraic approach which generalizes the Haag-Kastler algebras of local observables on Minkowski space [8]. There is a single C* algebra d together with distinguished subalgebras ~¢(O) corresponding to local regions of space-time. All reference to fields to supressed.

Abstract. On globally hyperbolic Lorentzian manifolds we construct field operators which satisfy the Dirac equation and have a causal anticommutator. Ambiguities in the construction are removed by formulating the theory in terms of C* algebras of local observables. A generalized form of the Haag-Kastler axioms is verified.Introduction. A Lorentzian manifold is a four dimensional manifold TV/ together with a pseudo-Riemannian metric g of signature ( + , -, -, -). Such manifolds are widely used as models for space-time. In particular in the absence of boundaries and gravitational fields one uses M = R4 and g = tj = Minkowski metric which has components {r)ab} = diag(l, -1,-1,-1).We are interested in formulating quantum field theories on general Lorentzian manifolds. This means giving up many ideas familiar from the usual Minkowski space treatments. Thus there are no Poincaré transformations on M, no vacuum, no particle states, and so forth. All one is left with are the field equations. These are to be solved taking as data a representation of the CCR or CAR over a space-like hypersurface (CCR = canonical commutation relations, CAR = canonical anticommutation relations). For linear equations this quantum problem can be reduced to a corresponding classical problem for which solutions can be constructed in favorable cases. In the nonlinear case very singular formal solutions can be exhibited which one might hope to make rigorous.The procedure sketched above is fraught with ambiguities. What hypersurface should one take? what representation of the CCR/CAR? and so forth. In general there is no natural choice and one should show that all choices lead to the same theory. To do this in a fundamental way it seems necessary to abandon the field

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We develop the quantization of the electromagnetic field on an arbitrary globally hyperbolic Lorentzian manifold with a compact Cauchy surface.

We study the sine-Gordon model in two dimensional space time in two different domains. For β > 8π and weak coupling, we introduce an ultraviolet cutoff and study the infrared behavior. A renormalization group analysis shows that the the model is asymptotically free in the infrared. For β < 8π and weak coupling, we introduce an infrared cutoff and study the ultraviolet behavior. A renormalization group analysis shows that the model is asymptotically free in the ultraviolet.