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Quantization Domains

We study the quantization of certain classical field theories using reflection positivity. We give elementary conditions that ensure the resulting vacuum state is cyclic for products of quantum field operators, localized in a bounded Euclidean space-time region O at positive time. We call such a domain a quantization domain for the classical field. The fact that bounded regions are quantization domains in classical field theory is similar to the "Reeh-Schlieder" property in axiomatic quantum field theory.