A local spectral condition for axiomatic quantum fields on the de Sitter surface
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The Wightman axioms provide an intuitive and mathematically rigorous approach to quantum field construction on flat Minkowski space, and much work has been done to extend these axioms to curved space-time manifolds. A key axiom in the Wightman framework states that the generator of time translation must be a positive semi-definite operator, which guarantees non-negative energy for a quantum field. One of the major difficulties in extending these axioms to curved manifolds arises when the manifold in question has no global time-like isometry, and hence no global notion of time and energy. I propose a set of Wightman-like axioms for the de Sitter surface which includes a solution to this problem by introducing a local non-negative spectrum condition which gives a local meaning to energy, in essence a local Hamiltonian, and guarantees it to be a non-negative operator. I then construct a mathematically rigorous free massless scalar field on two-dimensional de Sitter space and prove that it satisfies these axioms.