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SOME MATHEMATICAL TOOLS
FIELDS AND VACUUM EXPECTATION VALUES
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A. S. Wightman algebra analogous analytic continuation anti-commute anti-unitary argument asymptotic completeness axioms boundary values bounded operators Chapter coherent subspace complex Lorentz complex variables components cone construction convergence cyclic vector defined definition dense domain equation equivalence class Euclidean example exists extended tube follows free field given Haag's theorem Hamiltonian hermitian scalar field Hilbert space holomorphic function implies infinitely differentiable function integral interaction invariant irreducible set Jost point Laplace transform linear Lorentz group Lorentz transformations mass Math matrix momentum monomial neighborhood normal commutation relations open set particles PCT theorem Phys physically realizable polynomial proof prove quantum field theory quantum mechanics representation restricted Lorentz satisfying scalar field scalar product Section self-adjoint sequence smeared fields space-like separated space-time spinor subset super-selection rules Suppose symmetry tempered distribution test functions transformation law unique unitary vacuum expectation values vanishes vector space wedge theorem zero