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Local Wick polynomials and time ordered products of quantum fields in curved spacetime. (English) Zbl 0989.81081
Let \({\mathfrak A}(M,g)\) be the *-algebra generated by the identity and the smeared free field operators \(\varphi(f)\) on a spacetime \((M,g)\), and \(\omega:{\mathfrak A}(M,g)\to C\) a quasi-free Hadamard state. Let \(W_n(t)\) be the operator given by the smearing of \(W_n(x_1,\dots, x_n)=:\varphi(x_1) \cdot \cdots \cdot\varphi(x_n):_\omega\) with \(t=f_1\otimes\cdots \otimes f_n\), and \({\mathfrak W}(M,g)\) the *-algebra generated by 1 and \(W_n(t)\) containing Wick polynomials. Let \(\chi\) be an isometric causality preserving map from \((N, g')\) into \((M,g)\) \((g'=\chi^*g)\), and \(\iota_\chi: {\mathfrak W}(N,g')\to{\mathfrak W} (M,g)\) the corresponding homomorphism. If \(\iota_\chi (\Phi[\chi^*g] (f))= \Phi [g](f\cdot \chi^{-1})\) for \(\forall f\in{\mathfrak D}(N)\) holds for a quantum field \(\Phi\), \(\Phi\) is said to be local and covariant. Let \(\{\varphi^k(x)\}\) and \(\{\widetilde\varphi^k(x)\}\) be two sets of local Wick products satisfying some requirements. Ambiguity of the local products is given by the finite dimensional curvature terms \({\mathcal C}_{k-i}(x)\) in \(\widetilde \varphi^k(x) =\varphi^k (x) +\Sigma_{i=0 \sim k-2} {_kC_i} \cdot{\mathcal C}_{k-i}(x) \varphi^i(x)\). Similar expression for the local time order products \(T(\Pi_{i=1 \sim n} \varphi^{k_i} (x_i))\) is also obtained.

81T20 Quantum field theory on curved space or space-time backgrounds
81T05 Axiomatic quantum field theory; operator algebras
46L60 Applications of selfadjoint operator algebras to physics
46N50 Applications of functional analysis in quantum physics
47L90 Applications of operator algebras to the sciences
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