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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.

##### MSC:
 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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