×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI arXiv