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Generally covariant quantum field theory and scaling limits. (English) Zbl 0626.46063
The authors try to eliminate the rigid metric structure of space-time and to characterize the generally covariant quantum field theory in terms of the allowed germs of families of states. Here general covariance means that the field equations are independent of the coordinate system used to describe 4-dimensional smooth manifold $${\mathcal M}$$. Let X be a vector field on $${\mathcal M}^ n$$ with the components $$X^{\mu}(x)=x^{\mu}- \vec x^{\mu}+0(x-\bar y)^ 2$$, x($$\lambda)$$ an orbit $$dx^{\mu}(\lambda)/d\lambda = \lambda^{-1}X^{\mu}(x(\lambda))$$ with $$x(1)=P$$, and $$\eta^ xP$$ the tangent vector of x($$\lambda)$$ at $$\lambda =0$$. Let $$f^{(n)}$$ be a smooth function (monomial) on $${\mathcal M}^ n$$ to $${\mathfrak C}$$ with support in $${\mathcal O}$$ in every argument, $${\mathcal A}({\mathcal O})$$ formal linear combinations of the $$f^{(n)}$$’s of different degrees n, $$\hat {\mathcal A}(\hat {\mathcal O})$$ the similar tensor algebra of test functions with support in $$\hat {\mathcal O}=\eta {\mathcal O}$$ and $$\hat f^{(n)}\in \hat {\mathcal A}(\hat {\mathcal O})$$. They derive two transformations $(\beta \hat f^{(n)})(P_ 1,...,P_ n)=\hat f^{(n)}(\eta P_ 1,...,\eta P_ n)$ and $(\beta^{- 1}\alpha_{\lambda}\beta \hat f^{(n)})(z_ 1,...,z_ n)=\hat f^{(n)}(\lambda^{-1}z_ 1,...,\lambda^{-1}z_ n).$ Next let a state $$\omega$$ on $${\mathcal A}$$ denote a linear functional on $${\mathcal A}$$ with $$\omega (1)=1$$ and $$\omega (A^*A)\geq 0$$. The state is equal to a vacuum expectation of a representation $$\pi_{\omega}(A)$$ of $$A\in {\mathcal A}$$, and can give a detector when $${\mathcal M}=R^ 4$$. A folium $${\mathcal F}_ i({\mathcal O}_ i)$$ means the equivalent class of a representation $$\pi_ i$$ of $${\mathcal A}({\mathcal O}_ i)$$, and a scaling limit state at $$\bar P\in {\mathcal M}$$ is the one on $$\hat {\mathcal A}$$ given by $$\omega_{\bar P}^{(n)}(\hat f^{(n)})=\lim_{\lambda \to 0}N(\lambda)^ n\omega^{(n)}(\alpha_{\lambda}\beta \hat f^{(n)})$$, where $$N(\lambda)\geq 0$$ is a monotone function for $$\lambda\in (0,\infty].$$
The authors’ first result is the following: Uniformly defined scaling limit states are covariant under dilations and invariant under translations. Their basic main problems are the following (i) and (ii):
(i) May an open covering $${\mathcal O}=\cup {\mathcal O}_ i$$ derive different folia $${\mathcal F}({\mathcal O})$$ whose restrictions to $${\mathcal O}_ i$$ are given $${\mathcal F}_ i({\mathcal O}_ i)?$$
(ii) What are the compatibility conditions for the set $$\{{\mathcal F}_ i({\mathcal O}_ i)\}$$ so that an extension $${\mathcal F}({\mathcal O})$$ may exist? Relating to the problems they show that the Borchers algebra $${\mathcal A}({\mathcal O})$$ is free.
Reviewer: H.Yamagata

##### MSC:
 46N99 Miscellaneous applications of functional analysis 81T05 Axiomatic quantum field theory; operator algebras 83C99 General relativity
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