The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes
The authors discuss properties of algebraic quantum field theories on globally hyperbolic Lorentz manifolds. M. Radzikowski (1992) interpreted the global Hadamard criterion in terms of wave front sets by the use of techniques of microlocal analysis (the links with quantum field theory had been observed by Duistermaat and Hörmander in 1972 but not taken up). Namely, the wave front set of the two point distribution of any physically reasonable state should be contained in the set \(\{(x_1,k_1), (x_2, k_2) \in T^*(M)^2\setminus \{0\}: (x_1,_1)\sim(x_2,k_2)\), \(k_1^0\geq 0\), \(\sim\) means that a light-like geodesic links \(x_1t_0x_2\}\). He also proposed that a wave front set spectrum condition should exist for the higher \(m\)-point distributions. Unfortunately, it was soon discovered that not only the \(m\)-point distributions \(m>2\) associated to a quasi-free Hamadard state of a scalar field on a globally hyperbolic spacetime do not satisfy this condition, but also that the distributional product of two different fields gives rise to counterexamples to the two point condition. The authors sucessfully modify this condition in the following way: a state \(w\) with \(m\)-point distributions \(w_n\) is said to satisfy the microlocal spectrum condition \(\mu\)SC iff for any \(m\), \(WF(w_m)C\Gamma_m\) is the set \(\{(x_1,k_1),\dots,(x_m,k_m)\in T^*(M)^m\setminus \{0\)\} and \(x_i,k_k\) correspond to a finite graph immersion \((x,y,k_e)\), \(x\) maps vertices of \(G\) to points of \(M,\gamma\) of \(G\) to curves \(\gamma(e)\), \(\nabla k_e=0\) piecewise on curves \(\gamma\) and \(k_e\) is directed to the future whenever \(\upsilon< \upsilon^i\), \(x_i=x(i)\), \(k_i=\sum_{e,s(e)=i}\) \(k_e(x_i)\), with source \(s (\gamma (e))=\gamma(s(e))\). The authors show that the \(\mu\)SC condition is compatible with the usual Minkowski space spectrum condition, and nontrivial examples for physical states satisfying this new condition are presented. They establish that the Wick monomials of the free Klein-Gordon field on a globally hyperbolic spacetime with respect to any quasi-free state \(w\) satisfying the \(\mu\)SC condition are well defined Wightman fields on the GNS-Hilbert space of \(w\) with core \(D_w\) and dense invariant domain \(D\) generated by applying finitely many smeared Wick monomials to \(R\).
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