# zbMATH — the first resource for mathematics

The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes. (English) Zbl 0923.58052
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 $$R$$.

##### MSC:
 58J47 Propagation of singularities; initial value problems on manifolds 81T20 Quantum field theory on curved space or space-time backgrounds 81T05 Axiomatic quantum field theory; operator algebras 83C47 Methods of quantum field theory in general relativity and gravitational theory 35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
Full Text:
##### References:
 [1] [Bor62] Borchers, H.J.: On the structure of the algebra of the field operators. Nuovo Cimento24, 214 (1962) · Zbl 0129.42205 [2] [DB60] DeWitt, B.S., Brehme, R.W.: Radiation damping in a gravitational field. Ann. Phys.9, 220–259 (1960) · Zbl 0092.45003 [3] [DH72] Duistermaat J.J., Hörmander, L.: Fourier integral operators II. Acta Math.128, 183 (1972) · Zbl 0232.47055 [4] [Dim80] Dimock, J.: Algebras of local observables on a manifold. Commun. Math. Phys.77, 219–228 (1980) · Zbl 0455.58030 [5] [Dim82] Dimock, J.: Dirac quantum fields on a manifold. Trans. Am. Math. Soc.269, 133–147 (1982) · Zbl 0518.58018 [6] [Dim92] Dimock, J.: Quantized electromagnetic field on a manifold. Rev. Math. Phys.4, 223–233 (1992) · Zbl 0760.53049 [7] [Dui73] Duistermaat, J.J.: Fourier Integral Operators. Courant Institute of Mathematical Sciences, New York University, 1973 [8] [FH87] Fredenhagen, K., Haag, R.: Generally covariant quantum field theory and scaling limit. Commun. Math. Phys.108, 91 (1987) · Zbl 0626.46063 [9] [Fre92] Fredenhagen, K.: On the general theory of quantized fields. In: K. Schmüdgen, editor, Mathematical Physics X, Berlin: Springer Verlag, 1992, pp. 136–152 · Zbl 0947.81517 [10] [FSW78] Fulling, S.A., Sweeny, M., Wald, R.M.: Singularity structure of the two-point function in quantum field theory in curved spacctime. Commun. Math. Phys.63, 257–264 (1978) · Zbl 0401.35065 [11] [Ful89] Fulling, S.A.: Aspects of Quantum Field Theory in Curved Space-Time. Cambridge: Cambridge University Press, 1989 · Zbl 0677.53081 [12] [Haa92] Haag, R.: Local quantum physics: Fields, particles, algebras. Berlin: Springer, 1992 · Zbl 0777.46037 [13] [Hep69] Hepp, K.: Théorie de la renormalisation. Number 2 in Lecture Notes in Physics. Berlin, Heidelberg: Springer Verlag, 1969 [14] [HK64] Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys.5, 848 (1964) · Zbl 0139.46003 [15] [HNS84] Haag, R., Narnhofer, H., Stein, U.: On quantum field theory in gravitational background. Commun. Math. Phys.94, 219–238 (1984) [16] [Hör71] Hörmander, L.: Fourier integral operators I. Acta Math.127, 79 (1971) · Zbl 0212.46601 [17] [Hör83] Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Berlin: Springer, 1983 · Zbl 0521.35001 [18] [Jun95] Junker, W.: Adiabatic vacua and Hadamard states for scalar quantum fields on curved spacetimes. PhD thesis, University of Hamburg, 1995 [19] [Köh95] Köhler, M.: New examples for Wightman fields on a manifold. Class. Quant. Grav.12, 1413–1427 (1995) · Zbl 0823.53057 [20] [KW91] Kay, B.S., Wald, R.M.: Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate killing horizon. Phys. Rep.207(2) 49–136 (1991) · Zbl 0861.53074 [21] [Rad92] Radzikowski, M.J.: The Hadamard condition and Kay’s conjecture in (axiomatic) quantum field theory on curved space-time. PhD thesis, Princeton University, October 1992 [22] [Sat69] Sato, M.: Hyperfunctions and partial differential equations. In Proc. Int. Conf. on Funct. Anal. and Rel. Topics. Tokyo: Tokyo University Press, 1969, pp. 91–94 · Zbl 0192.58601 [23] [Sat70] Sato, M.: Regularity of hyperfunction solution of partial differential equations. Actes Congr. Int. Matl Nice2, 785–794 (1970) · Zbl 0205.15902 [24] [Uh162] Uhlmann, A.: Über die Definition der Quantenfelder nach Wightman und Haag. Wiss. Zeitschrift Karl Marx Univ.11, 213 (1962) · Zbl 0119.43804 [25] [Ver94] Verch, R.: Local definiteness, primarity and quasiequivalence of quasifree Hadamard quantum states in curved spacetime. Commun. Math. Phys.160, 507–536 (1994) · Zbl 0790.53077 [26] [Wal78] Wald, R.M.: Trace anomaly of a conformally invariant quantum field in curved spacetime. Phys. Rev. D17(6), 1477–1484 (1978) [27] [Wa194] Wald, R.M.: Quantum field theory in curved spacetime and black hole thermodynamics. Chicago lectures in physics. Chicago, USA: Univ. Chicago Press, 1994 [28] [WG64] Wightman, A.S., Gårding, L.: Fields as operator valued distributions in relativistie quantum theory. Ark. Fys,23(13) 1964 [29] [Wo192] Wollenberg, M.: Scaling limits and type of local algebras over curved spacetime. In: W.B. Arveson et al., editors. Operator algebras and topology. Putman Research notes in Mathematics270, Harlow: Longman, 1992
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.