Gauge Problem in Quantum Field Theory. II. Difficulties of Combining Einstein Equations and Wightman Theory

The results of a previous paper on the gauge problem in quantum electrodynamics are generalized to the gravitational case. In particular, the difficulties connected with the quantization of the gravitational field are analyzed in the framework of axiomatic field theory. For convenience, the simple case of weak gravitational field in vacuo is discussed. Even in this simple case, inconsistencies arise if one wants to combine the Einstein equations and quantum field theory. Under the assumptions: (1) existence of the vacuum, invariant under the Poincaré group, (2) existence of a representation of the Poincaré group such that the fields have tensor transformation properties, and (3) analyticity of the two-point function in the forward tube, it is proved that the Einstein equations for the gravitational potential have no solution apart from the trivial one $R_{\mu \nu \rho \sigma }=0\mathrm{}$. The result is obtained without assuming either local commutativity or the spectral condition or a positive metric in the Hilbert space. Thus the difficulties which arise in the quantization of the gravitational field have very little to do with the Hilbert-Lorentz condition, indefinite metric, etc.; rather, they are strongly connected with the definition of the Riemann tensor in terms of the gravitational potential. As a corollary of the above result, the representations of the Poincaré group for massless spin-2 particles in quantum field theory are shown to be essentially different from the corresponding ones of the classical case.