The energy momentum spectrum and vacuum expectation values in quantum field theory, II

We prove that the P(φ{symbol})₂ quantum field theory satisfies the spectral condition. The space time translation a=(x, t) is implemented by the unitary group U(a)=exp(itH-ixP), and the joint spectrum of the energy operator H and the momentum operator P is contained in the forward cone. We also obtain bounds on certain vacuum expectation values of products of field operators. Our proofs involve an analysis of the limit V→∞ for approximate theories in a periodic box of volume V. Assuming the existence of a uniform mass gap, we are able to establish all the Wightman axioms with the exception of the Lorentz invariance of the vacuum.