A Alternate Constructive Approach to the Phi -4(3) Quantum Field Theory, and a Possible Destructive Approach to PHI-4(4)

I study the construction of (phi)('4) quantum field theories by means of lattice approximations. It is easy to prove the existence of the continuum limit (by subsequences); the key question is whether this limit is something other than a (generalized) free field. I use correlation inequalities, infrared bounds and field equations to investigate this question. For space-time dimension d less than four, I give a simple proof that the continuum -limit theory is indeed nontrivial; it relies, however, on a conjectured correlation in inequality closely related to the (GAMMA)(,6) conjecture of Glimm and Jaffe. Moreover, the Euclidean invariance of the continuum theory is an open question within the present approach. For space-time dimension d greater than or equal to four, I argue--but do not prove--that the continuum limit is inevitably a (generalized) free field, irrespective of the choice of charge renormalization. The argument is based on old ideas of Landau and Pomeranchuk, improved through the use of correlation inequalities applied to the exact field equations.