Φ4 Theory Is Trivial

Perturbative study of many quantum field models has lead to many impressive results: beta functions for QED and for the Φ4 theory are known up to the fifth order whereas that of QCD up to the fourth order [1]-[7]. These studies however are based on the expansion in the small coupling constant such that the behavior at large couplings is completely missed. Although the possibility of a perturbative expansion in the strong coupling constant has been explored over the years [8]-[14] no definite method emerged. Of significant importance is the Φ4 theory at strong coupling whose triviality [15]-[21] has been the subject of intensive debate. In essence the triviality of the Φ4 theory means that the renormalized coupling constant λR vanishes in the limit of large cut-off and the model behaves like a non-interacting field theory. Many previous theoretical studies suggested that the Φ4 theory is trivial claiming it in d 6= 4 [22], [23], [24], in computer simulations [25] or for O(N) symmetric model [26]. In [20], [21] it was proved, based on non perturbative methods, that the Φ4 theory is trivial for a large bare coupling constant. In the present work we will give a proof of triviality of the Φ4 theory valid for any value of the the bare coupling constant. Thus we will show that the all order propagator of the theory is that of the free Lagrangian: