Nelson's symmetry and the infinite volume behavior of the vacuum inP(φ)2

Abstract

LetHl be the Hamiltonian in aP(φ)2 theory with sharp space cutoff in the interval (−l/2,l/2). LetEl=infσ(Hl), α(l)=−El/l, and let Ωl be the vacuum forHl. discuss properties of α(l) and Ωl. In particular, asl→∞, there are finite constants β∞<0 and α∞ such that α(l)↑α∞, (α(l)−α∞)l↓β∞, and hence α(l)=α∞+β∞/l+o(l−1). Moreover exp(−c1l)≦∥Ωl∥1≦exp(−c2l) forc1,c2 positive constants, where ∥Ωl∥1 is theL1(Q, dμ0) norm of Ω1 with respect to the Fock vacuum measure. We also present a new proof of recent estimates of Glimm and Jaffe on local perturbations ofHl in the infinite volume limit.