Tadeusz Balaban

(Higgs)

This is the second part of the paper entitled, “(Higgs)2,3 Quantum Fields in a Finite Volume.” The proof of an upper bound for vacuum energy is completed with the exception of some technical estimates.

_{2,\,3}

This paper continues the analysis of the low temperature expansions for classicalN-vector models started in [1]. A main part of it is a derivation of renormalization group equations and a construction of their solutions. To do this we have to introduce “a fluctuation integral” connected with a next renormalization transformation, and to make its preliminary analysis. The results of the paper are summarized in theorems stating that the renormalization transformation preserves the space of densitites, or actions described inductively in [1].

quantum fields in a finite volume. II. An upper bound

This paper is a contribution to a program to see symmetry breaking in a weakly interacting many boson system on a three-dimensional lattice at low temperature. It provides an overview of the analysis, given in Balaban et al. (The small field parabolic flow for bosonic many-body models: part 1—main results and algebra, arXiv:1609.01745, 2016, The small field parabolic flow for bosonic many-body models: part 2—fluctuation integral and renormalization, arXiv:1609.01746, 2016), of the ‘small field’ approximation to the ‘parabolic flow’ which exhibits the formation of a ‘Mexican hat’ potential well.

A low temperature expansion for classical

Expansions of the type described in the inductive hypothesis (H.5) in the paper [1] are constructed for local functions of the “background” configurations, i.e., solutions of the variational problems studied in the previous paper [3]. A main part of this construction is a further analysis of a local structure of the solutions.

N

We obtain a convergent multi-scale expansion for a class of low temperature classical vector spin models, of the type of lattice “λ|φ|4” field theory, in dimensions d ≥ 3. With the help of this expansion we prove main statements of the so called “spin wave picture”, like the existence of a continuum of phases parametrized by vectors of the unit sphere in the space of spins φ, and the existence of Goldstone bosons, i.e. free massless decay of truncated transversal two-point correlation functions.

-vector models. II. Renormalization group equations

In a previous paper, we developed a power series representation and estimates for an effective action of the form ln[∫ef(α1,…,αs;z∗,z)dμ(z∗,z)/∫ef(0,…,0;z∗,z)dμ(z∗,z)]. Here, f(α1,…,αs;z∗,z) is an analytic function of the complex fields α1(x),…,αs(x),z∗(x),z(x) indexed by x in a finite set X and dμ(z∗,z) is a compactly supported product measure. Such effective actions occur in the small field region for a renormalization group analysis. We illustrate the technique by a model renormalization group flow motivated by the ultraviolet regime in many boson systems.