Lon Rosen

Legendre transforms and r-particle irreducibility in quantum field theory: the formalism for r = 1,2

Abstract We analyze the first and second Legendre transforms Γ ( r ) ( r = 1, 2) of the generating functional G for connected Greens functions in Euclidean boson field theories. By using Spencers idea of t -lines we define and prove irreducibility properties independently of perturbation theory. In particular we prove that Γ ( r ) generates r -irreducible vertex functions, r -irreducible expectations and r -field projectors; moreover, Γ (2) generates (generalized) Bethe-Salpeter kernels with 2-cluster-irreducibility properties.

Legendre Transforms and R Particle Irreducibility in Quantum Field Theory: The Formal Power Series Framework

Abstract We study the higher Legendre transforms Γ ( r ) { A } of the generating functional G for connected Greens functions in Euclidean boson field theories. To analyze Γ ( r ) { A } rigorously even when it does not make sense as an ordinary functional, we develop the framework of formal power series in A . For r = 1, 2 we isolate regularity conditions on G that ensure the existence of Γ ( r ) as a formal power series and we verify these conditions for the weakly coupled P ( φ ) 2 model. We also establish the improved regularity of the functional Φ ( r ) obtained by subtracting from Γ ( r ) its trivial singular part.

Dimensional regularization and renormalization of QED

We give an χ-space definition of dimensional regularization suited to the tree expansion method of renormalization. We apply the dimensionally regularized tree expansion to QED, obtaining sharp bounds on the size of a renormalized graph. Subtractions are made with the Lagrangian counterterms of the tree expansion, not by minimal subtraction techniques, and so do not entail a knowledge of the meromorphic structure of a graph as a function of dimension. This renormalization procedure respects the Ward identities, and the counterterms required are gauge invariant.

Higher Legendre transforms and their relationship to Bethe–Salpeter kernels and r‐field projectors

We analyze the structure of the higher Legendre transforms Γ(r){A}(r?1) of the generating functional G of the connected Green’s functions Gn in Euclidean boson field theories. In addition to the vertex functions, Γ(r) generates a variety of objects of interest for their r‐irreducibility in certain channels, e.g., r‐irreducible expectations, rth order Bethe–Salpeter kernels, and r‐field projectors. Our analysis is independent of perturbation theory, our definition of r‐irreducibility being based on Spencer’s idea of t‐lines. We derive formulas for ∂ntΓ(r){A;t} (in terms of either δnAΓ(r){A;t} or the Gn’s) to be used as input in the proofs of r‐irreducibility. For the case of the weakly coupled P(φ)2 model, we establish the existence of the moments δnAΓ(r){0;t} and their regularity in t.

Construction of the Yukawa2 field theory with a large external field

We consider the Yukawa2 model with (relativistic) interaction density λψΓψφ+μφ, where Γ=1 or γ5. For sufficiently large μ, we apply the Glimm–Jaffe–Spencer cluster expansion to construct the infinite volume theory satisfying the Wightman and Osterwalder–Schrader axioms including a positive mass gap.

On the infinite volume limit of the strongly coupled Yukawa2 model

Using the FKG inequality, we construct infinite volume expectations of products of boson fields and fermi currents (ψ)ren for the scalar Yukawa2 model with arbitrary coupling constant. These expectations satisfy the Osterwalder–Schrader axioms with the possible exception of clustering.

Renormalization Theory and the Tree Expansion

Perturbative renormalization theory may not be a topic of CQFT, but it does have a long and rich history and is the basis of every textbook account of QFT. Still, few students really understand it. This is hardly surprising since no text really explains it. I believe that it is possible to give a simple and complete account of perturbative renormalization, and it is fitting that I try to do so in a school on CQFT; for the approach I shall advocate shares a number of ideas with current work in CQFT, and an understanding of the cancellation of ∞’s in perturbation theory is important for an understanding of what happens in the “small field region” of CQFT.

The second Legendre transform for the weakly coupled P(φ)2 model

We prove the existence and analyticity of the second Legendre transform of the generating functional for Euclidean Green’s functions in the weakly coupled P(φ)2 model. The proof involves a bound on the partition function with a nonlocal quadratic source term. This bound also implies bounds on the Schwinger functions Sn( f1,..., fn) that are optimal with respect to both topology and n‐dependence.