Guy Battle

A NOTE ON CLUSTER EXPANSIONS, TREE GRAPH IDENTITIES, EXTRA 1/N! FACTORS!!!*

We draw attention to a new tree graph identity which substantially improves on the usual tree graph method of proving convergence of cluster expansions in statistical mechanics and quantum field theory. We can control expansions that could not be controlled before.

Phase space localization theorem for ondelettes

It is proven that an orthonormal basis of ondelettes can never have exponential localization in both position space and momentum space.

A new combinatoric estimate for cluster expansions

We state and prove a new and previously unsuspected tree graph inequality, which is significantly stronger than the one commonly applied to cluster expansions. The older inequality controls the counting problem in the convergence proof of such an expansion, but the new inequality does more: it also exhibits extra 1/n! factors that can be applied to the cancellation of number divergences. The proof of this new combinatoric estimate is completely elementary.We state and prove a new and previously unsuspected tree graph inequality, which is significantly stronger than the one commonly applied to cluster expansions. The older inequality controls the counting problem in the convergence proof of such an expansion, but the new inequality does more: it also exhibits extra 1/n! factors that can be applied to the cancellation of number divergences. The proof of this new combinatoric estimate is completely elementary.

The FKG inequality for the Yukawa2 quantum field theory

We establish the FKG correlation inequality for the Euclidean scalar Yukawa2 quantum field model and, when the Fermi mass is zero, for pseudoscalar Yukawa2. To do so we approximate the quantum field model by a lattice spin system and show that the FKG inequality for this system follows from a positivity condition on the fundamental solution of the Euclidean Dirac equation with external field. We prove this positivity condition by applying the Vekua-Bers theory of generalized analytic functions.

Cardinal spline interpolation and the block spin construction of wavelets

Abstract . The construction of wavelets that are inter-scale orthogonal (pre-ondelettes) but not necessarily intra-scale orthogonal is a major part of the block spin approach to wavelets. We review this in the case of Lemarie wavelets and relate it to the cardinal spline approach of Chui and Wang. We advertize the computational value of the pre-ondelette basis as opposed to that of the orthonormal basis of wavelets.

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.

A gradient-projective basis of compactly supported wavelets in dimension n > 1

A given set W = {WX} of n-variable class C1 functions is a gradient-projective basis if for every tempered distribution f whose gradient is square-integrable, the sum

Momentum-Entire Wavelets with Discrete Rotational Symmetries in 2D

\sum\limits_\chi {(\int_{\mathbb{R}^n } {\nabla f \cdot } \nabla W_\chi ^* )} W_\chi

Pseudoscalar interaction of coupled quantum‐mechanical oscillators with independent Fermi systems

converges to f with respect to the norm

Gradient-Orthonormal Bases of Momentum-Entire Wavelets in Odd Dimension

\left\| {\nabla ( \cdot )} \right\|_{L^2 (\mathbb{R}^n )}