The functional integral in quantum field theory (QFT) and its analogue the
partition function in statistical physics are used to give unified,
concise, but usually only heuristic, characterizations of elementary particle
and condensed matter physics models and their symmetries. The
Brydges-Yau-Dimock-Hurd constructive method is one coherent rigorous and mathematical
approach to understanding functional integrals. In ten years of development, it
has been applied to numerous fundamental problems in QFT and statistical physics,
many of which have never been rigorously treated by other means: the dipole gas, the
Coulomb gas, sine--Gordon QFT, quantum electrodynamics, the
phi^4_d model in dimensions d=3 and d=4-epsilon. I am
currently working on such problems as correlation functions, the construction of a
jointly long-distance/short-distance convergent massless model, and the
construction of the Coulomb gas at the famous Kosterlitz-Thouless transition
point.
J. Dimock & T.R. Hurd, ``Sine-Gordon revisited'',
Annales Henri Poincaré,
2000.
The papers ``A renormalization group analysis of the
Kosterlitz-Thouless phase'' and "Construction of the two-dimensional sine-Gordon theory for beta < 8 pi" (Dimock and Hurd 1991 and 1993) solve two complementary open problems, using an early
version of the Brydges--Yau constructive method. The first constructed
the so--called KT phase of the Coulomb gas up to the conjectured
critical temperature
T_{KT}=(8\pi)^{-1}. The second constructed the sine--Gordon QFT
above T_KT. In 1996, my coauthor J. Dimock and I discovered a
technical flaw in the method which cast both results into uncertainty.
It required enormous effort to find improvements to the method which
would overcome the error, and restore all of our results. The above
paper does all of this.
D. Brydges, J. Dimock & T.R.
Hurd, ``A non--Gaussian fixed point for phi^4
in 4-\epsilon Dimensions'', Commun. Math. Phys. 198,
111-156, 1998.
Wilson, the originator of the constructive renormalization group (RG),
proposed this problem in the early '70s as a non-physical model which
is of non-renormalizable type (and hence by naive arguments should be
inconsistent), but in fact should have a non-trivial infrared fixed
point and be constructable. Taking the parameter epsilon > 0 as a small
quantity, my two coauthors and I give a complete construction of
the model, including the fixed point and associated stable and unstable
manifolds in a Banach space of measures, thus solving the problem
proposed by Wilson. The result requires a detailed analysis of a
single RG step, coupled with a non-trivial extension of the stable
manifold theorem from dynamical systems theory.
D. Brydges, J. Dimock & T.R. Hurd,
``Estimates on renormalization group transformations'', Can. Jour.
Math. 50, 756--793, 1998.
This paper gives a complete model independent treatment of the BYDH
method suitable for application to scalar field theories for which the
interaction is unbounded but stable.
D.H.U. Marchetti, T.R. Hurd & P.F. da Veiga, ``The1/N expansion as a perturbation around mean field theory: a
one-dimensional Fermi model'', Commun. Math. Phys. 179, 623--646,
1996.
The N-component two--dimensional Gross--Neveu (GN) model was
proposed in 1974 as one which although massless to all orders in
perturbation theory, should nevertheless for large N develop a
non--zero mass by spontaneous symmetry breaking. In this paper, we
present a complete combinatorial treatment of the one--dimensional
model on a lattice. We verify mean-field predictions, such as
exponential decay of correlations, for all values of the coupling
constant, for large N.
D. Brydges, J. Dimock and T.R.
Hurd, ``The short distance behaviour of ( phi^4)_3'', Commun. Math.
Phys. 172, 143--186, 1995.
This paper was the first to address via the BYDH method a model with a
``large--field'' problem caused by an unbounded interaction potential.
The model treated, the ultraviolet phi^4 model in three dimensions,
has been solved independently by several earlier methods. Our paper
provides a foundation for the further models we have since addressed.
