Noncovariant gauges in particle physics, renormalization in Yang-Mills theory, quantum gravity, and Chern-Simons theory.
Present Research Activities:
The main research activities are in quantum field theory, specifically in the unification and mathematical treatment of gauge theories. It seems that all viable field theories are gauge theories. For instance, the theory of gravity is a gauge theory and so is the theory describing the strong interactions. The theory of superstrings which endeavours to combine Einstein's general relativity with quantum mechanics, is the most radical gauge theory.
Some of the most recent gauge theories are not only mathematically complicated but also conceptually difficult to follow. In order to extract useful information from these theories, researchers need to use clever techniques which reduce the horrendous algebra and thus make the inherent properties transparent. One such approach is to replace covariant gauges like the Feynman gauge by physical (or noncovariant) gauges such as the light-cone gauge, or the Coulomb gauge. Since 1983 physical gauges have become increasingly popular, especially in such sophisticated theories as quantum chromodynamics and superstrings.
Current research aims at developing mathematically rigorous procedures for physical gauges. The latter are used to calculate basic quantities like self-energies of elementary particles, and to derive nonlocal counterterms in Yang-Mills and Chern-Simons theory at the two-loop level.
During the past two years, emphasis has been on the quantization of non-Abelian gauge theories in the noncovariant Coulomb gauge, which has perplexed theorists for 30 odd years. To solve the problem, a new gauge invariant technique, called split dimensional regularization, was developed, which leads to consistent, ambiguity-free Coulomb-gauge Feynman integrals.
G. Leibbrandt. 1997. Split Dimensional Regularization in the Coulomb Gauge. XXVIII International Conference on High Energy Physics, Warsaw, Poland, July 25-31, 1996. World Scientific, Singapore), Vol. II, 1639-1641.
G. Leibbrandt and J. Williams. 1996. Split Dimensional Regularization for the Coulomb Gauge. Nuclear Physics, B 475, 469-483.
G. Leibbrandt and M. Staley. 1996. QCD Pressure at Two Loops in the Temporal Gauge. Nuclear Physics, B 461, 259-274.
G. Leibbrandt and J.D. Williams. 1995. Two-Loop Quark Self-Energy in a New Formalism (I). Overlapping Divergences. Nuclear Physics, B 440, 573-602.
S. Fachin and G. Leibbrandt. 1995. Technique for Deriving Feynman Integrals in Axial-Type Gauges. International Journal of Modern Physics, A 10, No. 19, 2747-2767.
G. Leibbrandt and M. Staley. 1994. Finite Temperature Calculations in the Temporal Gauge. Nuclear Physics, B 428, 469-484.
G. Leibbrandt and M. Staley. 1994. Debye Screening Length in the Imaginary-Time Formalism in the Temporal Gauge. Proc. Third Workshop on Thermal Field Theories and their Applications, edited by F.C. Khanna, R. Kobes, G. Kunstatter and H. Umezawa (World Scientific Publ. Co. Pte Ltd., Sinapore 1994), pp. 249-254.
G. Leibbrandt and C.P. Martin. 1994. The Light-Cone Gauge, Chern-Simons Theory and Topologically Massive Yang-Mills Theory. Nuclear Physics, B 416, 351-376.
G. Leibbrandt, C.P. Martin and M. Staley. 1993. Nonlocalities in Field Theory. Nuclear Physics, B 405, 777-796. G. Leibbrandt and K. Allan Richardson. 1992. QED in a Unified Axial-Gauge Formalism with a General Gauge Parameter. Physical Review, D 46, 2578-2584.
BOOK:
G. Leibbrandt. 1994. Noncovariant Gauges: Quantization of Yang-Mills and Chern-Simons Theory in Axial-type Gauges. (World Scientific Publishing Co. Pte Ltd., Singapore).