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Lattice Field Theory

Lattice Field Theory Quantum field theory has proven a very successful theoretical framework for understanding the interactions of elementary particles. One very convenient formulation of such theories is given by the so-called path integral prescription originally due to Richard Feynman. Within this formalism one can compute the vacuum expectation values of arbitrary products of fields at different spacetime points by integrating them over a distribution given by the exponential of a function of the fields called the action.

Unfortunately in most cases this integral cannot be done exactly and we must resort to an approximation called perturbation theory to get answers. This technique can only be applied in so-called weakly coupled theories where the characteristic interaction parameter is small. For large couplings there is generically only one way to proceed - we must imagine defining the fields of the problem only over a finite set of grid points in space-time and evaluate the integrals using a numerical method known as Monte Carlo simulation.

A special formalism, and a variety of techniques and methodologies have arisen to treat such lattice field theories. The group at Syracuse has contributed specifically to the application of these ideas to problems involving quantum gravity and the fluctuations of random discrete manifolds. Much of the latter work exhibits strong connections to statistical mechanics and the behavior of condensed matter and biological membranes. We are also studying the problems associated to the study of so-called supersymmetric theories on lattices. These latter theories are thought to be important for understanding the unification of the fundamental forces and many of the open questions in such theories relate to the structure of the theories at strong coupling - hence the importance of good lattice formulations.

Students working in this field need to acquire both a good working knowledge of quantum field theory and perhaps current ideas about supersymmetry and quantum gravity but also experience in numerical simulation, data analysis techniques and computer programming. This wide knowledge can be a benefit after graduation leading to a wide variety of possible careers - from academic physicist, to software engineer to Wall street analyst!

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Lattice Supersymmetry and Technicolor (Simon Catterall)

Prof. Catterall is currently interested in theoretical and computational lattice studies of theories which attempt to go beyond the Standard Model of particle physics.

In the past he has worked on discrete theories of quantum gravity and string theory but more recently has been interested in studying supersymmetric theories on the lattice. He has developed new lattice formulations which allow an element of supersymmetry to be preserved exactly at non-zero lattice spacing and has begun studying these theories using Monte Carlo simulation. Such studies can potentially cast light on conjectured dualities between supersymmetric gauge theories and gravity and have impact and application in string and M-theory.

He has also recently begun investigations of lattice gauge theories with fermions in higher dimensional representations. Such models are conjectured to develop conformally invariant phases as the number of fermion flavors is increased. Close to these points the theory exhibits a slow evolution of the coupling with energy scale -- the theory walks. We have been examining the minimal model with 2 colors and 2 flavors which can form the basis of a technicolor model for breaking the electroweak theory with a light composite Higgs.

Prof. Catterall is a member of the USQCD collaboration and has access to multi Teraflop scale supersomputers at Fermilab and Jlab for carrying out these studies.