Elementary Particles

Computational 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!

Discrete Quantum Gravity (Catterall)  

Our research in recent years has focused on a class of discrete models for Euclidean quantum gravity in which continuous manifolds are replaced by finite, simplicial meshes. This dynamical triangulation approach has seen spectacular success when applied to two dimensional gravity - it has been possible to predict the effect of gravitational fluctuations on matter fields and to uncover typical features of the quantum geometry. The goal of this research has been to uncover which of these features survive to higher dimension, whether it is indeed possible non-perturbatively to formulate a quantum theory of four dimensional Einstein gravity and to extend some of the methods of two dimensional quantum gravity to three and four dimensions. We have utilized extensive Monte Carlo simulations to explore these issues in a non-perturbative setting.

The models follow from attempts to write down a (Euclidean) lattice path integral for gravitational systems. It is hypothesized that the functional integral over physically distinct metrics with some fixed topology can be approximated by a sum over distinct, equilateral triangulations with the same topology. A triangulation of a manifold is just a simplicial, piecewise linear decomposition of the manifold. For example, in two dimensions it just amounts to building a surface out of triangles.

The simulations attempt to sample the most important contributions to this sum over manifolds by utilizing a Monte Carlo algorithm to explore the space of triangulations. This is accomplished by finding a set of ergodic, local moves which change the triangulation in the vicinity of some point. Such moves are known and generalize to arbitrary dimension. This has allowed us to write a single code for simulating an arbitrary dimension random manifold. Furthermore, it is straightforward to extend this code to tackle models in which matter fields are coupled to the fluctuating geometry - one simply assigns them to subsimplices of the triangulation using a matter action which couples fields locally according to the triangulation in question.

In two dimensions we were able to exhibit the fractal structure of typical quantum manifolds using a finite size scaling technique. Perhaps rather surprisingly we found that the quantum effective dimension of the manifold could be quite different from its classical (smooth) dimension. We also developed a real space renormalization group for models of random geometries which allowed us to follow the flows of both matter and gravitational operators under a simultaneous blocking of the matter fields and coarsening of the discrete geometry. Indeed, it was possible in two dimensions to obtain the dressed anomalous dimensions of c=1/2 matter coupled to gravity this way - in good agreement with the theoretical predictions from conformal field theory. This method has also been applied in three and four dimensions. Unfortunately, the original goal of defining a four dimensional quantum theory of gravity close to some non-perturbative fixed point of Einstein gravity has proven elusive - indeed it appears that in pure gravity without higher derivative operators this is impossible - lattice models only possess first order phase transitions. This has led to an investigation of higher derivative theories which is our current principal line of research in this area.

We are currently extending our work on four dimensional triangulations to a wider class of model whose action contains, in addition to the usual Einstein-Hilbert piece, a term depending on the square of the scalar curvature, with coupling constant b.  We know from previous work that for b less than or equal to zero this lattice model possesses only first order phase transitions as the linear scalar coupling K is varied. For K<Kc(b) the partition function  is dominated by triangulations with (locally) negative curvature and large effective dimension. These so-called crumpled configurations contain lattice artifacts - vertices which are shared by a number of simplices which diverges with the total volume. Conversely, for K above the critical value Kc(b), the mean curvature is driven to large positive values and a branched polymer phase is entered, characterized by a effective dimension of two. The use of a DT lattice imposes kinematic constraints on the smallest and largest curvature possible and we can understand each phase as representing geometries which saturate these kinematic bounds. The discontinuous nature of the phase transition at Kc(b) thus reflects the drastic rearrangement of the geometry which must be effected on passing between the two phases.

However, the use of a positive coupling b restricts the fluctuations of the local curvature and may render the model insensitive to these lattice bounds. Our preliminary results indicate that the latent heat may be decreasing with increasing b yielding the hope that for some finite b it may vanish - the first order line perhaps terminating on a second order critical point. This would be an exciting prospect as it offers the possibility of constructing a continuum limit for this discrete model. If so it is possible that this model may realize an old conjecture due to Weinberg - that perhaps a theory of higher derivative quantum gravity might find a sensible definition near an I.R unstable fixed point.

Lattice Supersymmetry (Catterall)

Supersymmetry is thought to be a crucial ingredient in any theory which attempts to unify the separate interactions contained in the standard model of particle physics. Since low energy physics is manifestly not supersymmetric it is necessary that this symmetry be broken at some energy scale. Issues of spontaneous symmetry breaking have proven difficult to address in perturbation theory and hence one
is motivated to have some non-perturbative method for investigating such theories. The lattice furnishes such a framework. Unfortunately, there are several barriers to a lattice formulation of supersymmetric theories.

Firstly, supersymmetry is a spacetime symmetry which is explicitly broken by the discretization procedure. Furthermore, unlike the usual Poincare symmetry, lattice discretizations of supersymmetry generically leave no subgroup of supersymmetry unbroken. This is quite different from Poincare symmetry where the lattice leaves intact a subgroup of discrete rotations and translations. In the latter case the existence of this remaining symmetry is sufficient to prohibit the appearance of relevant operators in the long wavelength effective action which violate the full  symmetry. This leads to a restoration of the full invariance without fine tuning  in the continuum limit. Since generic latticizations of supersymmetric theories do not have this property their effective actions typically contain supersymmetry breaking counter terms. Thus to achieve a supersymmetric continuum limit will then require tuning the parameters of the lattice action to eliminate
these symmetry breaking counter terms - typically a very difficult proposition.

Secondly, supersymmetric theories necessarily involve fermionic fields which suffer from so-called doubling problems when we attempt to define them on the lattice. Most methods of eliminating the doubled modes serve to break supersymmetry also.

Until recently these problems have prevented the lattice formulation of even a single, interacting supersymmetric field theory. However, we have recently found an example of a theory which does admit a lattice formulation which preserves an appropriate subgroup of the continuum supersymmetry - the N=2, two-dimensional Wess Zumino model. Furthermore the formulation explicitly eliminates the unwanted doubled fermion modes in a way compatible with supersymmetry. The presence of this lattice supersymmetry ensures that the energy of the vacuum is identically zero and the boson spectrum is degenerate with the fermion spectrum for any  lattice spacing. It also guarantees that the full N=2 symmetry is restored automatically as the lattice spacing is taken to zero.

From a computational point of view supersymmetric theories, even when formulated in this way, are very demanding. The presence of fermionic (anticommuting) fields leads to a highly non-local effective bosonic theory. A great deal of work has been done on algorithms to simulate such models and current approaches utilize the Hybrid Monte Carlo method. In this algorithm each of the system variables is paired with a momentum and subsequently evolved according to an auxiliary Hamiltonian composed of the original action plus kinetic terms. Ergodicity is ensured by frequently restarting the classical trajectory with new momenta drawn from a gaussian distribution. Integration errors are eliminated by subjecting an entire trajectory to a metropolis test before refreshing the momenta. In this way the amount of computation required to generate a new configuration can be made to scale approximately linearly with system volume. Furthermore the bulk of the computation time is taken up with the solution of a sparse linear matrix problem for which very efficient iterative algorithms exist.

Our numerical studies have so far concentrated on a (0+1)-dimensional example -- supersymmetric quantum mechanics. Using a Fourier accelerated HMC algorithm we were able to generate one million trajectories on lattices as large as L=256. On a 300 MHz pentium II machine this takes about 7 days. We are now conducting numerical studies of the N=2 two-dimensional Wess Zumino model.

In addition we would like to see whether some of the arguments leading to an exact lattice supersymmetry can be generalized to gauge models - principally super Yang Mills theories possibly coupled to matter. Even if the theoretical difficulties inherent in such a procedure could be surmounted we would clearly need a very substantial amount of computing power to carry out such studies.