Recently, a mathematical subject has appeared under the name quantum groups. An interesting question is whether those are relevant for quantum field theories. In this regard, some progress has been made in that quantum group extensions of the Poincare group have been found. Closely related to quantum groups are the quantum plane and its higher-dimensional analog, quantum hyperspaces. An interesting arena for future research is to define field theories invariant under q-deformed Poincare transformations.
The concept of a group lattice has been developed by one of us. There are many different kinds of group lattices, and applications seem particularly promising. For example, the molecule C_60, also known as buckminsterfullerene, is an example of a group lattice. Group-lattice methods, combined with experimental information, have lead to an accurate determination of the electronic energy levels of C_60.
Another interesting group lattice is the twisted two-dimensional group lattice. It is closely related to the quantum plane. Unlike the case of quantum groups, no difficulty exists in constructing field theories on group lattices. Furthermore, if the action is quadratic then the theory is solvable. This means that interacting field theories can be defined and treated perturbatively. Since the discreteness of the group lattice renders the system ultraviolet finite, these theories are of interest as possibilities for finite quantum versions of gravity. Twisted group lattices offer a straightforward, competitive, yet mathematically similar approach to quantum hyperspaces.
Roughly speaking, group lattices are more regular than random lattices but more random than regular lattices. Enough structure is preserved so as to maintain solvability. The structure is determined by a group G. There is a one-to-one corresondence between group elements and lattice sites. Nearest neighbor sites are determined by a subset of G denoted by NN.
A precise definition is as follows. Starting with G, one associates a site with each element in G. The origin is the identity element, e. Let NN be a subset of elements of G with NN^(-1)=NN. A site g' is a nearest neighbor site to g if g'g^(-1) is in NN. In other words, the nearest neighbors to g are hg for h in NN. In summary, a group lattice is specified by G and NN. The concept of a group lattice is quite general.
Twisted group lattices are special group lattices based on semidirect products of cyclic groups. The two-dimensional twisted group lattice begins with a square two-dimensional lattice and ``adds a certain twist to it''. The lattice is finite and one has the analog of periodic boundary conditions. Group lattices can be associated with non-back-tracking walks on regular lattices in which certain paths are identified. The group is defined by the relations, xy=zyx, zy=yz, zx=xz and z^N=e. On a two-dimensional square lattice, associate x with taking a step to the right, let y represent taking a step upward and let the inverses of x and y be steps in the opposite direction. Consider a path which goes around a plaquette. It corresponds to the commutator [x,y]=xyx^(-1)y^(-1). Hence, for the twisted group lattice, one does not return to the identity element but instead arrives at an N-th root of it. The integer N controls the amount of ``twisting'' in the lattice. More generally, a closed path starting at the origin does not return to the origin, e, unless the area enclosed by the path is 0 mod N. This introduces a kind of ``holonomy'' structure to the lattice.
The partition function for a quadratic field theory on a two-dimensional twisted group lattice factorizes into a product of partition functions. The quantum plane is defined as the free algebra generated by two elements, x and y, modulo the relation xy=qyx, where q is a complex number. When q=exp(2 pi i k/N), the quantum plane becomes the k sector of the twisted group lattice in the infinite size limit. This shows how one can use twisted group lattices to define field theories associated with quantum groups. The three-dimensional version of the twisted group lattice has already been solved.
The research program is as follows. The first step is to solve the four-dimensional twisted group lattice. If one knows the irreducible representations of G, then this is straightforward. However, it is not clear whether one can find these representations. Fortunately, there exists an alternative method, based on the construction of a ten-dimensional lattice, by which the partition function and propagators can be obtained.
The second step is to define field theories on the four-dimensional twisted lattice. This again is straightforward. At this stage there are two directions in which one can proceed. The first possibility is to perform a factorization of the partition function to obtain the corresponding field theory on a quantum four-space. The second direction is to consider different field theories on the four-dimensional twisted group lattices to determine whether any of these have applications in particle physics or gravity.
Finally, there is the mathematical challenge to generalize all of the above to arbitrary dimensions.
Olaf Lechtenfeld and Stuart Samuel:
The solution of the d-dimensional twisted group lattices
hep-th/9412197 (Int. J. Mod. Phys. A11 (1996) 3227)
Olaf Lechtenfeld and Stuart Samuel:
Four-dimensional twisted group lattices
hep-th/9411001 (Nucl. Phys. B449 (1995) 406)