Susanne Schander, Perimeter Institute
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By Jorge Pullin, LSU
Geometrodynamics is the name John Archibald Wheeler gave to the description of space-time completely in terms of geometry and its eventual quantization. The description of space-time is in terms of a metric of space that evolves in time.
An approach to quantization that has been successful for the kind of theories that describe particle physics, like chromodynamics -which describes the strong interactions inside nuclei-, is the use of lattices. In it, one approximates the differential equations of the theory by finite differences. This has two upshots. On the one hand, infinities that tend to arise associated with the differential equations are eliminated. On the other hand, the resulting equations are amenable to be solved on a computer. The resulting approach is known as lattice gauge theory. Its application to the theory of strong interactions, lattice quantum chromodynamics, allows for instance to compute the mass of the proton.
Since the gauge theories of particle physics are typically represented in terms of a vectors like the potentials that appears in electromagnetism, attempts to apply lattice techniques to gravity have usually started from formulations of the theories in terms of potentials. The formulation used to set up loop quantum gravity would be an example. In this talk the use of lattices was explored with the traditional formulation of gravity used in geometrodynamics. Among the issues discussed was how to keep the metric of space yielding positive distances in the quantum theory. Moreover, a method to represent the symmetries of the theory on the lattice was given. Also the issue of the continuum limit, that is, how to retrieve from the discrete theory the continuum behavior we observe in space-time at large scales was addressed.