Models in Two Space–Time Dimensions Without Interactions
Gerard ’t Hooft · Fundamental theories of physics · 2016
Sooner or later, we shall have to contemplate multi-particle systems and quantized field theories. These naturally arise whenever quantum mechanics confronts Einstein’s Special Relativity. The prototype is the theory of bosons in one space-, one time dimension. First, we consider the massless, non-interacting case. This allows for a mapping onto a deterministic system: the (continuum limit of a) cellular automaton. Now, both classically and quantum mechanically, these models are trivial and featureless, so we have to ask how mass terms and interactions are to be introduced, and how to go from 1 to 3 space-like dimensions (if not more). This is where the mathematics becomes hard, but there is something else we can do: (super-)string theory is based on a two-dimensional world: the string world sheet. At first sight, our results may seem to be spectacular: strings are classical automata living on a lattice in target space, a $3+1$ dimensional lattice with lattice links of length $2\pi\sqrt{\alpha'}$ . As yet, however, this only seems to apply to the bulk behavior of strings, not their interaction mechanism, which, in this formalism, is quite complex. Knowing that strings also generate the gravitational force, this should not have come as a surprise.