文档介绍：Nuclear Physics B342 (1990) 471—485
North-Holland
QUANTIZATION OF DISCRETE DETERMINISTIC THEORIES
BY HILBERT SPACE EXTENSION
Gerard ‘t HOOFT
Institute for Theoretical Physics, Princetonplein 5, . Box 80006,
3508 TA Utrecht, herlands
Received 30 April 1990
Quantization of a theory usually implies that it is being replaced by a physically different
system. In this paper it is pointed out that ifa deterministic theory pletely discrete, such as
a classical gauge theory on a lattice, with discrete gauge group, then there is an essentially trivial
procedure to quantize it. The equations for the evolution of the physical variables are kept
unchanged, but are reformulated in terms of the evolution of vectors in a Hilbert space. This
transition turns a system into a conventional quantum theory, which may have more symmetries
than can be seen in the original classical theory. This is illustrated in a cellular automaton, of
which only the quantum version is time-reversal symmetric. Another automaton shows self-dual-
ity only after Hilbert space extension.
We discuss the importance of such observations for physics. The procedure can also be used
to construct pletely finite and soluble quantum gravity model in 2 + I dimensions.
1. Introduction
Investigation of the problem of quantizing black holes has led to the suspicion
that space-time at the Planck scale, and all relevant physical variables there, are
discrete. The finiteness of the entropy of a black hole implies that the number of
bits of information that can be stored there is finite and determined by the area of
its horizon [1]. This gave us the idea that Nature at the Planck scale is an
information processing machine like puter, or more precisely, a cellular
automaton [2].
Since Quantum Mechanics dominates our present view of the known laws of
Nature it is natural to think of such a discrete theory as a discrete quantum theory.
In some sense all quantum theories are dis