Field Theory of Galois' Fields
Y. Nambu · World Scientific series in 20th century physics · 1995
The motivation for the present work comes from various sources which, however, need not be elaborated on here. I will be exploring a class of quantum field theories defined over finite sets of integers. Essentially these are the familiar Z„ lattice theories, but carried to their logical extremes. Generally speaking, a set of integers modulo m constitutes a residue system ring Zm of characteristic m (i.e., ma = 0 for any a), which is closed under addition and multiplication, but not necessarily admitting division. Actually I will introduce three kinds of Zm's for three different physical quantities. Thus the lattice is taken to be periodic, and its coordinates x take values in Zi, where I may be different for each space dimension. However, the length of the time dimension will be left open. Next, the field Fat each lattice site takes values in another set Zk, as does the action functional which depends on F. Finally, the partition function W (quantum or statistical) belongs to the third set Zh, for reasons that will become clear in a moment. At any rate, one is thus dealing with mappings from Z, to Zk to Zr,. Although these parameters are arbitrary integers, the case of prime numbers has a special significance. For then the sets become finite (Galois) fields which admit, like the ordinary numbers that appear in physics, all the