Introduction

We have now reached the point where we can start discussing classical and quantum discrete time (DT) field theories. Such theories are fundamentally different in character from the theories we have discussed so far, in the following specific ways.

1. (i) We can no longer maintain a particulate view of the world, by which we mean that we can no longer view matter as consisting solely of localized point-like particles running along worldlines in spacetime.

2. (ii) We now have to develop techniques for dealing with mechanical systems involving continuously many degrees of freedom.

3. (iii) Some degrees of freedom, such as quark fields, might not correspond to directly observable degrees of freedom.

4. (iv) Special relativity emphasizes certain spacetime symmetries between inertial frames, but simultaneity is not one of those symmetries. The problem we face in DT mechanics is that the formalism explicitly breaks Lorentz symmetry, raising the question of the compatibility of DT field theory with relativity.

The first three points in this list are encountered in continuous time (CT) field theories but the fourth is specific to our subject: temporal discretization is done in our approach in a given inertial frame, the preferred frame, and this breaks Lorentz covariance explicitly. We shall address this fundamental point later in this book. We shall develop the equations of DT classical and quantum field theory (QFT) as seen from the preferred frame in this and the next few chapters.