We now show how we can rewrite the above CT equations in DT terms. All these DT equations can be derived from first principles, that is, from the DT calculus of variations [Jaroszkiewicz, 2014].

The important DT operator is the temporal displacement operator U_{n}: for any function f_{n} indexed by discrete time n, we have Uf_{n} = f_{n+1}. The inverse temporal operator is denoted U_{n}, which has action U_{n}f_{n} = f_{n-1}. With these operators we define the operators T_{n} = T^{—} (U_{n} - 1), T_{n} = T^{—} (1 - U_{n}), S_{n} = 6^{-1} (U_{n} + 4 + U_{n}), and D_{n} = T_{n}T_{n}-S_{n}V^{2}. Here T is the chronon, or shortest interval of discrete time. In the limit T ^ 0, we have the following effective limits

Temporal discretization naturally involves nodes and links. The former are temporal points of zero duration whilst the latter are the temporal intervals between those points. In our discretization scheme, the links have temporal duration T. Here we show what happens for fixed chronon duration: it is possible to develop DT mechanics based on variable chronon duration, something considered by Lee in his DT path integral approach [Lee, 1983].

Some DT fields are associated with temporal nodes whilst others are naturally associated with the temporal links. Table 23.1 shows the associations we have used in our approach to DT Maxwell electrodynamics [Jaroszkiewicz, 2014].

Table 23.1 The association of the various fields with nodes or links.

Node n

Link n

Potential fields

^{A}n

A^{0}

Physical fields

= ^{V}X ^{A}n

En = -VA^_{n} - TnA_{n}

Charge fields

^{j}n

fn = ^{p}n

Gauge fields

Xn

Note that the operator T_{n} converts a node field into a link field and vice-versa, whilst v leaves the association unchanged.

The physical fields E_{n}, B_{n}, = (p_{n}, j_{n}) are invariant to the DT gauge

transformation

The DT equivalent of Maxwell’s equations are then

which give the DT continuity equation T_{n}p_{n} + V • j_{n} = 0.

The fields satisfy the second-order DT equations

where A_{n} = T_{n}A°_{n} + S_{n}V -A_{n}. A DT Lorentz gauge is one where the gauge function X_{n} satisfies the relation D_{n}x„ = -A_{n}. Then in such a gauge, A'_{n} = 0 and we have the equations for the transformed potentials:

All of these DT equations collapse back to the CT Maxwell’s equations in the limit of the chronon T going to zero.