The discrete-time theory

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_nf_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_nT_n-S_nV². 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	^An	A⁰
Physical fields	= ^VX ^An	En = -VA^_n - TnA_n
Charge fields	^jn	fn = ^pn
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_np_n + V • j_n = 0.

The fields satisfy the second-order DT equations

where A_n = T_nA°_n + S_nV -A_n. A DT Lorentz gauge is one where the gauge function X_n satisfies the relation D_nx„ = -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.