 # Discrete time

There can be no direct proof that time is either continuous or discontinuous. The nature of time is contextual, that is, how it is looked at. There are advantages and disadvantages in any mathematical image of time.

Continuous time has had a very successful history by any measure of the word success. However, concerns can be raised at its continued exclusive use in fundamental theory. We need only look at the concept of temperature to see the dangers of keeping to a traditional intuitive perspective in physics. The concept of temperature is a remarkably useful one in many fields but it is now understood as an effective concept that can break down, such as for SUOs not in thermal equilibrium. The continuum based notions of classical thermodynamics have been replaced successfully by quantum statistical physics, which is based on discrete principles.

With this in mind, a number of theorists have explored the notion that time is discrete, either viewing discrete time as a numerical approximation approach to continuous time [Bender & Strong, 1985] or else as an intrinsic concept in its own right [Caldirola, 1978]. A review is given in [Jaroszkiewicz, 2014].

The mathematics of discrete time is particularly rich in phenomena, richer than for continuous time. The reason is that there is a hierarchy of assumptions that leads to traditional Newtonian mechanics: with each step up in the hierarchy, there is a new constraint on what the mathematics can do for us, so it does less. Let us look at this hierarchy from the ground and work our way up.

In the first instance, we may model time simply as a finite index set, such as {a, b, c}. Then different events can be given different times, or indices, such as events Ea, Eb, Ec. No ordering need be assumed, the temporal index serving merely to implement Cumming’s view that ‘time is what keeps everything from happening at once’ [Cummings, 1922].

The next step would be to impose an ordering on the index set, such as a < b < c. Then Eb is later than Ea and earlier than Ec. A first-order discrete time mechanics would be a rule that allowed an observer to use a knowledge of Ea to determine what Eb should be, and then use Eb to determine what Ec should be. Essentially, we would have Eb = Uba(Ea), where Uba is some evolution function that takes us from the set of possible events at time a to the set of possible events at time b. This process could then continue, in the form Ec = Ucb(Eb), where Ub need not have any relationship with Uba: they are in principle completely different functions, with the domain of Ub being the range of Uba. Discrete time mechanics was presented in this way by Maeda and collaborators [Ikeda & Maeda, 1978; Maeda, 1980, 1981].

We may bypass time b in this approach by writing Ec = Ub о Uba(Ea), where the symbol о denotes the composition of two functions. In general, we would have to ensure that domains and ranges of functions in a composition make it meaningful. This is usually not an issue because domains and ranges in applications to space-times invariably assume that all domains and ranges are copies of the same domain. This need not be the case however, a prospect that arises when processes of observation are looked at carefully [Jaroszkiewicz, 2010].

A second-order discrete time mechanics would require a knowledge of Ea and, independently, a knowledge of Eb in order to work out Ec, that is, a rule of the form Ec = Uba(Eb, Ea). When Newtonian continuous time mechanics is discretized in time in some chosen way, it turns out that the resulting discrete time mechanics usually takes on this second-order form.

There is no natural way of temporal discretization, however, for the very good reason that discretization represents a step back down the mathematical hierarchy, with a consequent increase in generality.

If we have a discrete-time mechanics based on integer time, then we can go up the hierarchy and move to (say) rational time, that is, take the time index parameter to be a rational number. Now we have a dense set of states to play with, but there would still be the ‘holes’ where the irrationals were. Moreover, we would not have a linear continuum. The next step in the hierarchy would be to include the irrationals, so that finally our time would be a continuum. Even so, the dynamics we were using could be different to standard Newtonian mechanics. We could have continuous time but dynamical variables that were not differentiable functions of time, as in the case of Brownian motion, or even discontinuous functions of time. Newtonian mechanics generally supposes that dynamical variables are at least twice-differentiable functions of time except possibly at a finite number of times where impulses occur. 