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Calling Time on Digital Clocks?

David Sloan

University of Cambridge

The measurement of time has always been performed using physical devices - Clocks - be they artificial (such as a pendulum based clock) or natural (such as celestial observations). Since we are interested in repeated intervals, physical clocks have involved two parts - a cyclic portion whose interval is used to determine a fixed ”tick”, defined to be equal measurements of time interval, and a digital portion counting the number of ”ticks” passing. Throughout this process, the clock is thought to measure the passage of time, a continuous parameter.

In modern quantum mechanical treatments of systems, it is often found that variables that are continuous at the classical level become discrete, a prime example being the energy levels of a hydrogen atom. Indeed in some cosmological treatments such as Loop Quantum Cosmology, some of the fundamental parameters describing the evolution of the universe can be thought to live on a lattice-like structure within the real line, called a superselection sector. If this behavior holds within the full theory, we will therefore only have access to ”digital” clocks - the continuous, cyclic portion of the physical system is no-longer relevant. This effect which can result from treating polymer quantization as fundamental for all of physics, or at least for all clocks, leads to a change in the understanding of temporal ordering. In a world which has continuous time, there is a freedom to reparametrize a given time variable, with the only stipulation being that the reparametrization is monotonic in the given time variable. The reparametrization may also be used to map between solutions of the theory under, for example, time reversal.

A digital time structure can be overlaid on to a continuous structure, leading to a series of ”snapshots” of the configuration. This can be done in two distinct ways: The first is to take the approach used in the production of video. Here one leaves the shutter of ones camera open for the interval represented by equal clock values, leading to a smearing of configuration variables across their prior values during the prescribed time interval. In the second approach, one views instantaneous configurations (as described by the underlying continuum time). One can then play the following game. Suppose that the snapshots are shuffled. Can we reconstruct the correct time ordering? I will show that in the case of smeared configurations this is indeed possible, however in the case of instantaneous config- urations, for certain systems every possible ordering of the snapshots is a valid solution to the equations of motion.

If one is to take the position that digital time is indeed fundamental, one is forced to take more seriously the ”instantaneous” picture of temporal configurations. I will discuss how in the case of finite systems this leads to a requirement that sets of solutions be identified as indistinguishable, and in order to reconstruct a meaningful evolution new principles must be introduced.