Finding Time For Evolution

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Time is a concept that is taken for granted in most areas of human life. It is treated casually without much thought, and we only seem to run into trouble when trying to take the concept apart to see what exactly makes it tick.

But tearing apart concepts is what theoretical physicists do best, and our failure to explain the concept of time remains one of the outstanding issues in modern-day physics. Although somewhat of a philosophical quandary, this ignorance affects many different areas of investigation and has come to be known as the problem of time.

The problem, as such, is usually separated into three main concerns. However, these are, in our opinion, all manifestations of the same basic issue. Most physical theories typically incorporate time as an abstract dimension or external parameter while ignoring the emergent experience of our passage through time. This immediately leads us to one aspect of the usually stated problem; time evolution is universally restricted to a single direction, past to future. Meanwhile, time-reversal invariance is rampant in physical theories, particularly those presumed to be of a fundamental nature. One notable exception is thermodynamics, which picks out a time direction by virtue of the second law of thermodynamics.

A second aspect of the three-pronged problem is the lack of consensus over the treatment of time in different physical theories. The main question here is how to settle upon a common ground between the absolute time in Newtonian and quantum physics and the ambiguous, dynamical time of general relativity.

The last of the three concerns is the apparent timelessness of the Universe. This was brought to the forefront by Wheeler and DeWitt in 1967 and is nicely illustrated by their self-named equation:

Ĥ | ψ ⟩ = 0 ,

where |ψ⟩ describes the quantum state of the universe and Ĥ is a quantum Hamiltonian which happens to have a pedigree in general relativity. The zero on the right is a consequence of the Universe having no net energy, but this Schrödinger-like equation is really highlighting the fact that isolated systems must naturally conserve energy and freeze the dynamics as a result. The implication is the Universe, as viewed by a hypothetical exterior observer, must be in a timeless state. This is in direct contradiction to our own interior vantage point and leaves us with the challenge of reconciling these two pictures.

This strange state of affairs was taken up by Page and Wootters in 1983. Their method, which is sometimes known as the Conditional Probability Interpretation (CPI), restores time to an otherwise timeless universe by separating it into two subsystems: a clock and the remainder. Given that the two subsystems are strongly entangled (in the quantum sense), an eigenvalue of the clock system, such as its center-of-mass position, can be viewed as a time variable for the rest of the Universe. Although the subject of some controversy, the CPI approach provides an obvious resolution to the timeless Universe paradox and, perhaps, the potential for addressing the other two aspects of the problem (arrow of time and disagreement between theories).

In Search For A Realistic Clock For The CPI Framework

In an attempt to drive the conversation forward, we have recently investigated the type of clock systems that could be used in the CPI description of evolution. According to the CPI strategy, the method requires a good clock. Along with the requisite entanglement with the rest of the Universe, such a clock is defined as a weakly interacting system with many distinguishable states; basic features of any standard measuring device. In the relevant literature, the ante is usually raised from a good clock to an ideal one; meaning a clock that experiences no interactions whatsoever with the other subsystem. Our initial concern was that the ideal case is not at all realistic; all systems within the Universe are necessarily subject to some interactions.

With this setup in mind, we looked for a realistic clock that could be tested within the CPI framework, building upon an avenue of investigation that was initiated by Dolby in 2004. To help simplify the problem, we made the harmless assumption that the clock system would be small enough for the effect of the interactions on the other subsystem to be neglected. We settled on a semi-classical clock for our first investigation: the coherent-state description of a damped harmonic oscillator. The central idea was to identify the optimal clock, which minimizes uncertainty in the time measurements subject to the constraint that the damping is suitably weak as per the aforementioned assumption.

Our initial expectation was that an optimal clock would indeed be ideal (as any decent thesaurus would have one believe). However, it seems that a Pandora’s box was opened with the inclusion of interaction effects. In an unexpected twist, we found that taking our description to the ideal limit of no interactions actually maximized the uncertainty, which can be shown to decrease monotonically with damping strength. Ensuring that the interactions are suitably weak, one finds a compromise that sets the damping factor roughly equal to the inverse of the desired running time, an obviously finite-valued quantity. In other words, the usual choice of a clock in the CPI framework may be ideal but it fails to be realistic and not even in a limiting sense. The complete analysis and a detailed discussion can be found in our recently published article, “Realistic Clocks for a Universe without Time.”

We are now attempting to understand this outcome from a physical perspective and then connect it to the other aspects of the problem of time. Our first order of business is to see what transpires when the clock is a fully-quantum system and not just semi-classical. While it might be argued that our conclusions for damped clocks can be avoided by allowing for an infinite runtime, as this removes all damping and regains the ideal limit, such a counter-argument appears to fall apart when the clock is indeed quantum. In light of these new investigations, we also have good reason to believe that the CPI description of time requires interactions as a strict matter of principle and not only in practice. This new perspective on clocks could end up resolving at least some of the issues at hand, but only time will tell.

These findings are described in the article entitled “Realistic Clocks for a Universe Without Time,” recently published in the journal Foundations of Physics. This work was conducted by K. L. H. Bryan and A. J. M. Medved from Rhodes University and the National Institute for Theoretical Physics.

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