*An exploration of the quantum conundrum of the Wheeler-DeWitt Equation*

**By Nicholas Folidis**

Time. It rules our lives and we all experience it on a daily basis. A moment in time often goes so fast that we almost miss it but every now and then it can feel as an eternity.

Time has been a matter of debate and fascination among scientists for years.

Albert Einstein believed that time is a relative concept and is not as constant as everybody thinks. According to him, the notions of past, present and future are nothing more than a stubborn illusion. Einstein fused time and space together to create a four-dimensional continuum called spacetime, in which every event that has happened or will ever happen, since the beginning of the Universe, already exists and as a result we are all travelling through time and space and experiencing all these different events as they occur. Therefore, reality and the flow of time is a result of all the individual moments, an array of “nows”, we live in as we process them from our subjective viewpoint within spacetime. But what is time and, for that matter, what is reality?

There are two contradicting theories to explain reality. Quantum mechanics describes the world of subatomic particles and all small-scale events whilst general relativity describes the whole Universe and all cosmological events. Physicists, including Einstein, have tried for years to combine these two theories together to create a so called “theory of everything”; a unified field theory that would explain all small and large scale events within our Universe.

In the mid-1960s, John Wheeler and Bryce DeWitt took a step towards creating a quantum gravity theory by successfully, mathematically describing the previously uncombined ideas of gravity and quantum mechanics in one, quite complicated, field equation that we now call the Wheeler-DeWitt equation.

In their effort to solve one problem, Wheeler and DeWitt introduced an even bigger one, the so-called ‘problem of time’. In their field equation, the quantity of time simply does not exist. In effect, the Wheeler-DeWitt equation tells us that nothing ever happens in the Universe. The Universe itself is determined by a set of laws that are absolute and do not change with time. Put simply, the quantum state of the Universe is forever frozen. That mind-boggling prediction is at odds with every observational evidence we have ever collected. Many scientists disagree with the mathematical computation of the cosmos that the Wheeler-DeWitt equation provides, but nobody was able to prove it wrong.

That changed in 1983 when theoretical physicists Don Page and William Wootters came up with a solution on the quantum conundrum of time. Their novel proposal is based on the quantum entanglement phenomenon. According to quantum entanglement, it is possible for two particles to exist in a way that the quantum state of one particle cannot be described independently of the state of the other, even when they are spatially separated. Page and Wootters showed mathematically that an entangled pair of particles evolve in a way that can serve as a clock, since measuring the properties of one will change those of the other particle. This change gives the illusion of time.

Hypothetically if a clock, entirely independent of our Universe could exist, an external ‘god-like’ observer would not be able to observe any change. For them the Universe would appear completely static and unchanging, just as the Wheeler-DeWitt equation predicted. Time then, according to that idea, is an emergent phenomenon that exists due to the quantum entanglement and can only exist for observers within our Universe.

Marco Genovese’s team in Turin, Italy, performed an experiment in 2013 that supported Page and Wootters’ theory. The team created a toy Universe consisting of just a pair of entangled, polarised photons. In short, the observer can measure the evolution of the system in one of two ways. The first way is by treating photon A as an internal clock –due to its ability to alternate between horizontal and vertical polarisation– and trying to measure its difference in polarisation, thereby becoming entangled with it, and compare its polarity with that of photon B. The second way is by acting as an external observer, using a clock completely independent of the two photons, and comparing their properties as a whole against that external clock.

As Page and Wootters predicted, in the first instance, the polarisation of photon B varies with time, while in the second instance time does not emerge, as there is no observable change in the system.

The Page-Wootters mechanism proves that the Wheeler-DeWitt ‘problem of time’ goes away when viewed through the lens of entanglement and that quantum mechanics and general relativity are not that incompatible after all, giving hope for a unified theory. Scaling up the experiment and trying to prove the idea at a macroscopic scale would be the next challenge we have to tackle.

Only time will tell…

*From Issue 16*

## Comments