What does it mean that 'time' does not exist in quantum gravity theory?

Physicists sometimes say that 'time does not exist.' This is often in the context of quantum gravity theory, which aims to merge quantum mechanics and general relativity. Let's explain what that means.

First, let's consider special relativity, which describes the properties of classical flat spacetime. In Figure 1, the horizontal axis is the position coordinate, and the vertical axis is the time coordinate. In relativity, the vertical axis is represented by multiplying time t by the speed of light c. The blue curve in the figure is the world line of a particle. In this classical spacetime, time certainly exists, and it is an object that can be measured using various clocks.

Time t also appears, of course, in quantum mechanics regarding this particle. In the Schrödinger equation, which is its fundamental equation, it appears as a parameter of the state vector representing the particle state. To describe the time evolution of that state, a time derivative term is also required on the left side of this equation.

However, because quantum gravity theory is a theory of quantum spacetime (generally curved spacetime), we can consider a quantum superposition of spacetimes with different shapes. Within each of the superposed spacetimes, there is a specific time axis or time coordinate, but it may turn out that each one is different from the time axis or time coordinate system of the other superposed spacetimes. This is because since the shape of each spacetime is different, the way time flows within that spacetime also differs. Clocks that measure time also move differently within each spacetime. In such a situation, the overall quantum state where spacetimes are superposed cannot select a flow of time in a specific spacetime and use it as an external time.

So, what happens to the Schrödinger equation for the quantum state of superposed spacetimes? The fundamental equation of quantum gravity theory is as follows.

If we dare to write this like the Schrödinger equation, it looks like this.

This is called the Wheeler-DeWitt equation, and it is an equation for the wave function of spacetime, which also describes the superposition of spacetimes, that is, the wave function ψ of the quantum universe. That wave function includes the metric tensor h, which indicates the shape of the universe, as an argument (parameter), but it does not include time. Since the time t of any spacetime cannot appear, time does not appear in ψ. The time derivative term that was present in the original Schrödinger equation has also disappeared in the Wheeler-DeWitt equation. In other words, time has vanished from this theory. This is the content of 'time does not exist.'

But in a sense, this is natural. After all, we have superposed different spacetimes that have various flows of time. However, has physical time really disappeared as well? Many physicists do not think so.

If we add the degrees of freedom of a physical clock that measures time to the theory, couldn't we extract information about time? For example, it is often discussed to have the parameter a, which represents the size of the universe and is one of the degrees of freedom of h, act as a physical clock. We also add a matter field φ to the theory and discuss what values that field has. In that case, we interpret φ=φ(a) as 'the value of the field at time a.' Then, it becomes possible to discuss the arrow of time that appears due to the second law of thermodynamics in such a setting. It is a method of reading out physical time from the correlation, or relationship, of various physical quantities. In such methodologies and interpretations, it can be said that true time has not disappeared. However, it is not known whether a normal probability interpretation can be performed using this time. Among the superposed universes, there may be universes where the size of the universe itself turns to contraction midway. In that universe alone, time reversal would occur. Then, it seems doubtful whether probability conservation as a whole holds in the first place.

Physicists have various concepts of time. When they say 'time does not exist,' if you first ask 'which time are you referring to?', I think that the misunderstanding, rather than 'time,' will disappear.