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Do we have the right mathematical tools to quantize space-time?

TL;DR: Quantum mechanics quantized the count of configurations per degree of freedom. To quantize space-time we need to quantize the number of degrees of freedom. We have no tools for that.

If you follow our project, you may know that, while we work on the foundations of physics, our primary goal is not to produce a theory of everything, a theory of quantum gravity, that merges quantum mechanics with general relativity. Our focus is to rederive the current theories, the ones we already have, from clear physical assumptions, while understanding exactly what each mathematical symbol represents physically. Therefore, while we do not work on quantum gravity specifically, we are qualified to answer the following question: what mathematical structures are suitable for quantum gravity? And the answer is: not the current ones.

We could show how various mathematical structures fail to apply to those regimes, but this would only be interesting at a technical level. It is more interesting to understand what happens at a more conceptual level. If we want to merge quantum mechanics and general relativity, for example, we really want to understand two things: what exactly is quantization? What exactly is the geometry of space-time? Only then would we really understand what it would mean to quantize space-time. To answer these question, we are going to concentrate on something seemingly simple: counting.

If you studied statistical mechanics, you know that entropy \(S\) is \(S = \log W\), the logarithm of the count of states \(W\). Technically, this is the highest information entropy achievable by a distribution over a region \(V\) that contains \(W\) states. But how do we count states? If we are in a discrete space, the count is just the number of states \(W=\#V\). Mathematically, this is the cardinality of the set. But what about the continuum? What about if states are identified by position and momentum?

In that case, cardinality doesn’t work. Suppose we take a finite volume \(V=\Delta x \Delta p\) of phase space. It contains infinitely many points, infinitely many states. If we double the volume, we still have infinitely many states. Therefore, \(W(2V) = W(V) = \infty\). And this makes no sense: if the volume increases there should be more states. Physically, the correct way to count states is to use the volume in phase space measured in units of position times momentum. Mathematically, what we have to use is called a measure. This measure is additive in the sense that if we have two non-overlapping regions \(U\) and \(V\), \(W(U+V) = W(U) + W(V)\). However, this presents a problem.

Suppose we start with a finite region \(V\) and we cut it in half. Each side will have half the number of states. Suppose we repeat the process over and over. At some point the count of states will be less than one: the region will have less than one state. This does not make much sense. If we look at the entropy \(S = \log W\), this would be negative because the logarithm of one is zero, and the logarithm of a number less than one is less than zero. Negative entropy also does not make much sense. Therefore, requiring the state count to be additive is somewhat problematic. It works fine when the regions are big, but not when the regions are small.

This is the problem that quantum mechanics solves. Quantization is imposing a lower limit of one to the count of states on the continuum. You can’t have less than one state. You can’t have less than zero entropy.

If you take a set of quantum states \(V\), you can ask what the maximum entropy achievable with a distribution over the region \(V\) is. What you find is that the entropy has a lower bound to zero, because the entropy of a single quantum state is zero. Using \(S = \log W\) in reverse, each state counts as one. You will also find that the count of states \(W\) is not additive, but it is additive for orthogonal subspaces, partially recovering classical behavior in particular circumstances.

So quantization is really setting a lower limit of one to the count of states, or better, the count of joint configurations of position and momentum for a given degree of freedom. What about the geometry of space-time?

Typically, people see the geometry of space-time as characterizing distances and angles. However, general relativity is not really a theory about objects whose state is characterized by positions in space. General relativity is a field theory, so the state of the system is characterized by the configuration of all the fields at all points. Therefore we should be asking two questions: how do we count the configurations of the fields? How does that counting relate to the geometry of space-time?

The value of each field (and its conjugate) form an independent degree of freedom. For example, the value of the electromagnetic field at one point, together with its vector potential at the same point, form an independent degree of freedom because, in principle, knowing the value of a field at one point tells us nothing about the value of the same field at a different point. If we have a volume of space \(V\), then, there are many independent degrees of freedom… but how many?

Again, if we just count the points, we would say infinite, but we would have the same problem: double the volume, you have the same number of degrees of freedom, which does not make sense. Again, we can use a measure, so that if we double the volume, we have twice as many degrees of freedom. This is what happens in general relativity: the spatial volume essentially counts the number of independent degrees of freedom. My conjecture is that that’s the only thing general relativity does. Space-time is flat when doubling the space in any direction doubles the number of independent degrees of freedom. Space-time is curved when that is not the case. It is a conjecture in the sense that it makes sense, but I am not yet able to prove it. Regardless, since we need to be able to define entropy, and the entropy has to be finite in a finite volume, the count of degrees of freedom has to be finite and, in a field theory, this count is additive. This is also true in any quantum field theory.

Ok, we use a measure to count the number of independent degrees of freedom for both classical and quantum field theories. And you should be able, now, to see the problem: if we keep dividing volumes of space, we will end up with a volume that has less than one independent degree of freedom. Which does not make sense. We should expect that, at some point, the region of space contains one independent degree of freedom and no less. The count of independent degrees of freedom must be quantized. This is, to me, what Plank’s scale is.

To recap, in a classical field theory, both the count of degrees of freedom and configurations per degree of freedom are additive; in a quantum field theory the count of configurations per degree of freedom is quantized, but the count of degrees of freedom is still additive. However, to be precise, we need to quantize both counts.

So, how do we quantize both? It does not seem to me that we have the right math do that. I say that because we haven’t even found the right math to describe the quantized measure in simple quantum mechanics, and we had to start developing our own. Clearly, quantum mechanics does it, but exactly how it does it and whether it is the only way to do it is an open question. On top of that, the lack of additivity for the count of independent degrees of freedom means that, at a fine scale, they are not independent. That is, the total number of states is no longer the product of the configurations of each degree of freedom times the number of degrees of freedom. This is not something that the current formalism of quantum mechanics can handle. I suspect that the leap needed to handle this case is as big as the leap from classical to quantum mechanics.

I hope you see both the simplicity and the necessity of this argument. Regardless of what a theory says, it must at least allow us to count configurations and degrees of freedom. Having fewer than one configuration or fewer than one degree of freedom does not make sense. We need the proper math to count and, suprisingly enough, we do not have it.