In the following I would like to discuss this question: when we search for a fundamental theory, what relationship should there be between the quantum and the classical? Or put differently: which one is primary, the classical or the quantum?
The operational arrows of quantum vs. classical
Let me try to explain these questions by asking a simpler one: how do we obtain a quantum theory? Well, normally the answer to that question is that we use the procedure called quantisation, which is a word that lacks a precise definition but which roughly means that we take a classical theory and replace all variables with their quantum counterparts, i.e. we replace quantities such as position and momentum with the corresponding operators, or we replace classical fields with their quantum cousins, operator-valued distributions.
Once we have a quantum theory we must carry out various tests to check that the theory we have found meets our expectations. For instance, we must check whether various classical symmetries also hold in the quantum theory. If some of these tests fail the theory will be regarded as meaningless. If for instance the theory we are working with is a gauge theory — this is the mathematical framework that describes the fundamental interactions found in the standard model of particle physics — then an essential task is to ensure that the gauge symmetry is preserved at the quantum level. If the gauge symmetry is broken, if for instance there is an anomaly, then the quantum theory does not make physical sense.
Also, we must check whether the quantum theory has a classical limit, where the classical theory reemerges. A quantum theory must include the classical theory, what we experience in our everyday lives, as a limiting case. Quantum mechanics includes classical mechanics as a limiting case, quantum electro-dynamics includes classical electro-dynamics as a limiting case, the classical theory must be included in the quantum ditto.
This means that we have a set of operational arrows, which start in the classical and takes us to the quantum and then back again to the classical. We begin with a classical theory which we quantise to obtain a quantum theory and we then check that we can get back to the classical theory.
This is the case for essentially all the theories that we work with in high-energy physics. The structural input in the standard model of particle physics is essentially classical, the spinors, the gauge theory, all of it; the standard model is a quantum theory, which we obtain by quantising its classical counterpart.
This is also the case in loop quantum gravity, which is an attempt to quantise general relativity by using certain loop variables related to a formulation of Einsteins theory due to the physicist Abhay Ashtekar. The mathematical structure found in loop quantum gravity only makes sense because it structurally matches its classical counterpart. If we did not know Einsteins theory and were presented with the mathematical formalism of loop quantum gravity alone, i.e. if we only look at the quantum side of the aisles, then it makes very little sense. It is highly contrived. You would never think of this construction if it was not for the fact that its structure matches that of the classical theory of general relativity. This means that in the case of loop quantum gravity the operational arrows are solidly rooted in the classical realm.
The sames can be said of essentially all frameworks that I know of in theoretical high-energy physics. It can be said of string theory and it can be said of the non-commutative formulation of the standard model due to Ali Chamseddine and Alain Connes. In the latter case even more so since the spectral triple formulation, that Chamseddine and Connes uses, is simply classical, the quantum field theory part of the story is added in a secondary step and only applied to the standard model part — gravity is left out. So all the operational arrows in high-energy physics are rooted in the classical realm; the quantum realm is secondary.
Quantum or classical: which is primary?
But this raises the question which is primary, the quantum or the classical? According to the operational arrows, that I have just described, the classical must be primary. But that does not really make sense — surely, the quantum must be more fundamental than the classical? Or what?
We know that both realities exist. So which one of them is emergent from the other one? Is the classical a product of the quantum or the other way around?
Surely the answer must be that the quantum is primary. The classical world, that we experience, is an emergent proporty of the underlying quantum reality, that we encounter in the laboratory and in textbooks on quantum mechanics and quantum field theory. That must be the answer.
But if that is the case, then shouldn’t the operational arrows, that I have just discussed, in the case of a fundamental theory — a final theory — start in the quantum? If a fundamental theory is truly fundamental, shouldn’t it involve operational arrows that are rooted in the quantum and points towards the classical? If its structure is justified by and taken from a classical theory, then how can it be fundamental?
Think of Einsteins theory of relativity. Einsteins theory is beautiful and by beautiful we usually mean that it makes sense by itself. It is simple, it is elegant, it does not borrow its structure from Newtonian physics, it makes sense by itself. Wouldn’t we expect the same of a fundamental theory? That it makes sense by itself? If a fundamental theory is truly fundamental, then it must be ‘quantum’ in a way that makes sense by itself without being justified by a classical theory. It must have a mathematical structure that does not require a classical motivation.
So this is the program that I advocate: the fundamental theory that we seek should not be a quantisation of something else. We are not searching for a quantisation of general relativity. Such a theory would have the operational arrows point in the wrong direction. What we are looking for is a quantum theory that makes sense by itself and which produces its classical counterparts — say, general relativity — as a secondary spinoff.
We are searching for something, which makes sense by itself. At a conceptual level. We want something akin to Einsteins theory of general relativity, which has the confidence to stand alone; a fundamental theory that does not hide behind its mothers skirts. What we want is the “yes, of course!” feeling that strikes you when you study general relativity for the first time. It makes sense.
The point that I am advocating may seem obvious but if you look at what is going on in theoretical high-energy physics today then you’ll see that this is not what is going on there. There is a large number of projects and programs, which take their point of departure in existing theories, incremental improvements of what we already know or believe we know. The framework of quantum field theory is dominating — this is understandable, QFT is after all an extremely successful framework — and that implies a process of quantisation and a set of operational arrows, which are solidly rooted in the classical.
My own work
The line of thinking, which I have presented here, forms the foundation of my work with the mathematician Johannes Aastrup. In our approach we start out with a framework, which at a conceptual level is neither classical nor quantum. It is simply the mathematical machinery that describes how extended objects (could be a spinor, i.e. an electron) are translated in space. The objects themselves are not important (they emerge via representation theory), what matters is the moving itself. The action.
This framework does involve the key mathematics of a quantum field theory — the canonical commutation relations are hidden within a fundamental relation between key elements in our framework — but it is, at least conceptually, not a quantisation of a classical framework. In fact, one can argue that our framework from the get-go has very little to do with quantum theory, it is simply the framework you get when you consider moving stuff around.
Our idea is to analyse what theory such a framework will produce. This work is still ongoing — its a complicated question — but the point here is that with such a starting point the theory that emerges from it will have operational arrows that are not rooted in a classical theory. If our program is successful it will involve structures, which are not justified by a classical theory. It will stand on its own.