Say you cook up a model about a physical system. Such a model consists of, say, a system of differential equations. What criterion decides whether the model is classical or quantum-mechanical?
None of the following criteria are valid:
Partial differential equations: Both the Maxwell equations and the Schrödinger equation are PDE's, but the first model is clearly classical and the second one is not. Conversely, finite-dimensional quantum systems have as equations of motion ordinary differential equations, so the latter are not restricted to classical systems only.
Complex numbers: You can use those to analyse electric circuits, so that's not enough. Conversely, you don't need complex numbers to formulate standard QM (cf. this PSE post).
Operators and Hilbert spaces: You can formulate classical mechanics à la Koopman-von Neumann. In the same vein:
Dirac-von Neumann axioms: These are too restrictive (e.g., they do not accommodate topological quantum field theories). Also, a certain model may be formulated in such a way that it's very hard to tell whether it satisfies these axioms or not. For example, the Schrödinger equation corresponds to a model that does not explicitly satisfy these axioms; and only when formulated in abstract terms this becomes obvious. It's not clear whether the same thing could be done with e.g. the Maxwell equations. In fact, one can formulate these equations as a Dirac-like equation $(\Gamma^\mu\partial_\mu+\Gamma^0)\Psi=0$ (see e.g. 1804.00556), which can be recast in abstract terms as $i\dot\Psi=H\Psi$ for a certain $H$.
Probabilities: Classical statistical mechanics does also deal with probabilistic concepts. Also, one could argue that standard QM is not inherently probabilistic, but that probabilities are an emergent property due to the measurement process and our choice of observable degrees of freedom.
Planck's constant: It's just a matter of units. You can eliminate this constant by means of the redefinition $t\to \hbar t$. One could even argue that this would be a natural definition from an experimental point of view, if we agree to measure frequencies instead of energies. Conversely, you may introduce this constant in classical mechanics by a similar change of variables (say, $F=\hbar\tilde F$ in the Newton equation). Needless to say, such a change of variables would be unnatural, but naturalness is not a well-defined criterion for classical vs. quantum.
Realism/determinism: This seems to depend on interpretations. But whether a theory is classical or quantum mechanical should not depend on how we interpret the theory; it should be intrinsic to the formalism.
People are after a quantum theory of gravity. What prevents me from saying that General Relativity is already quantum mechanical? It seems intuitively obvious that it is a classical theory, but I'm not sure how to put that intuition into words. None of the criteria above is conclusive.
Answer
I think this is a subtle question and I think it depends somewhat on how you choose to represent quantum mechanics. To see one extreme of this, consider the viewpoint put forth by Kibble in [1]. For simplicity I will be thinking of finite-dimensional quantum systems here; there are some subtleties in infinite dimensions but as far as I know the basic picture still holds. In this, he shows that if we describe the theory in terms of physical states (rays in the Hilbert space), then the dynamics of Schrödinger evolution correspond exactly to Hamiltonian evolution via the symplectic form from the Kähler structure on the projective Hilbert space (which is to say, the evolution is that of a classical system). However there are two distinctions which make quantum mechanics different from classical mechanics:
- The phase space must be a projective Hilbert space (as opposed to just a symplectic manifold), and the Hamiltonian is restricted to being a quadratic form in the homogeneous coordinates on projective space. In classical mechanics any (sufficiently smooth) function is admissible as a Hamiltonian.
- Composite systems are described differently. In classical mechanics the phase space of a composite system is the Cartesian product of the phase spaces. In quantum mechanics, it is the Segre embedding (which descends from the tensor product of Hilbert spaces). This is parametrically different; if the phase spaces of the two subsystems are $2m$ and $2n$, then in classical mechanics the composite system has dimension $2m+2n$, whereas in quantum mechanics it has dimension $2(n+1)(m+1)-2$. The extra states are the entangled states. Virtually all the observable consequences of QM come here, e.g. Bell inequalities. Of course if we consider identical particles things get even a bit more complicated.
If you ignore the second point, and focus only on a single quantum system, the surprising conclusion is that every quantum mechanical system is a special case of classical mechanics (with the provision that again I haven't checked the details in infinite dimensions but it is at least morally true). However, part of the structure of quantum mechanics is how it describes composite systems so you can't just ignore this second point. A mathematician would say that this gives an injective functor from the category of quantum mechanical theories to the category of classical theories which is not compatible with the symmetric monoidal structures on the two.
I want to point out that this is emphatically not how we typically think of the correspondence principle in quantum mechanics. That is, it is a mapping from a finite-dimensional quantum mechanical system to a finite-dimensional classical system (of the same dimension). Normally, if we think about e.g. a free particle in one dimension, the Hilbert space for that quantum system is infinite dimensional, yet it corresponds to a 2-dimensional classical phase space. But the point is that, at least in this question, we can't restrict to the ordinary notion of correspondence since we don't have a physical interpretation for the system of equations describing the theory.
Additionally, despite the above example, whether a theory is classical or quantum has essentially nothing to do with where the states live. Indeed, if we just want to consider a free particle in one dimension again, we would typically describe its state as a self-adjoint trace class unit trace operator $\hat \rho$ on the Hilbert space $L^2(\mathbb R)$. In contrast, in classical mechanics we would describe a state as a probability distribution $\rho$ on the phase space $\mathbb R^2$ (note that in the above example we had only pure classical states i.e. only those described by a $\delta$ function on the phase space whereas now we have mixed states). However we could just as easily describe the quantum state by its Wigner function, in which case it lives in exactly the same affine space as the classical distribution. However the Wigner function satisfies slightly different inequalities than the classical probability distribution; in particular it can be slightly negative and cannot be too positive. The details of this were first worked out in [2]. In this case, it is the dynamics that give away the quantum nature. Specifically, to go from classical to quantum mechanics, we must replace the Poisson bracket by the Moyal bracket (which has $O(\hbar^2)$ corrections), indicating the failure of Liouville's theorem in the phase space formulation of quantum mechanics: (quasi)probability density is not conserved along trajectories of the system.
All of this is to say that it seems difficult (and maybe impossible) to try to find a single distinguishing feature between classical and quantum mechanics without considering composite systems, so if that is what you want, I'm not sure I have an answer. If you do allow for composite systems though, it is a pretty unambiguous distinction. Given this, it is perhaps not surprising that all the experimental tests we have which demonstrate that the world is quantum and not classical are based on entanglement.
References:
[1]: Kibble, T. W. B. "Geometrization of quantum mechanics". Comm. Math. Phys. 65 (1979), no. 2, 189--201.
[2]: H.J. Groenewold (1946), "On the Principles of elementary quantum mechanics", Physica 12, pp. 405-460.
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