Can We have Quantum Formalism Without Classical Formalism?

Question

Looking at special relativity (which gives a real geometry of space-time) which reduces to Newtonian mechanics when $v\ll c$, It seems that special relativity is possible without Newtonian mechanics. At least we have an explanation for things.

But is it the same for quantum mechanics? Is Quantum mechanics formalism possible without classical such?

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It may seem to be opinion based but I just want to know what the current status of science tells us.

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Edit: One more thing; is it possible to have a world that works in a quantum regime? Like we can assume a high-speed world where everything is working at a special relativity level. So that there is no approximation at all. Is the same is true for quantum mechanics? Do we not get into trouble with interpretation?

Answer 1

You are probably asking about problematic aspects of the classical limit. The classical world, described by classical limits, is that part of the quantum world where several telltale quantum features, mostly interference, are not directly apparent, and so their phenomena are effectively describable by classical mechanics. But the only extant consistent underlying theory is the quantum one.

If you were asking whether it is always easy to find suitable classical limits, the century-old answer is "no".

(How do you work out finite-dimensional classical limits of small spin systems?)

In an ultra-relativistic system, you'd have to work real hard to prevent components not exchanging energy and momenta to slow down, and thereby leave the relativistic world.

In our solidly quantum world, you may maintain coherence of some macroscopic quantum systems, such as superfluids or superconductors, even if they involve lots of subsystem actions $S\gg \hbar$. None of them, however, are the dispositive ones preventing these systems from decohering to classical ones.

Still, the vast majority of $S\gg \hbar$ systems are the classical-looking cars and engineering devices and horses of our deceptively Newtonian everyday world.

My strong sense is you'd profit from reading up on decoherence.

Answer 2

The measurement postulate of quantum mechanics requires contact between a quantum and a macroscopic (i.e., classical) object. Humans are operating in classical terms and making a measurement available to a human observer necessarily requires translating it in "classical/human" language.

Perhaps, the same point could be made for relativity, but I have never seen it discussed explicitly.