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Which of these is understood to be the relationship between classical and quantum mechanics? Or is it a mixture of these, or something else entirely?

  1. The classical world comprises of measurement outcomes that emerge from quantum mechanics. The universe is fundamentally a quantum state, from which measurement outcomes emerge through a process which is not completely understood yet (some approaches can be decoherence or objective collapse theories). And the classical world is just the set of measurement outcomes.

  2. The classical world corresponds to coherent states of the quantum mechanical wavefunction. In this approach too, the classical world emerges from quantum mechanics, but the correspondence is at the wavefunction level and doesn't have to do with measurements.

  3. The classical world is assumed to exist as part of the definition of Quantum Mechanics. This is because measurements require an interaction between a quantum system and a classical environment. This means both classical and quantum mechanics co-exist and are equally fundamental.

I want to know the modern understanding of this, where we stand currently in understanding the classical world.

Ken Wharton
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Ryder Rude
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Classical physics describes a system in terms of variables like $x(t)$ for the $x$ position of a particle. When you measure $x(t)$ you also get the value $x(t)$.

The equations of motion of quantum theory describe a system in terms of operators called observables that can be represented by Hermitian matrices. The possible results of measuring an observable are its eigenvalues. Quantum theory predicts a probability for each of the possible outcomes that depends on the situation in which the measurement is being conducted, specifically on the outcomes of past measurements as summarised in the quantum state, which records which observables were measured and the values obtained.

In general the outcome of a quantum experiment will depend on what happened to all of the eigenvalues of the measured observable during the experiment: this is called quantum interference. For an example, see Section 2 of this paper:

https://arxiv.org/abs/math/9911150

The description I've given so far is fairly uncontroversial but what is controversial is what is happening in reality to bring about the predictions of quantum theory. This controversy is related to the relationship between quantum and classical physics. The different accounts of what is happening in reality to bring about the predictions of quantum theory are called interpretations of quantum theory.

Some of the interpretations say there is no account of what is happening in reality such as the Copenhagen or statistical interpretations of quantum theory. Classical physics works in everyday life and you're not allowed to ask about what is happening in quantum experiments. In an experiment you try to test a theory by setting up a situation which results in particular predictions when described in terms of that theory. So if your theory provides no description of reality, as with these interpretations, it doesn't allow you to test whether an experiment is set up correctly if you take it seriously. I don't see why anyone would adopt such a theory but some people do.

There are other theories that modify the equations of motion of quantum theory such as spontaneous collapse theories, which say that all of the different versions of a system regularly get cut down to just one version:

https://arxiv.org/abs/2310.14969

Or pilot wave theories, which say that the real world consists of a bunch of particles that interact with the observables according to their own equation of motion:

https://arxiv.org/abs/2411.10782

These theories don't currently reproduce the predictions of relativistic quantum theories that provide the bulk of experimentally tested predictions of quantum theory:

https://arxiv.org/abs/2205.00568

Some physicists are doing research on improving and testing these theories.

Another way of thinking about quantum theory is to apply its equations of motion to reality in the way we would apply any other scientific theory. Interference and other quantum phenomena can be explained in terms of the existence of multiple versions of each system.

When information is copied out of a quantum system interference is suppressed: this is called decoherence:

https://arxiv.org/abs/1911.06282

Objects in everyday life that are large enough for you to see at all have information copied out of them a lot faster than the timescales over which they change significantly. As a result decoherence suppresses interference very effectively for those objects. This tends to result in the versions of the objects you see around evolving independently of one another in layers each of which acts approximately like the universe as described by classical physics:

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/quant-ph/0104033

This is called the many worlds interpretation of quantum theory. Some physicists don't like it. A selection of criticisms:

https://arxiv.org/abs/2210.05377

https://arxiv.org/abs/0811.0810

https://arxiv.org/abs/0905.0624

alanf
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None of the above because there is no "classical world", there is only the quantum. But due to a number of effects (some which make for nice calculations, others which seem hopelessly intractable) there are wavefunctions which appear very sharply peaked over a macroscopic scale and would be infinitely sharply peaked if we took $\hbar \to 0$. This is enough to "trick" humans into developing classical mechanics first as a way of modelling these quantum phenomena.

Connor Behan
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A simple but provably limited approach is the direct correspondence between classical and quantum theory occuring in Dirac's procedure of canonical quantization which relates the Poisson bracket of Hamiltonian Mechanics to the commutators of quantum theory.

$$\{A,B\} \rightarrow \frac{1}{i\hbar} [\hat{A},\hat{B}]$$

One of the least appreciated aspects of current theory is that it is the merging of basic single particle quantum mechanics with special relativity which gave rise to quantum field theory via the need for the second quantization of the quantum theory (e.g. counting of particles obeying pauli exclusion principle and their respective energies)

Special relativity in some sense breaks the very stable world described by the Schrodinger equation by forcing time to mix with space. In the non-relativistic world described by the Schrodinger equation there is no ambiguity that time and space are seperate. The wave equation in the Schrodinger world is perfectly deterministic and evolves deterministically. It however doesn't describe a world full of the particles that we see every day.

It is only with the incorporation of special relativity that we see something resembling the world we know.

This relationship between spin and the classical world is still truly not fully understood and was already noted as being important to the quantization of gravity as early as 1939 by Fierz and Pauli as they probed the nature of spin-2 particles in weak gravitational fields.

It is an intriguing possibility that if we better understood how spin statistics actually arose we might close the gap between the quantum and the classical.

Last American Hero
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