Are the postulates of QM independent of interpretation, or based on Copenhagen?

Question

In particular, I'm talking about the measurement postulate (or at least part of it, I have seen different formulations, here there's one), in which, after measurement of an observable $A$ with eigenbasis $\{|a\rangle\}$, the new quantum state is the projection of the old quantum state, $|\psi\rangle$ onto one of the eigenvalues' subspace:

$$|\psi\rangle \to \frac{\Lambda_a |\psi\rangle}{\sqrt{\langle \psi|\Lambda_a|\psi\rangle}} $$

For a non-degenerate subspace, $\Lambda_a=|a\rangle\langle a|$ and the new state is simply the eigenstate $|a\rangle$.

But this postulate is essentially just the "collapse", that is, fundamentally a Copenhagen postulate. I assume an Everettian or Pilot-Wave approach would reject this. Is this correct? Or do all interpretations share the same postulates, but with different interpretations of the mathematical formalism?

Answer 1

We cannot observe $|\phi\rangle$ directly, so the postulates of QM should, in principle, be phrased in terms of what we can observe i.e. measurement outcomes and their probabilities (okay, we cannot observe probabilities directly either but if we do enough experiments then we can infer them with as much precision as we like). Rather than saying categorically what $|\phi\rangle$ is, it would be better to say that a system behaves as if it has a wave function which changes in such-and-such a way.

If the postulates of QM were phrased in this way, they would be independent of interpretation. In practice, I think that physicists are willing to grant that they could be phrased in this way if necessary, and so do not get too hung up on this point.

Answer 2

I would say that the postulates of quantum mechanics were created independently of the Copenhagen interpretation but are heavily tied to it. The idea of the "measurement postulate" or Bohm rule is a key part of that interpretation.

Even if all current interpretations provide the same results, you need extra considerations. To get Bohmian mechanics there is no collapse and you need an extra equation that it is not in the postulates. To get many worlds you need to remove the Born rule (if that makes even sense). To get objective-collapse you need an extra postulate too.

However all that said, Copenhagen is often considered as incomplete for not being clear on what a measurement device is.

Answer 3

Blockquote Or do all interpretations share the same postulates, but with different interpretations of the mathematical formalism?

The interpretations do not all share the same postulates. The state postulate says the state function represents all the information about a state. Whereas, contextual hidden-variable interpretations think there is more going on than the state/wave function tells us.

Answer 4

The controversy over the interpretation of quantum is, confusingly, not about interpreting the equations of quantum theory. An interpretation would describe how those equations describe reality. The Copenhagen and statistical interpretations say that those equations don't describe reality, but they provide no alternative and keep using the equations anyway. They would add a collapse postulate but give no explanation of how or collapse happens. As such, their predictions are vague and ad hoc and there's no reason to consider them further.

The pilot wave theory takes the equations of motion of quantum mechanics and supplements them with an additional equation describing the motion of particles in addition. Under some circumstances those particles are supposed to obey different probability rules than quantum theory:

https://arxiv.org/abs/1906.10761

Since this theory has different predictions than quantum theory it isn't an interpretation of quantum theory, it's a candidate to replace quantum theory.

There are theories called spontaneous collapse models that modify the equations of quantum theory to include collapse to produce one specific outcome:

https://arxiv.org/abs/2310.14969

These theories also have different predictions than quantum theory and so are an alternative to quantum theory, not an interpretation of it.

It should be noted that pilot wave and spontaneous collapse theories can't currently reproduce the predictions of quantum field theories that are used to make the vast bulk of actual predictions made using quantum theory:

https://arxiv.org/abs/2205.00568

The many worlds interpretation (MWI) takes the equations of quantum theory as a description of how reality works and works out their consequences. In this theory each measurable quantity has multiple possible values and those values can intefere with one another. When information is copied out of a system that suppresses interference between the different possible values of the measured quantity, this process is called decoherence:

https://arxiv.org/abs/quant-ph/0306072

As a result of this process, on the scale of everyday life reality as described by quantum theory looks a bit like a collection of parallel universes:

https://arxiv.org/abs/1111.2189

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

When you do a measurement, the result you see won't interfere with other versions of the same measurement as a result of decoherence so you will only see one of them. As a result you can adjust the state to reflect that lack of interference:

https://arxiv.org/abs/2008.02328

This is why the MWI was originally called the relative state interpretation, the state is relative to what outcome happens in your branch. The collapse rule works to the extent that it works at all just describes resetting the relative state to take account of measurement outcomes.

I should note that one of the answers above states that the MWI discards the Born rule. This is misleading. The standard quantum probability rule has been explained by a couple of different arguments in unmodified quantum theory:

https://arxiv.org/abs/0906.2718

https://arxiv.org/abs/quant-ph/0405161

So the MWI currently provides the only known explanations of how the probability rule arises and of the circumstances under which it works: it requires decoherence.