Is the "wavefunction collapse" interpretation consistent with relativity?

This can be viewed as a follow up to this/this questions about an apparent inconsistency between the notion of wavefunction collapse and relativity.

The setup is simple: two entangled systems are located at spacetime points $A$ and $B$ which are spacelike separated, and are both being measured by respective observers. Now there are reference frame in which $A$ occurs before $B$, so we would say that the measurement of $A$ causes the wavefunction to collapse, and by the time it is measured by $B$ it is already in a collapsed state. But there are also reference frames in which the time order is opposite, so in those the collapse occurs at $B$ and not in $A$, hence the inconsistency.

The answers to the quoted questions generally argue that the collapse is not observable, and therefore there is no paradox. I think that this is indisputably true but also misses a point:

An interpretation necesserily says something about unobservable properties (otherwise we would call it a theory and not an interpretation). The collapse interpretation in particular implies that there is an objective wavefunction that describes a physical system at any given time (in fact, an objective wavefunction seems to be the only reason we need the collapse mechanism to begin with). But this is impossible within relativity as demonstrated by the example - we cannot assign an objective wavefunction to the system without violating Lorentz symmetry.

Therefore, is it not perfectly correct to argue that the wavefunction collapse interpretation is inconsistent with relativity ?

Here’s what I think is the best framework for discussing and analyzing about this kind of question: Zero in on what might exist in space and time, ask relativity-questions about those quantities in a block-universe framework, and then see if you can build other quantities out of those relativity-friendly pieces. I’ll unpack this below.

First of all, relativity concerns things that exist in space and time, at some particular spacetime location in a given reference frame. If some particular quantity doesn’t have a well-defined location (for example, the “total energy of a system”) it’s really hard to ask relativity-based equations about that quantity directly. Instead, you would want to reduce this quantity into other localized-quantities (associated with locations in space and time). In the “total energy” case, you’d start asking about the local energy density, or better yet the stress-energy tensor, which can indeed be assigned to any location. Then you could apply relativity to these “spacetime-based quantities”, see if they obey relativistic principles. Finally, you could then try to make sense of the original quantity in a relativistic way. (In the above example, you could choose a way to build up the total energy from any given perspective, given the local stress-energy tensors.)

Now, the quantum wavefunction is also not a function on ordinary space and time. So it’s really hard to ask relativity-questions about it. Let’s try the above strategy and see what happens. Given a wavefunction, what can we say about what is happening at some spacetime location? One idea would be to look at expectation values of local operators. Another idea works if you know that there’s one particle in a given region, and all the other particles are elsewhere. Then there’s a procedure for “tracing out” other particles, and generating something called a local density matrix for the particle in that area. There are other tricks you might play as well.

Whichever localized wavefunction-derived quantities you want to consider, now build up a block-universe history of all these localized quantities, and then implement a quantum collapse at some point. You’ll find that when analyzing your history, you’ll see special “planes of simultaneity” linking these localized parameters across great distances, with the planes chosen by how you implemented the collapse. (You can’t see them in real life, as an observer, but in the model you can see these planes.) Relativity is clearly broken; you’re never going to find a relativity-respecting account of these planes of simultaneity.

(Not to mention that unlike the total-energy example, in this case it’s impossible rebuild your original wavefunction from these localized quantities! You’ve thrown away information in the wavefunction when getting back down to spacetime, down to the level where you can analyze relativistic concepts.)

Interestingly, there is a different way to extract spacetime-local quantities from quantum theory. The idea is to use both the wavefunction as determined from the past preparation along with a different wavefunction that is effectively going to be measured in the future. These are called “local weak values”, and when analyzed in the same manner, the result is quite a bit different. In any block universe history, it turns out that they don’t show any evidence of “planes of simultaneity”. ( https://arxiv.org/abs/2412.05456 ) That’s basically because instead of having one wavefunction that collapses, you have two wavefunctions and neither one collapses. But that said, you also can’t rebuild the wavefunction(s) out of these local weak values. If you were going for a relativity-friendly account of quantum phenomena, it might make you want to look to the weak values on their own, without a wavefunction, as argued in the above link.

In conclusion, I’d say that if you are going to ascribe any sort of reality to the spacetime-localized quantities you can derive from a single wavefunction, then a collapse of that wavefunction would violate relativity. (I think there’s a caveat here if you carefully consider the relativistic-GRW picture, but that’s another question for another time.)

The answer to this is a bit complicated. If you look at a quantum physics textbook that mentions collapse, such as "Introduction to quantum mechanics" 3rd edition by Griffiths and Schroeter, you will find that the author doesn't bother to give an equation of motion for collapse. If you give no equation of motion for collapse it is easy to fool yourself about its implications because it is so vague. On its face collapse contradicts the equations of motion of quantum theory. If there was no theory that addressed this problem that would be a major problem with collapse theories.

Some physicists have tried to come up with theories that modify the equations of motion of quantum theory to include collapse and to come up with ways to test such theories. For a review see

https://arxiv.org/abs/2310.14969

These theories don't currently reproduce the predictions of relativistic quantum field theories so they simply aren't being tested in that regime

https://arxiv.org/abs/2205.00568

In addition, it's difficult to see how such a theory could avoid violating Lorentz invariance in the light of Hardy type experiments:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.68.2981

Hardy type experiments have been conducted and quantum theory has passed those tests

https://arxiv.org/abs/2401.03505

The lack of quantum interference and entanglement for objects in everyday life can be explained by decoherence without collapse

https://arxiv.org/abs/1911.06282

So quantum theory itself doesn't seem to require collapse. Since relativistic quantum field theories have Lorentz invariant equations of motion quantum theory without collapse doesn't clash with relativity. Quantum theory is consistent with relativity. Whether collapse is consistent with relativity is harder to say but there are unsolved problems with trying to make collapse theories that are consistent with relativity.

An interpretation necesserily says something about unobservable properties (otherwise we would call it a theory and not an interpretation).

That's certainly an interesting remark about the difference between theories and interpretations.

I would say the characteristic of "interpretations" is not necessarily that they allege "unobservable" properties, but that they arrange observations into different theoretical systems for which there is no known distinguishing observation that confirms one interpretation and demolishes the other.

The reason I draw attention to this subtlety is because often, within the terms of each interpretation when considered alone, the allegedly "unobservable" things are often clearly visible, at least to any usual standard of scientific evidence. It is the presence of a competing interpretation, that seemingly explains the same observations in a different way, which causes the ambiguity about what it is that we are observing.

That is, interpretations are not (or not usually) competing over the existence of utterly invisible or unobservable things, but competing over what the same set of visible and observable things mean and how to conceptualise and relate them appropriately.

To give an (intentionally risible) example of this, when a car driver hits a tree, there can be genuine physical ambiguity as to which ran into which. Conventionally we consider the bigger mass (the tree and the earth to which it is rooted) to be stationary, and the smaller mass (the car and driver) to be more physically agile and liable to human control, but if a particular interpretation is determined to reject that convention and substitute another, then the very same observation (the collision) can be explained in different ways, such as that the tree emerged from the shadows and struck the driver's vehicle.

This is still different from a "theory" in your scheme, since different theories might either account for different observations, or might allege that different things would be observable under the right conditions or with the right technology or apparatus.

The collapse interpretation in particular implies that there is an objective wavefunction

I don't think it does imply that. I think even the originators of these ideas had somewhat muddled ideas, or at least struggled to describe their thinking in a way that it would be received clearly.

Certainly, there appeared to be anti-realist strains of philosophy amongst prominent physicists in the 1930s, and it's not always clear that they made any distinction between their knowledge of the real world and the "objective" state of the real world. In such a case, what they know or how their knowledge evolves, is the objective thing to them - which can be very confusing for those who proceed from a different understanding and terminology.

Physicists are not always competent philosophers, and the fact that people who made important contributions to physics also had views or musings on the philosophy of physics, should not be taken as gospel.

For those of us who do distinguish between knowledge and the real world, the "collapse" is not characterising an objective physical change.

I think it's generally accepted that "collapse" is some kind of perceived event that triggers an alteration in how the physicist is analysing the physics or what he knows about the state of the physical system. For those physicists who make some kind of fundamental distinction between the classical and QM worlds, the "collapse" is when they shift mental gears back to the conventional, classical analysis of the physics.

First, I think that, in standard (non-relativistic) quantum mechanics, the "wavefunction collapse" is not an "interpretation", it is just a jargon to describe the fact that upon a (von Neumann) measurement, the wavefunction changes to a different one so that an immediately repeated identical measurement gives the same result. This change is not described by the Schrödinger equation.

In entangled systems, a measurement on one system also changes the wavefunction of the combined systems and thus the probabilities of the measurement results for the other system. Bell-type experiments on entangled systems A and B, like photon polarizations, done in one inertial system, give in accordance with standard quantum theory, statistical correlations between the measurement results that do not depend on the sequence of measurements on A and B already in the experimenters inertial system of A and B! Formally, already here the first "wavefunction collapse" of the combined system occurs at the first system measured. But the correlation results are symmetric whichever system is measured first.

This doesn't change for an observer in an inertial system moving at high speed relative to the inertial system in which the systems A and B are located. Depending on the velocity direction, the measurement on A can appear to happen before, simultaneously or after the measurement on B. This observer doesn't see any wavefunction, much less its "collapse". He only sees the measurement results in A and B happening either simultaneously, or before or after each other. Just like the experimenters in the inertial system of A and B who are chosing the timing.

A conceptual difficulty only arises if one thinks that the first measurement, say on A, causes the result on B and that the wavefunction is a physical object. This is already excluded by the fact that A and B are spacelike separated, i.e., the measurement on A cannot cause the result on B and vice versa.

These entanglement correlations are examples of statistical correlations without causation.

Quantum mechanics is Schrödinger's equation:

$$ \Big[ -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +V(x, t) \Big]\psi(x, t)= i\hbar \frac{\partial}{\partial t}\psi(x, t)$$

Note that this equation does not have a "$c$" in it. It has no reason to respect it as a speed limit.

When relativistic considerations are introduced to quantum mechanics, the concept of a "particle" is lost and one need a "field". That is a long story, but a minimum explanation is given in the introduction to Landau and Lifshitz's QED book: https://perso.crans.org/sylvainrey/Biblio%20Physique/Physique/Électromagnétisme/%5BLandau%2C%20Lifshitz%5D%2004%20Quantum%20Electrodynamics.pdf