# A Collapse in QM: in conflict with SR?

#### Auto-Didact

Summary
Collapse in QM is seen by some physicists as problematic because of its instantaneous effects and therefore inconsistency with SR. I show how this line of reasoning is not necessarily valid.
It is often claimed in these forums and beyond that collapse in QM is completely unnecessary for one of several reasons. To be explicitly clear, collapse seen purely as a probability theoretical concept of conditional updating - i.e. treating the wavefunction as an epistemic object - isn't generally seen to be problematic, so this thread will not focus on that topic at all.

As @DarMM eloquently puts it:
If you accept the wave-function as an ontic element and collapse as real then it would imply a non-local effect. However interpretations that view the wavefunction as ontic typically don't have collapse and interpretations that have collapse don't view the wavefunction as ontic. So usually this is a non-issue. Textbook QM (a form of Copenhagen) doesn't have the wavefunctions as ontic.
The most explicit reason for why collapse in QM is seen as problematic is the fact that collapse if real is a non-local effect, which obviously then seems to quickly run into trouble with SR-based QFT. In reality however, this inconsistency with SR is contingent i.e. it isn't necessarily the case that there is an actual problem: it would only necessarily be the case if such non-local effects can not be mathematically described without also directly violating the mathematical basis of SR.

The real issue about collapse in QM is therefore purely a mathematical constructive issue coming directly from the theory of complex partial differential equations, namely:
1. within the mathematical theory of analysis (or any constructive generalization thereof), does there exist a class of structures which describe non-local properties such that a local change has non-local consequences without directly violating the mathematical structures in SR which define locality?
2. and does the Schrödinger equation (or any experimentally indistinguishable explicit generalization thereof) and/or its solutions properly belong to that class of mathematical structures?

From the pure mathematics and mathematical physics literature, we can already answer the first question positively, i.e. we have evidence of at least one such a class of structures which does seem to exist: elements of sheaf cohomology. In other words if it can be demonstrated that wavefunctions are elements of sheaf cohomology then this would constitute a constructive solution to the 'collapse problem' by giving a mathematical description of collapse of the wavefunction in the following sense:

The mathematical reason for unique measurement outcomes in single particle wavefunctions is due to the non-local nature of the described system i.e. the presence of some cohomology element $\eta$: for any sufficiently small open subregion $G'$ of a region $G$, the cohomology element $\eta$ vanishes when restricted down to $G'$.

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#### Auto-Didact

I agree that they are in conflict; the question is are they necessarily in conflict? This is a pure mathematics and/or mathematical physics question about the consistency of the mathematics underlying SR and the mathematics underlying QM, i.e. a question about the mathematical consistency of on the one hand null cones in Minkowski space and the Poincaré group, and on the other hand the Schrödinger equation, its solution and its underlying state space structure.

If there is a form of mathematics which naturally unifies all of the above, then there possibly isn't a real inconsistency if relativistic QM is recast in the language of that form of mathematics; I see no a priori reason to exclude sheaf cohomology theory as a possible candidate for this underlying form of mathematics.

#### Michael Price

I agree that they are in conflict; the question is are they necessarily in conflict? This is a pure mathematics and/or mathematical physics question about the consistency of the mathematics underlying SR and the mathematics underlying QM, i.e. a question about the mathematical consistency of on the one hand null cones in Minkowski space and the Poincaré group, and on the other hand the Schrödinger equation, its solution and its underlying state space structure.

If there is a form of mathematics which naturally unifies all of the above, then there possibly isn't a real inconsistency if relativistic QM is recast in the language of that form of mathematics; I see no a priori reason to exclude sheaf cohomology theory as a possible candidate for this underlying form of mathematics.
The Schrödinger equation is non-relativistic, so is not bothered by violations of SR. So it is consistent with instantaneous collapse. But in QFT, SR holds, and so collapse becomes more of a problem.

#### Auto-Didact

@DarMM Sheaf cohomological collapse is ontic. It goes against the typical characterization of most interpretations where collapse is a non-issue, which is why I included your quote in the OP.
The Schrödinger equation is non-relativistic, so is not bothered by violations of SR. So it is consistent with instantaneous collapse. But in QFT, SR holds, and so collapse becomes more of a problem.
This thread isn't about QFT as that is already an SR-based construction which is mathematically muddied from the constructive mathematics point of view; QFT is just one theory within the class of relativistic QT. My claim is that it may be possible to construct a new form of relativistic QM using sheaf cohomology as the underlying mathematical basis, and so naturally unify SR and non-relativistic QM.

#### DarMM

Gold Member
Wavefunctions that are merely receptacles of our knowledge are not sufficient to model even simple systems, such as the double.slit experiment. You can call that a knee jerk reaction or over a hundred of experience of the collective scientific community. If it were otherwise there'd be no measurement problem.
You do know in the Copenhagen interpretation the wavefunction is just epistemic and that the epistemic view is the majority in Foundational studies? That's not to say it is correct, but that the issue is not as clear cut as you make it.

Of course there is still a measurement problem in an epistemic view as you have no account of how an individual outcome results.

#### Auto-Didact

@DarMM I can't recall any reviews from the top of my head which directly address the topic as illustrated in this thread. However, the book Geometric Quantization by Woodhouse probably contains these answers, but I'm not so sure it is the best place to start.

I know that at the end of that book, Chapter 10.6 in particular focuses on cohomological wavefunctions, but I'll look to see if I can also find any reviews in the literature which also treat the subject from the perspective of a physicist instead of a mathematician.

#### martinbn

1. within the mathematical theory of analysis (or any constructive generalization thereof), does there exist a class of structures which describe non-local properties such that a local change has non-local consequences without directly violating the mathematical structures in SR which define locality?
From the pure mathematics and mathematical physics literature, we can already answer the first question positively, i.e. we have evidence of at least one such a class of structures which does seem to exist: elements of sheaf cohomology.
How is this an answer to that question?

#### Auto-Didact

How is this an answer to that question?
Directly: sheaf theory contains structures, namely elements of the overlaps of transition functions of complex manifolds which have intrinsically non-local topological properties and are at the same time fully consistent with the more geometric definition of SR locality as encoded in the nullcone structure in Minkowski space.

Geometric quantization is a quantization procedure/research programme from mathematical physics, from which there has arisen the aim to unify QM and relativity in this manner. In other words, a physical theory based on cohomological wavefunctions might naturally solve the inconsistency between NRQM and SR (or even GR).

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#### martinbn

Directly: sheaf theory contains structures, namely elements of the overlaps of transition functions of complex manifolds which have intrinsically non-local topological properties and are at the same time fully consistent with the more geometric definition of SR locality as encoded in the nullcone structure in Minkowski space.
May it's just me, but I still don't follow. What is the meaning of non-local here?
Geometric quantization is a quantization procedure/research programme from mathematical physics, from which there has arisen the aim to unify QM and relativity in this manner. In other words, a physical theory based on cohomological wavefunctions might naturally solve the inconsistency between NRQM and SR (or even GR).

#### Auto-Didact

May it's just me, but I still don't follow. What is the meaning of non-local here?
That is the key question: the cohomology of a topological space is a fundamentally non-local invariant of that topological space. This kind of sophisticated mathematical non-locality is completely independent of relativity theory, whose form of locality is quite mathematically pedestrian in comparison.

In other words, defining (non-)locality based on cohomology from mathematics in this sense fully subsumes the meaning of locality in physics as a particular form of locality, e.g. locality as defined by Laplace in classical field theory i.e. based on vector calculus, potential theory and the theory of PDEs, and expanded to SR/QFT using the covariant derivative formalism.

On the other hand, non-locality in QM - i.e. entanglement and collapse - are mathematical constructions of this more sophisticated definition of non-locality from topology; this is the real reason why entanglement, a process with non-local effects, is not inconsistent with SR.

tl;dr the meaning of the word locality from physics was based on a prematurely settled mathematical definition, which has since been superseded by an expanded meaning of the word locality in mathematics. Physicists who have missed this update have tied themselves up in unnecessary logical knots.

#### Auto-Didact

May it's just me, but I still don't follow. What is the meaning of non-local here?
This paper (Abramsky et al. 2011) gives a sheaf theoretic treatment of non-locality, essentially explaining what I am trying (and probably failing) to explain to you.

@DarMM I just remembered I was lately re-exposed to this line of reasoning by John Baez; he might have written a lot more on this topic, but I haven't had the time to remain up to date.

For those unfamiliar with algebraic geometry, topology, pure mathematics and/or mathematical physics, here is a recent popular article of Baez in which he explains his constructivist mathematical physics vision for QG at an introductory level understandable to students and non-mathematicians alike.

#### PeterDonis

Mentor
This is a pure mathematics and/or mathematical physics question about the consistency of the mathematics underlying SR and the mathematics underlying QM, i.e. a question about the mathematical consistency of on the one hand null cones in Minkowski space and the Poincaré group, and on the other hand the Schrödinger equation, its solution and its underlying state space structure.
If you're looking at consistency with SR, you need to be looking at QFT, not the Schrodinger Equation. The Schrodinger Equation is non-relativistic and should not be expected to be consistent with SR.

#### PeterDonis

Mentor
Kid yourself all you like, but wavefunction collapse is in conflict with SR
I am afraid this is like so many of your answers, where you you throw in some irrelevant point to muddy the waters.
This attitude just got you a thread ban and a warning. It does not contribute anything useful to the discussion.

#### PeterDonis

Mentor
This thread isn't about QFT as that is already an SR-based construction which is mathematically muddied from the constructive mathematics point of view; QFT is just one theory within the class of relativistic QT. My claim is that it may be possible to construct a new form of relativistic QM using sheaf cohomology as the underlying mathematical basis, and so naturally unify SR and non-relativistic QM.
Then you need to find some published paper that proposes such a theory and use that as a basis for discussion in a new thread. In the absence of such a specific theory to discuss, this thread is closed.

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