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- 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:

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:

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'##.

As @DarMM eloquently puts it:

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.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 real issue about collapse in QM is therefore purely a mathematical constructive issue coming directly from the theory of complex partial differential equations, namely:

- 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?
- 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'##.