Abstract
The results of Cubitt et al. on the spectral gap problem add a new chapter to the issue of undecidability in physics, as they show that it is impossible to decide whether the Hamiltonian of a quantum many-body system is gapped or gapless. This implies, amongst other things, that a reductionist viewpoint would be untenable. In this paper, we examine their proof and a few philosophical implications, in particular ones regarding models and limitative results. In more detail, we examine the way these theorems model many-body quantum systems, and we question what, if anything, is the physical counterpart of the models used by Cubitt et al. We argue that these models are non-representational and that, even if they are so artificial that it is hard to imagine a physical system arising from them, they nonetheless offer an opportunity to learn about the world and the relation between mathematics and reality. On this basis, we draw the conclusion that their results do not undermine the reductionist viewpoint in a strong sense but leave the question open in a weak sense.
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Notes
A gapped system is robust against perturbations, which might change the state (in macroscopic bodies also called phase) of the system—it is far easier to mix states which are near each other in energy.
Richardson proved such a theoretical limit by showing that we cannot use an algorithm to establish if an arbitrary function taken by a certain class is equal to zero.
A self-referring code is an algorithm that encodes something like “I will proceed if it can be proven that I have stopped”. If we can prove that the code terminates, it must keep running, and if we can prove that it runs indefinitely, it must stop. Consequently, a supposedly universal deciding algorithm will not be able to claim either answer.
String theory is a well-known example of a unification of this kind.
For example, the axiomatic-deductive reasoning in mathematics in the case of Gödel’s theorems.
Of course, a limitative result can be read in a positive way. For instance, while Cantor’s original claim is negative (i.e. there is no enumeration of all sets of natural numbers), it can be used to establish a positive fact, for it offers a way to obtain a new entity, that is, a set that differs from all the enumerated sets.
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Acknowledgements
We are grateful to two anonymous reviewers of the Journal for General Philosophy of Science for their helpful comments.
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This work has been partially funded by the FFABR programme of the MIUR (Italian Ministry of Education, University and Research).
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Ippoliti, E., Caprara, S. Undecidability of the Spectral Gap: An Epistemological Look. J Gen Philos Sci 52, 157–170 (2021). https://doi.org/10.1007/s10838-020-09549-9
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DOI: https://doi.org/10.1007/s10838-020-09549-9