A Note on the Second Spectral Gap Incompleteness Theorem
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866909667558424576 |
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| author | Cubitt, Toby S. |
| author_facet | Cubitt, Toby S. |
| contents | Pick a formal system. Any formal system. Whatever your favourite formal system is, as long as it's capable of reasoning about elementary arithmetic. The First Spectral Gap Incompleteness Theorem of [CPGW15] proved that there exist Hamiltonians whose spectral gap is independent of that system; your formal system is incapable of proving that the Hamiltonian is gapped, and equally incapable of proving that it's gapless.
In this note, I prove a Second Spectral Gap Incompleteness Theorem: I show how to explicitly construct, within the formal system, a concrete example of a Hamiltonian whose spectral gap is independent of that system. Just to be sure, I prove this result three times. Once with Gödel's help. Once with Zermelo and Fraenkel's help. And finally, doing away with these high-powered friends, I give a simple, direct argument which reveals the inherent self-referential structure at the heart of these results, by asking the Hamiltonian about its own spectral gap. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_09854 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A Note on the Second Spectral Gap Incompleteness Theorem Cubitt, Toby S. Quantum Physics Pick a formal system. Any formal system. Whatever your favourite formal system is, as long as it's capable of reasoning about elementary arithmetic. The First Spectral Gap Incompleteness Theorem of [CPGW15] proved that there exist Hamiltonians whose spectral gap is independent of that system; your formal system is incapable of proving that the Hamiltonian is gapped, and equally incapable of proving that it's gapless. In this note, I prove a Second Spectral Gap Incompleteness Theorem: I show how to explicitly construct, within the formal system, a concrete example of a Hamiltonian whose spectral gap is independent of that system. Just to be sure, I prove this result three times. Once with Gödel's help. Once with Zermelo and Fraenkel's help. And finally, doing away with these high-powered friends, I give a simple, direct argument which reveals the inherent self-referential structure at the heart of these results, by asking the Hamiltonian about its own spectral gap. |
| title | A Note on the Second Spectral Gap Incompleteness Theorem |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2105.09854 |