Quantum logic is undecidable

Maarten Van den Nest, Hans J. Briegel

Bernard d’Espagnat

Massimo Tessarotto, Claudio Asci, Alessandro Soranzo +2 more

The problem is posed of establishing a possible relationship between a new type of Multi-verse representation, Gödel undecidability theorems and the logic of classical, quantum mechanics and quantum gravity. For this purpose example cases of multi-verses are first discussed in the context of non-relativistic classical, quantum mechanics and quantum gravity. As a result, it is confirmed that thanks to Gödel theorems non-relativistic classical and quantum mechanics, as well as quantum gravity theory are incomplete. As a consequence, they necessarily admit undecidable logical propositions and therefore obey a three-way boolean logical, i.e., a propositional logic with the three different logical truth values true, false and undecidable.

Andrew Stewart Caldin

FINDING: Undecidability of the spectral gap in quantum many-body systems is proven via reduction from the undecidable tiling problem, linking quantum physics to Turing's halting problem. MATH: Reduction: Halting problem ≤ Turing machine ≤ tiling problem ≤ spectral gap problem. Key constants: none directly; relies on logical undecidability, not numerical constants. CONNECTION: Tiling problem uses Wang tiles, which are directly related to quasicrystalline tilings and Penrose tilings (golden ratio φ = 1.618, 1/φ = 0.618). The undecidability proof for the spectral gap uses a Hamiltonian whose ground state encodes a tiling; the spectral gap is zero iff the tiling is aperiodic. Aperiodic tilings are intimately linked to the golden ratio and fivefold symmetry (crystallographically forbidden). DEPTH: 9 — This result shows that fundamental physical properties (energy gaps) can be logically undecidable, implying limits to what can be computed or known about quantum systems. It bridges comp Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Mladen Pavičić

Leopoldo Rom�n, B. Rumbos