Mathematical Physics
Rigorous mathematical proofs of statements that physics relies on.
arXiv: math-ph
15 topics
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An electron moving through a crystal with random impurities can be trapped by wave interference (Anderson localization) or can spread through the whole sample. Physicists expect both in 3D at weak disorder, but only the trapping has been proven.
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At low temperature a gas of bosons (particles that can share one quantum state) should collect a finite fraction of its particles in a single state, a Bose-Einstein condensate. This is proven for non-interacting particles but not for repelling particles in an infinite volume.
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Crystalline order at positive temperature
5 problemsAt low but nonzero temperature the atoms of a crystal vibrate around lattice sites, yet the lattice as a whole should stay in register over infinite distances. No one has proved this long-range order for any realistic system of particles moving in continuous 3D space.
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At zero temperature atoms arrange into regular crystals, yet for no realistic interaction in three dimensions has anyone proved that the lowest-energy arrangement of many particles is periodic. Two-dimensional proofs exist only for special forces.
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Rigorous Fermi liquids and superconductivity
4 problemsElectrons in a metal repel each other, yet they often behave like nearly free particles (a Fermi liquid) or bind into pairs and superconduct. Proving either behavior from the basic equations is open in most cases of physical interest.
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The equations of fluid flow predict how velocity changes in time, and it is not known whether a smooth 3D flow always stays smooth or can develop infinite velocity in finite time (blowup). Since September 2026 several computer-generated blowup constructions have been claimed and are under examination.
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Maximal ionization of atoms
5 problemsAn atom with nuclear charge Z is neutral when it holds Z electrons, and no real atom is known to hold more than one extra electron. Nobody has proved from the Schrodinger equation that the number of extra electrons stays bounded as Z grows.
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The Ising model, a grid of tiny magnets that point up or down, has a sharp phase transition whose 3D critical exponents are known to many digits from computers, yet mathematicians cannot prove that they exist. The predicted behaviour of such models, and of random paths that never cross themselves, is proven mainly in four or more dimensions, where it is close to that of free fields.
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Nearly regular motion such as planetary orbits is kept in order by invariant tori found by Kolmogorov, Arnold and Moser (KAM theory), but with three or more degrees of freedom orbits can slowly drift through the gaps between tori (Arnold diffusion). How general this drift is, and how fast it acts, is not settled.
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The Boltzmann equation describes a gas through the statistics of molecular collisions, and fluid equations follow from it; deriving both from Newton's laws for individual molecules is Hilbert's sixth problem. For a gas of hard spheres a derivation was posted in 2025 (Deng, Hani and Ma) and is under review, while other physically central cases remain unproven.
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The Lieb-Thirring inequality bounds the total binding energy of all bound states in a potential well by a simple integral of the potential, and it is a key step in proving that ordinary matter does not collapse. The best numerical constant in this bound is still unknown in three dimensions.
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Quantum chaos and the Riemann zeros
7 problemsWhen a classical system is chaotic, its quantum energy levels repel each other with the statistics of random matrices, and its standing waves spread evenly over the available space. The zeros of the Riemann zeta function, which control the primes, show the same statistics, as if they were energy levels of an unknown chaotic system.
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In a chain or lattice of quantum magnets, the energy of the lowest excitation may stay finite (a gap) or vanish as the system grows. Physicists predict which happens, but proofs exist only for specially designed models.
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Mass gap in two-dimensional sigma models
3 problemsThe O(N) model describes arrows of fixed length on a grid, each with N components, that prefer to point the same way as their neighbours. In two dimensions physicists predict that for N of three or more the alignment is always lost beyond a finite distance, however cold the system is, but nobody has proven it.
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Quantum field theories describe particles as excitations of fields that fill space, but no interacting one has been built with full mathematical rigor in four-dimensional spacetime. The main test case is Yang-Mills theory, the theory of gluons, which should produce particles with a smallest nonzero mass.