The Spectral Gap
The most intimate level of existence is the boundary between something and nothing. The spectral gap is the energy difference between the ground state and the 1st state of excitement in a physical system.
Mathematically modeling the spectral gap is central to quantum physics. The central issue is whether there is a quantifiable gap in the spectral gap, or whether it is gapless (perfectly continuous). The Planck quantum of action suggests discrete discontinuities, but certain transitions indicate otherwise. It looks like the spectral gap is itself situational.
The spectral gap plays a key role in various manifestations of the material world. That there is a small spectral gap in the energy needed to shove an electron from a low-energy state to a more excited one is the central property for semiconductors.
When the spectral gap closes, a phase transition becomes possible. A material becomes superconducting with a closing spectral gap.
Mathematically characterizing the spectral gap has proved insolvable.
There exists no algorithm to determine whether the spectral gap is gapped or gapless. ~ English theoretical physicist Toby Cubitt, Spanish mathematician David Pérez-García & German mathematician Michael Wolf
This means a general method to determine whether matter described by quantum mechanics has a spectral gap, or not, cannot exist; which limits the extent to which we can predict the behaviour of quantum materials, and potentially even fundamental particle physics. ~Toby Cubitt
The Standard Model assumes a spectral gap. That is nothing more than wishful thinking.
The reason this problem is impossible to solve in general is because models at this level exhibit extremely bizarre behaviour that essentially defeats any attempt to analyze them. ~ David Pérez-García