In atomic physics quantum numbers n,l,j are used. But is it correct to use the field of rational numbers for quantization of the electromagnetic and gravitational fields? Maybe it is appropriate to introduce a continuum field, isomorphic, for example, to the lengths of a segment on a fractal line? I would be grateful if someone would tell me if this idea has been considered before.
Well, if you use rational numbers as quantum numbers, that would essentially just be a rephrasing of the concept of a quantity with a continuous spectrum.
So, the concept isn't new, you just commonly call it differently.
Dear Nikolay,
In standard quantum field theory, quantisation uses operators on Hilbert spaces over the field of complex numbers, not natural or rational numbers. Quantum numbers like n, l, j are discrete because of boundary conditions and symmetries, such as spherical symmetry in atoms.
For fields like the electromagnetic and gravitational fields, quantisation leads to continuous spectra, such as photon momentum and graviton modes, not limited to natural or rational numbers.
The idea of using a continuum based on fractal geometry or non-standard number fields has been explored in some specialised theories, such as p-adic string theory, fractal spacetime models, and non-commutative geometry.
However, no mainstream quantum field theory or quantum gravity framework uses rational numbers or fractal number fields as the fundamental field for quantisation, though such ideas have appeared in speculative and mathematical physics research.
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