Quantization Viewed as Galois Extension

TL;DR

This paper explores a novel perspective on quantization by framing it as a Galois extension of fields, revealing new quantization schemes based on algebraic invariance conditions.

Contribution

It introduces a new quantization scheme derived from Galois theory, linking algebraic field extensions with quantum state invariance.

Findings

Different Galois extensions lead to distinct quantization schemes.

Invariance under Galois group imposes physical state conditions.

Normalization involves sums over products of wave functions for different roots.

Abstract

Quantization is studied from a viewpoint of field extension. If the dynamical fields and their action have a periodicity, the space of wave functions should be algebraically extended `a la Galois, so that it may be consistent with the periodicity. This was pointed out by Y. Nambu three decades ago. Having chosen quantum mechanics (one dimensional field theory), this paper shows that a different Galois extension gives a different quantization scheme. A new scheme of quantization appears when the invariance under Galois group is imposed as a physical state condition. Then, the normalization condition appears as a sum over the product of more than three wave functions, each of which is given for a different root adjoined by the field extension.