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.
Highlights
In a macroscopic world the dynamics is controlled by classical mechanics, while in a microscopic world, it is described by quantum mechanics
This paper considers how the space of wave functions is extended in the procedure of quantization, so that it may be consistent with the additive periodicity in the field space
The wave function is defined in terms of path integral for each root
Summary
In a macroscopic world the dynamics is controlled by classical mechanics, while in a microscopic world, it is described by quantum mechanics. A classical particle becomes a wave in quantum mechanics. Waves propagating along different paths may interfere, resulting to be strengthened or weakened. The path integral quantization manifestly represents this wave interference [1]. A wave function ψ(q, t) in quantum mechanics gives the “probability amplitude” of a particle located at position q and time t. The transition of a wave function ψ(q0, 0) at time 0 to that ψ(q, t) at time t can be described by a transition amplitude U (q, q0; t, 0), ψ(q, t) = U (q, q0; t, 0)ψ(q0, 0),
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