Group theoretical derivation of the minimal coupling principle

Abstract

The group theoretical methods worked out by Bargmann, Mackey and Wigner, which deductively establish the Quantum Theory of a free particle for which Galileian transformations form a symmetry group, are extended to the case of an interacting particle. In doing so, the obstacles caused by loss of symmetry are overcome. In this approach, specific forms of the wave equation of an interacting particle, including the equation derived from the minimal coupling principle, are implied by particular firstorder invariance properties that characterize the interaction with respect to specific subgroups of Galileian transformations; moreover, the possibility of yet unknown forms of the wave equation is left open.

Group Theoretical Derivation of Consistent Free Particle Theories

“…In [12] we have proved that the following statement holds. Proposition 2.1 If G is a connected group, then every continuous generalized projective representation U of G is a projective representation, i.e.…”

“…This methodology avoids the shortcomings arising with canonical quantization, because the eventual occurrence of inconsistencies, whenever ascertained, would be the proof of the failure of the starting assumptions, which should be accordingly modified. This approach has turned out to be very effective in developing non-relativistic quantum theories of an interacting particle [11][12][13], by making use of group theoretical methods. Here we undertake the approach for the relativistic quantum theory of an isolated system and of a "massive" free particle in particular.…”

Group Theoretical Settling of Spin Zero Relativistic Particle Theories

“…Then there are the following six inequivalent representations U (1) , U (2) ,...,U (6) with the same s, µ.…”

“…(S.1) Every S g : Ω(H) → Ω(H) is bijective. Properties (S.1) and (S.2) were sufficient [6] to prove that each quantum transformation S g is an automorphism of the lattice Π(H) of the projection operators; therefore, according to Wigner theorem [6], [11], a unitary or anti-unitary operatorŨ g must exist such that S g [A] =Ũ g AŨ −1 g , for every A ∈ Ω(H).…”

“…Hence, in general the correspondence U : P → V(H), g → U g realized according to these prescriptions is a generalized projective representation. We assume that the correspondence g → S g from P ↑ + into the automorphisms of Π(H) is continuous, according to Bargmann topology [6]; this assumption can be motivated by the idea that a small transformation g ∈ P ↑ + determine a small change from A to S g [A]. This continuity implies [9] that the restriction U : P ↑ + → U (H) can be made continuous by redefining the free phase θ(g), and therefore it is a continuous projective representation, according to Prop.…”