/sci/ - Science & Math

>>11296408
Solutions [math]A[/math] to the strong EL equation [math]d\ast F = j[/math] of the [math]U(1)[/math]-Yang-Mills action [math]S[A] = \int (F\ast \wedge F - j\ast A)[/math], coupled to a current [math]j[/math], determines the radiative pattern. Relativity postulates that [math]S[/math] is Poincare invariant, but shifting into non-inertial reference frames is manifestly [math]not[/math] part of the Poincare group. The difference [math]\delta S= S_\text{inertial}-S_\text{non-inertial}[/math], or classically its EL term [math]\partial_{dA} \delta S[/math], contributes to radiation amplitudes due to acceleration. Now [math]\delta S_\text{lin}[/math] for a linearly accelerating particle is still Poincare invariant in the plane perpendicular to the motion, while [math]\delta S_\text{rot}[/math] for a circularly accelerating particle is only Poincare invariant along the axis, hence they [math]must[/math] take different forms; in fact, assuming no quantum phase transition occurs in the system, this symmetry argument extend to the solutions [math]A[/math] and hence to the radiation pattern.
>>11301737
General it is more straightforward to define the fields via the curvature 2-form [math]F[/math] of a principal [math]U(1)[/math] bundle (or an associated vector bundle thereof), at least geometrically speaking. However if your base manifold [math]M[/math] is not simply connected, then the relation [math]F = dA[/math] need not necessarily hold. In the case of a magnetic monopole, for instance, [math]F=dA[/math] holds only on separate patches of the sphere (together with gauge equivalence/patch transition conditions on overlappnig regions). Physically, a monopole can be considered as one end of a semi-infinite solenoid, in which [math]A[/math] is not well-defined. In this case [math]M = \mathbb{R}^3 \setminus \mathbb{R}[/math] and your fields [math]A,E,B[/math] are [math]multiply[/math]-valued. In this case it may be more straightforward to consider a solenoid.