Uh, pardon the stupid question, but isn't it safer to consider the fine structure constant to be the IR fixed point of \alpha (i.e. \alpha(\mu = m_e)^-1 = 137..

Uh, pardon the stupid question, but isn't it safer to consider the fine structure constant to be the IR fixed point of \alpha (i.e. \alpha(\mu = m_e)^-1 = 137...)? \alpha(\mu = M_Z) is about 128. But m_0(photon), h, and e are the same at both energy scales (we can determine the numerical values experimentally). To the best of our ability to measure, none of them (including, if we ignore the main posting, \alpha_em) varies anywhere in a background fixed by the isometry group of SR (where we find c as the sole free parameter, corresponding to the speed of a particle where m_0(particle) = 0). So I'm having trouble understanding your assertion that "Alpha is the fundamental physical constant: c, e and \hbar are the derived quantities" (in your other comment, but also reflected in your second paragraph above). I think we both totally agree that varying \alpha_em at different points in space or time leads to a mess.