  The Born-Infeld form of the hydrogen atom has a spectrum that can be used to
determine the physical viability of the theory, and place an experimentally
relevant bound on the single parameter found in it. We compute this spectrum
using the relativistic Dirac equation, and a form of the Born-Infeld potential
that approximates the self-field corrections of the electron. Using these
together, we can establish that if the Born-Infeld nonlinear electrodynamics is
to be physically relevant, it must contain a fundamental constant that is well
below the original value proposed by Born. This work extends the original
Schrodinger spectrum from Carley and Kiessling (2006) [1] for the self-field
correction, and shows that using the Dirac equation introduces minor
corrections - but also gives access to a range for the fundamental constant
that is below that attainable from non-relativistic considerations.
