  The relentless efforts of the physics community has not yet availed us the
solution of how to unify the Quantum Mechanics with General Relativity, a
puzzle that has engaged the minds of the physicists for almost a century. The
insufficiency of today's and foreseeable future's technology for a direct reach
into the Planck energies at which the fundamental theory, the Quantum Theory of
Gravity, lies has lead to the search of the low energy effects of that
fundamental Planck level theory irregardless of the details of it. In this
thesis, one of the leading candidates of such an exotic effect, that is the
violation of Lorentz and CPT symmetries is analyzed. The action level effective
field theoretical framework for such an analysis called Standard Model
Extension has already been in the literature for the last two decades; here,
the nonminimal photon sector of such a framework is examined from a quantum
field theoretical point of view. All possible polarization vectors for
different kinds of CPT violations, the generic forms of the dispersion
relations that these polarization vectors satisfy, and the corresponding
propagators of the photon field are explicitly calculated. Special models of
Lorentz violations are introduced, and a particular one called vacuum
orthogonal model is analyzed extensively. It is found that this particular form
of Lorentz violation cannot induce any effects that is detectable in the vacuum
propagation of the photon. Isotropic and leading order cases of the Lorentz
violation is studied and this found result is explicitly shown.
