  It is commonly assumed that zero and non-zero photon mass would lead to
qualitatively different physics. For example, massless photon has two
polarization degrees of freedom, while massive photon at least three. This
feature seems counter-intuitive. In this paper we will show that if we change
propagator by setting $i \epsilon$ (needed to avoid poles) to a finite value,
and also introduce it in a way that breaks Lawrentz symmetry, then we would
obtain the continuous transition we desire once the speed of the photons is
"large enough" with respect to "preferred" frame. The two transverse
polarization degrees of freedom will be long lived, while longitudinal will be
short lived. Their lifetime will be near-zero if $m \ll \sqrt{\epsilon}$, which
is where the properties of two circular polarizations arize. The $i \epsilon$
corresponds to the intensity of Mensky's "continuous measurement" and the short
lifetime of the longitudinal photons can be understood as the "conversion" of
quantum degrees of freedom (photons) into "classical" ones by the measurement
device (thus getting rid of the former). While the "classical" trajectory of
the longitudinal photons does arize, it plays no physical role due to quantum
Zeno effect: intuitively, it is similar to an electron being kept at a ground
state due to continuous measurement.
