"In particular, if we define "mass" the way thoughtful experts
do (and have done for many decades), then photons are massless
and it is trivial to come up with examples where the center of
mass hops around, even in the absence of externally-applied
forces. Just convert massive electron-positron pairs to
massless photons.
When somebody starts making center-of-mass arguments involving
the mass of photon, it's hard to take the argument seriously."
In this respect I agree with Sciama, not with John. The photons
are massless - in the sense that they have zero rest mass;
however, they have a non-zero relativistic mass m. If you reverse
annihilation process, it is non-zero relativistic mass of each photon,
from which the REST mass of emerging electron-positron pair buils up.
Moreover, there are situations when a single photon in free space
has A NON-ZERO REST MASS! Such a situation can be created in a quantum
optics experiement when a single photon is passed through a
beam-splitter and no attempt is made to locate path chosen by the
passed photon; in this case the meaningful characteristic of the photon
is its average momentum, which is, in this state, NOT equal to E/c.
Accordingly, its rest mass is not equal to zero in such a state.
Also, the concept of the center of mass has, at least for a system of
non-interacting particles, rigorous definition in relativistic domain
(See, e.g., Landau & Lifshits, The Field Theory). The center of mass
of a relativistic system does not hop around if the system is not subject
to external forces and is observed from one fixed inertial RF; I would
like to learn of an example to the opposite.