Except for our differing interpretations of "kinetic energy" which
leads to inconsequential differences in values, JD's answers agree
with mine (given below) and our methods appear to be essentially
identical as well. I conclude, as I expected, that our viewpoints
are operationally equivalent. Furthermore, it seems that we both
agree that one cannot arbitrarily add a constant to the
"interaction energy" term. JD would have that interaction energy
be embodied in the associated fields and I would not disagree. I
note, however, that it can be pretty difficult to determine the
field energy directly (i.e., as an integral over all space of the
field energy density) while it is generally pretty easy to
calculate its value as the potential energy of the configuration
as long as one agrees to set the zero point "correctly"--i.e., at
infinite separation, at least for the gravitational and
electrostatic forces. (The force between quarks, for instance, is
a horse of a different color, so to speak.)
For what it's worth here were my answers:
1. What is the initial momenergy four-vector, P = [E,px,py,pz],
for either one of the two electrons?
P_1_o= [ m, 0, 0, 0]
2. How did you obtain the answer given in 1?
An electron at rest has only its rest mass energy and no
momentum.
3. What is the initial momenergy four-vector, P = [E,px,py,pz],
for the two electron system?
P_sys_o= [ 4m, 0, 0, 0]
4. How did you obtain the answer given in 3?
I used the answer to question 7 and the fact that both energy
and momentum for this isolated system are conserved. (Note
that I cannot add the two momenergy four vectors for the
individual electrons because that does not properly take into
account the interaction energy of the system.)
5. What is the (asymptotically-approached) final momenergy
four-vector for the single electron that is moving in the +x
direction?
P_1_f= [ 2m, sqrt(3)m, 0, 0]
6. How did you obtain the answer given in 5?
The problem statement specifies the final kinetic energy to be
m. Adding the rest mass energy we get an energy of 2m. The y-
and z-components of the momentum are zero because the initial
separation was along the x-axis. The x-component of the
momentum is determined from the requirement that the invariant
mass, (E^2 - p^2)^(1/2), be equal to m.
7. What is the (asymptotically-approached) final momenergy
four-vector for the two electron system?
P_sys_f= [ 4m, 0, 0, 0]
8. How did you obtain the answer given in 7?
I added the momenergy four vectors for the two electrons.
(Note that, in contrast to the situation in problem 1, I *can*
do this now because there is no interaction energy to account
for.)
9. How much of the system energy given in the answer to 3 is
interaction energy?
Initial interaction energy = 2m
10. How did you obtain the answer given in 9?
I subtracted the energies of the two individual electrons in my
answer to problem 1 (m each) from the energy of the system in
my answer to problem 3 (4m). The difference is the result of
the interaction.