Hello Hanson:
Sorry for the delay in replying... [about]

Anyway.

give some examples of your calc that do occur in, or do
reflect our real world applications. (mech/dynamics/electr)
I ask this because the EM force carrier is supposed to be of
a spin 1 nature, whereas the gravitational force mediator is
supposed to be a spin 2 entity.

You are correct, the photon is spin 1, and the graviton which we won't
be detecting any time soon is spin 2. A fundamental property of EM is
that like charges repel. That is consistent with the force mediating
particle being spin 1. Likewise, in gravity like charges attract, so
the force mediating particle must be spin 2. Brian Hatfield gave a
good explanation of this in the introduction to "The Feynman Lectures
on Gravity". Force mediating particles must have integral spin. To
always act one way, the gravity mediating particle must be even spin.
Since light is bent by a gravity field, a spin 0 particle will not
work (anyone want to provide the reason, I've seen that written a few
times, but am not clear on the logic). Ergo the simplest particle
would be spin 2.

In EM where like charges repel, the photon must be odd spin. I don't
know that I have ever heard a spin 3 particle discussed, but a photon
is spin 1.

A Lagrange density describes all the ways a system can trade energy
per unit volume. Integrate a Lagrangian over space and an arbitrary
amount of time. If you can find something that can be varied without
changing the integral, that is a conserved quantity for the action.

From the Lagrangian, one can crank out the force equations by varying

the 4-velocity keeping the 4-potential fixed, or the field equations
by varying the 4-potential and keeping the 4-velocity fixed.

In the real world of EM, how can we look at the Lagrangian and tell
that like charges repel and the force mediating particle is spin 1?
Here is the Lagrangian:

LEM = - rho_m/gamma (1)
- Jq^u A_u/c (2)
- 1/4c^2 (d_u A_v - d_v A_u)(d^u A^v - d^v A^u) (3)

To generate the Lorentz force equation, only (1) and (2) matter since
they are the ones that have a 4-velocity inside. Because they have
the same sign, the resulting force law will indicate that like charges
repel.

To generate the Maxwell field equations, only (2) and (3) matter since
they have the 4-potentials. Here again, because they have the same
sign, like charges repel.

There are 2 ways you can spot the spin 1 force mediating particle.
The first way focuses on (3). If one switches the order of the
indexes on the antisymmetric field strength tensor, the sign of the
field strength tensor will flip. That is a signal that the spin is
odd.

What I did not understand was how to look at term (2) and pick out a
spin 1 field. Section 3.2 does a great job of it. The idea is
transform the charge couping term into a charge-charge interaction.
Look at the product of a charge-charge interaction, and focus on the
phase. That takes 2 pi to get back to where it started as expect for
a spin 1 particle.

So to review, there are 4 reasons while like charges repel in the LEM
Lagrangian: the force equation (1&2), the field equation (2&3), the
spin 1 particle in the antisymmetric field strength rank 2 tensor (3),
and the spin 1 particle in the current-current interaction (2).
Nothing like logical consistency!

If you wanted to make a Lagrangian that could do the work of gravity,
all four of these chips must stack up. Here is a Lagrangian I play
with:

LG = - rho_m/gamma (4)
+ Jm^u A_u/c (5)
- 1/4c^2 (d_u A_v + d_v A_u)(d^u A^v + d^v A^u) (6)

The force equation will have like charges attract because the signs of
(4) and (5) are different.

The field equations will have like charges attract because the signs
of
(5) and (6) are different. These two are easy and well known.

Look at (6), and you seen an indication of an even spin field because
if the indexes are changed, the sign does not change. Because there
are indexes, the second rank field strength tensor cannot be spin 0.
The trace of this tensor would make a spin 0 field. Effectively there
will always be spin 0 field associated with this higher spin field.
Cool. If the trace is zero, then the particle characterized by (6)
will travel at the speed of light. If not, the particle will have a
non-zero mass.

In Misner, Thorne, & Wheeler, problem 7.2, they consider an
antisymmetric tensor for (6) which cannot work because it would
indicate an odd spin mediating particle, and thus not self-consistent.

The memorial day weekend calculation was about term (5). One has to
be able to spot a spin 2 particle in the current-current interaction.
That will have a phase that looks like 2 jx jy, so that in pi radians
it will get back to where it started. I used a different kind of
conjugate that many physicists are not aware of, basically one that
tosses in a pair of basis vectors.

So to review, there are 4 reasons while like charges attract in the LG
Lagrangian: the force equation (4&5), the field equation (5&6), the
spin 2 particle in the symmetric rank 2 field strength tensor (6), and
the spin 2 particle in the current-current interaction (5).

Steve was correct to complain about my lack of understanding about the
current-current/spin 2 issue. Hopefully I have made progress on it.

doug