The more I read on this topic, the less confident I get. If you wish to read a solid technical explanation, I recommend a reply by David Simmons-Duffin sent to me by Henry. It all comes down to Lorentz invariance and unitarity when considering propagators.

In technical tales, if so much as one step is not understood, the path gets lost. I'll point out a few places where I get confused. He writes down the potential between two stationary particles for an arbitrary spin-L force carrier in momentum space:

$\tilde{V}(\vec{k}) = q_1 q_2 P_{\mu 1,...\mu L;\nu 1,...\nu L}(\vec{k}) n^{\mu 1}...n^{\mu L}n^{\nu 1}...n^{\nu L}$

I better just quote the man's description of this beast:

where $P_{\mu 1,...\mu L;\nu 1,...\nu L}(\vec{k})$ is the propagator of the force_carrier, evaluated at a spacelike momentum $\inline k = (0, \vec{k})$ and $\inline n^{\mu}=(1,0,0,0)$ is a unit vector pointing in the time direction.  All these factors of n enter because the spin-L force-carrier couples to a symmetric tensor current, which is given by $\inline q n^{\mu 1}...n^{\mu L} \delta^3(x)$ for a stationary particle. The propagator is traceless in the $\inline \mu$'s and also in the $\inline \nu$'s since it is a two-point function of a traceless symmetric tensor (the force carrier).

Some of that makes sense, some doesn't.

Here is a problem: I don't know how to go from the potential in momentum space to a force in regular spacetime. With the field equations, the switch employed a sign change due to the D'Alembertian (a k2 in momentum space). There is no D'Alembertian in the force equation, so that trick is out, and I don't know the replacement.

While Simmons-Duffin worked out the odd spin case, the even one was skipped. That is too bad for me since it is the case of interest. He does mention the critical role of an odd number of metrics going into the propagator for spin 1. When I calculated the propagator for a tensor field in the previous blog, it too had an odd number of propagators.

He does provide a simple summary table:

Spin even:
q1 q2 > 0 attractive
q1 q2 < 0 repulsive

Spin odd:
q1 q2 < 0 attractive
q1 q2 > 0 repulsive

This looks simple, but upset me the more I thought about it. What was bothering me are the number of charges in the four fundamental forces of Nature.

For gravity, there is only one charge. That is a deep mystery to me. Granted gravity is super weak, but if there were two types of gravity charges roughly balanced in the Universe, there wouldn't be a Universe as we know it. I have a vague memory of a erudite discussion of general relativity where someone said that was the case in GR because only positive values of energy density were plugged in. Should I erase this vague memory, or is there any value in it? So at this point in my physics education, I accept that there is but one sign for the values of mass, but don't know why that is so.

At the other end is the strong force. There is quantum chromodynamics (QCD), with 3 colors and their anti-colors. These can be combined into 8 linearly independent states for the gluons. Due to the phenomena of confinement, we don't get to see any of these colors or gluons. They travel the speed of light all of 10-15m before giving in to the power of confinement. Perhaps the reason there is no "Coulomb-esque's law" for the strong force is there is no simple pair of charges like there is for EM.

The weak force with its non-Abelian SU(2) gauge symmetry looks too complicated to fit into the above model. I don't recall any discussions about the weak force being attractive or repulsive. The weak force is all about decay, decay, decay.

The only fundamental force that would have a q1 q2 product that could be positive or negative looks like EM.

After the exercise of these three posts that relate spin, charge, and attraction/repulsion, I still accept, from authority, that a spin 2 mediating particle is a requirement for a reasonable proposal for gravity. I feel better about the calculation going from spacetime to momentum space, then seeing the values of the polarization states of the transverse spacelike wave.

Doug

Snarky puzzle:
How would a ruling from the United States Supreme Court that mass had to be treated as -m instead of +m effect you personally or professionally?

Google+ hangout: arrange by email if interested.

Next Monday/Tuesday: A New Toy Model for the New Year