The first theme of this chapter is generalizing the idea of potential energy to the relativistic case. The track we pursue assumes that the potential energy U which arises in non-relativistic mechanics has a corresponding potential momentum Q in relativity. The potential 4-momentum is a four-vector combining these two components: q = (Q,U/c). The total four-momentum pi = q + p, where p is the ordinary, or kinetic four-momentum. Breaking into timelike and spacelike parts, we have E = U + m*c^2*gamma and Pi = Q + m*v*gamma. Recall that gamma = 1/(1 - v^2/c^2)^0.5 and that m*v*gamma is the kinetic momentum P.
The potential momentum Q doesn't change the free particle relationship between total energy, potential energy, and kinetic momentum: (E - U)^2 = P^2*c^2 + m^2*c^4. The only way potential momentum affects particles is via its effect on quantum mechanical wavelength. Wavelength is related to Pi, velocity is related to P. Thus, we get funny interference effects, as in the Aharonov-Bohm effect. We also see that the refraction of a particle in a potential momentum gradient represents a force on the particle which is proportional to the particle speed and perpendicular to its velocity. There is a final effect having to do with time-variable potential momentum which produces an additional force.
You don't have to derive this stuff, but you should be generally familiar with how it arises. You should also be able to use the derived relativistic force formula, equation (14.20), and the related equation (14.17). You should also understand how to use the Lorentz condition, equation (14.21), which relates the different components of the potential four-momentum, even if you don't understand where it comes from just yet. It wouldn't hurt to review 4-vectors (section 5.2) and the spacetime Pythagorean theorem (section 4.3), as well as relativity generally.
The second theme of this chapter is how force physically works in quantum relativity, as the transfer of momentum and energy by virtual intermediary particles. Momentum and energy are conserved when a virtual particle is emitted or absorbed by a real particle. A consequence of this is that the mass of a virtual particle is generally different from its real mass. This difference limits the lifetime (if the particle's worldline is timelike) or the range (if it is spacelike) via the mass-proper time version of the uncertainty principle (see section 7.6).
Feynman's idea that particles with negative energy are ``going back in time'' allows us to treat the emission of virtual particles with a spacelike worldline in a relativistically consistent way -- in other words, for purposes of ``before'' and ``after'' in 4-momentum conservation at a vertex, ``before'' means the arrow is pointing into the vertex, whereas ``after'' means the arrow is pointing away from the vertex, irrespective of whether the arrow points forward or backward in time.
Feynman's view also interprets a particle with energy -E and momentum p as an antiparticle with energy E and momentum -p. This corresponds to changing the direction of the arrow on the world line of the particle in a spacetime diagram.