# How to tailor-make fields

Hi folks,

Suppose my brother had experimental reasons to want a field with two charges where like charges attract and unlike charges repel. What kind of mediator would that need? Is there a recipe for cooking up fields to order like that?

I guess the reason they say a graviton has spin 2 is so that it always attracts, but maybe that's wrong. Is it?

I'm not sure why photons mediate a force where unlike charges attract and like charges repel. Is that something to do with the spin as well? Or is it an artifact of the photon and electron spins? Or something completely different?

A. Zee describes it in the first chapter of his book 'QFT in a nutshell'.

OK, I'm struggling through that book and I seem to have fallen at the first hurdle (p. 17-18).

This bit I get: he considers Feynman's argument about a hell of a lot of screens with a hell of a lot of holes in and derives that the amplitude for a particle at (0,0) to be at (x,t) is the integral of the action along all possible paths. So far so good.

Then comes this weird substitution where suddenly there's a particle at every point in some 2D lattice and in place of the old x we have some scalar which might be the vertical displacement of the particle from the lattice plane. First thing I notice is that we seem to have nailed the particles down to a particlar x,y position so they can't run around like they used to. They can just move up and down. That sounds like a severe change of scenario but I guess we can still calculate a Lagrangian with kinetic energy just going vertically and potential energy caused by nearest-neighbour pairs pulling each other to their own vertical positions. That seems to be what he does, and I'm still riding.

Having derived that Lagrangian and fiddled the factors nicely, he then shoves mass squared into it without explanation. He seems to appeal to the KG equation as an example of something invariant, but how come it didn't show up in the derivation above, what is mass after we switched to these phis and sigmas, and what are all those higher factors doing there if its supposed to be the KG equation? In fact, hasn't he just made his previous derivation redundant?

But what bugs me most of all is when he uses this action in a path integral where suddenly the path is integrated along phi, this vertical displacement! I know what he's thinking. He substituted q for phi. But where does that leave the concept of a path? Surely, a path is through space, or spacetime if you like. What on earth does it mean to integrate it along the value of this scalar field? In fact, what happens to Feynman's original motivation for the path integral if we've nailed all the particles down anyway?

The path integral is often initially explained for a single-particle system, where the "paths" are indeed paths of the particle through spacetime. But you can generalize the path integral to arbitrary systems, where the "paths" are through the configuration space of the system. So for a two-particle system each "path" really gives the path of both particles through spacetime. Zee considers a quantum mechanical system of a bunch of particles each constrained to move only in the vertical direction, with some interaction terms. Now each "path" describes the evolution of this whole system over time. Then you take the limit in which the spacing goes to zero, and the "paths" describe possible evolutions of phi, a continuous function of space and time.

OK, I'm struggling through that book and I seem to have fallen at the first hurdle (p. 17-18).

This bit I get: he considers Feynman's argument about a hell of a lot of screens with a hell of a lot of holes in and derives that the amplitude for a particle at (0,0) to be at (x,t) is the integral of the action along all possible paths. So far so good.

Then comes this weird substitution where suddenly there's a particle at every point in some 2D lattice and in place of the old x we have some scalar which might be the vertical displacement of the particle from the lattice plane. First thing I notice is that we seem to have nailed the particles down to a particlar x,y position so they can't run around like they used to. They can just move up and down. That sounds like a severe change of scenario but I guess we can still calculate a Lagrangian with kinetic energy just going vertically and potential energy caused by nearest-neighbour pairs pulling each other to their own vertical positions. That seems to be what he does, and I'm still riding.

Having derived that Lagrangian and fiddled the factors nicely, he then shoves mass squared into it without explanation. He seems to appeal to the KG equation as an example of something invariant, but how come it didn't show up in the derivation above, what is mass after we switched to these phis and sigmas, and what are all those higher factors doing there if its supposed to be the KG equation? In fact, hasn't he just made his previous derivation redundant?

But what bugs me most of all is when he uses this action in a path integral where suddenly the path is integrated along phi, this vertical displacement! I know what he's thinking. He substituted q for phi. But where does that leave the concept of a path? Surely, a path is through space, or spacetime if you like. What on earth does it mean to integrate it along the value of this scalar field? In fact, what happens to Feynman's original motivation for the path integral if we've nailed all the particles down anyway?

Phi is the field itself, not the position of a particle or position in the lattice. Think of it as a vector in the configuration space of the field (or many body system), as opposed to a vector in spacetime. Your "path" is not a path through spacetime, but a path through the configuration space of the field i.e. the infinite dimensional vector space with components associated with displacements at each lattice point.

As the system evolves, the state of the field (represented by phi) moves through this space as each point on the lattice changes in value in a similar way to how the position state of an object moves through a 3D cartesion space.