Avoiding Integrals with Feynman Diagrams
Feynman diagrams have come up now and again, even within this series. It's a framework for identifying particle interactions so well constructed, it becomes the main tool of computation at the point where they can replace the actual computation of integrals within quantum field theory. However first, they want to be defined.
Feynman diagrams consist chiefly of edges, nodes and loops. Edges describe the travel of particles, or rather the way the particle approaches the nodes, which stand in for particle interaction events. Loops are a little difficult, so I'll define them further down. The simplest Feynman diagrams then consist of only edges and nodes. The smallest process then would generally look like this:
We describe such a combination of edges connected to a single node as a vertex for simplicity. Within particle physics there are distinctions made between the different elementary particles, how they interact with others and how they decay. Generally, the elementary particles decompose into groups of fermions and bosons, the fermions decompose into (anti-)quarks and (anti-)leptons, while the bosons decompose into three kinds of gauge bosons and the higgs boson (which is the only scalar boson). We'll go over the particles before placing them into the Feynman diagrams. At the same time, we should keep in mind that combinations of these particles make new particles.
Quarks also carry a colour charge. This is a construction to accommodate anti-symmetric quark wave functions that had always had an extra degree of freedom compared to the symmetric ones. It features particles carrying three "colour" "charges" (red, blue, green) each with their own anti-color (anti-red, etc.). Their interactions across quarks/gluons open the field of quantum-chromodynamics (QCD). Without getting too far into it, colourlessness (that is, a net charge of zero) is a conserved quantity.
In terms of Feynman diagrams, we're really mainly interested in the different types of vertices they give. They are the "elementary building blocks" of particle interactions, if you will, and are their mathematical representations are given by the Feynman propagator.
To check which Feynman diagrams are even possible, one only has to check what restraints are usually imposed upon normal interactions. There are of course all the conserved quantities. The Feynman diagram is describing a closed system, but there's some of the usual suspects that there's no good way to check. There's no speeds or momenta noted, safe for their directions, and even that we should put a pin in for later. Instead, we take the conserved quantities of particle/nuclear physics. These are mostly quantum numbers, but also charges, so the majority of the problems are reduced to counting. Here's a list of what conserved quantities we can use here: energy [E] & mass [m] (or rather, 4-momentum [P]), spin [s], charge [q], colour [c].
As the 4-momentum is conserved, this is the chief point of complex calculation done when checking the validity of Feynman diagrams. There are some oddities that come from that, namely the antiparticles moving backward in time. This is because on it's face, the Feynman diagram has to treat particles the same as their anti-particle counterparts, whereas the mathematical approach separates particle and antiparticle contributions. Setting their "time-direction" to be backwards, happens to coincide nicely with the separate description of anti-particles with "negative" energy (or rather, energy states that annihilate with their counterparts).
Note the usage of photons, W/Z bosons and gluons as an intermediary in some of the Feynman diagrams. Sometimes, it's not plausible for a particle to decay directly into its end products, because of unresolved interactions. These are either electromagnetic (photon), weak (W/Z) or strong (gluon). They are also often byproducts of convenient energy. These are "gauge bosons" and on a mechanical level, their only function is the balancing of these scales, so to say. These will show up as interaction constants in the mathematical representation of the diagram.
This has been the very basics in particle physics, and to continue, we will have to do some quantum field theory. I'll skip a lot of it, because what I want to talk about is pretty far in there, and this piece is starting to look a little long already. We'll take as given that the propagator is a Green's function, and I'll say a few words on how things are constructed as we go along. When we refer to the propagator as a Green's function, then we have to express that a particle travels from position x to position y, and wrap it nicely, so that it behaves. This tells us that the expression better be grounded and that the order of spots really matters. We are also still doing quantum field theory, so at each spot we want to evaluate the Heisenberg field. To ground the expression, there's no better ground than the vacuum state, and to correctly order the spots the particle visits, we introduce a time-ordering operator. That will have to do for now, I've got other pieces going more into depth on here somewhere.
We should know the procedure to transfer the Heisenberg field operator from one representation to another using U(t, t') pretty well from quantum physics. Apply it to all the Heisenberg fields, then almost all of these matrices cancel to one and the Green's function comes out at least manageable, if not pretty.
The advantage of this form is that the operator has been thoroughly deconstructed until it's just the fields. To generalize the case, the expression shouldn't be explicitly time-dependent, so we take a limit to infinities in both directions. So far the expression carried the vacuum state, but that state can be associated with the energy-ground state using the time evolution operator. Using a few definitions the generalized Green's function comes out as follows.
This is the Gell-Mann-Low formula and writing it down makes me very anxious. It gets wilder though, but first a word on the vacuum states of the numerator. The divisor can be identified with the sum of all vacuum loops, meaning, the numerator of the Gell-Mann-Low formula is called the "full Green's function". We'll mark this with an F in the index. If we were to recover the operator representation, then the vacuum state would have to be identified into an in-vacuum (on the left) and an out-vacuum (on the right). They're still defined through the limits, but for future reference, these are the Moller operators.
The expressions are about to get really long, so there's merit in defining a few recurring terms. First is the generating functional Z. It can be defined identically for the full Green's function.
This is actually a really important form, because this is where one can actually read off Feynman rules, after some transformations. The time-ordering operator often gets dissolved through application of Wick's theorem
Where Δ(x - y) is the Feynman propagator. Unfortunately there's not a lot we can do to circumvent this part of the computation. Once this form has been derived, one needs to check what kind of Feynman diagram one is actually interested in, more specifically the order of the diagram. In regular quantum field theory, each order adds a factor in the interaction constant, usually denoted as g, meaning that each order of the Feynman integral is an internal point of the Feynman diagram. You'll notice those as the dots in the examples. These are the interaction points, and each kind of interaction has its own Lagrangian to arrive at the Green's function at. The amount of propagators connected to each internal point is the deciding factor in this. We refer to these distinctions as n-point interaction functions, and in the Lagrangian, they usually show up as the highest order of the potential. To compute the n-point interaction, one needs all derivatives of the generating functional with respect to the current J up to the n-th order, then identify the statistic factor of how many ways that diagram can be constructed, as well as a possible factor of -1 for diagrams with an odd number of propagators.
Of course, once this method is practiced, the Green's function is probably the better way of arriving to a solution, but for I find that trying to properly get to the point where writing Feynman diagrams instead of Green's functions primes the user for learning them anyways.