quantum field theory comes from starting with a theory of fields, and applying the rules of quantum mechanics

http://www.youtube.com/watch?v=ywZ5_YfwihI
quantum field theory comes from starting with a theory of fields, and applying the rules of quantum mechanics
Ken Wilson, Nobel Laureate and deep thinker about quantum field theory, died last week. He was a true giant of theoretical physics, although not someone with a lot of public name recognition. John Preskill wrote a great post about Wilson's achievements, to which there's not much I can add. But it might be fun to just do a general discussion of the idea of "effective field theory," which is crucial to modern physics and owes a lot of its present form to Wilson's work. (If you want something more technical, you could do worse than Joe Polchinski's lectures.).

So: quantum field theory comes from starting with a theory of fields, and applying the rules of quantum mechanics. A field is simply a mathematical object that is defined by its value at every point in space and time. (As opposed to a particle, which has one position and no reality anywhere else.) For simplicity let's think about a "scalar" field, which is one that simply has a value, rather than also having a direction (like the electric field) or any other structure. The Higgs boson is a particle associated with a scalar field. Following the example of every quantum field theory textbook ever written, let's denote our scalar field.

What happens when you do quantum mechanics to such a field? Remarkably, it turns into a collection of particles. That is, we can express the quantum state of the field as a superposition of different possibilities: no particles, one particle (with certain momentum), two particles, etc. (The collection of all these possibilities is known as "Fock space.") It's much like an electron orbiting an atomic nucleus, which classically could be anywhere, but in quantum mechanics takes on certain discrete energy levels. Classically the field has a value everywhere, but quantum-mechanically the field can be thought of as a way of keeping track an arbitrary collection of particles, including their appearance and disappearance and interaction.

So one way of describing what the field does is to talk about these particle interactions. That's where Feynman diagrams come in. The quantum field describes the amplitude (which we would square to get the probability) that there is one particle, two particles, whatever. And one such state can evolve into another state; e.g., a particle can decay, as when a neutron decays to a proton, electron, and an anti-neutrino. The particles associated with our scalar field will be spinless bosons, like the Higgs. So we might be interested, for example, in a process by which one boson decays into two bosons. That's represented by this Feynman diagram:.

3pointvertex.

Think of the picture, with time running left to right, as representing one particle converting into two. Crucially, it's not simply a reminder that this process can happen; the rules of quantum field theory give explicit instructions for associating every such diagram with a number, which we can use to calculate the probability that this process actually occurs. (Admittedly, it will never happen that one boson decays into two bosons of exactly the same type; that would violate energy conservation. But one heavy particle can decay into different, lighter particles. We are just keeping things simple by only working with one kind of particle in our examples.) Note also that we can rotate the legs of the diagram in different ways to get other allowed processes, like two particles combining into one.

This diagram, sadly, doesn't give us the complete answer to our question of how often one particle converts into two; it can be thought of as the first (and hopefully largest) term in an infinite series expansion. But the whole expansion can be built up in terms of Feynman diagrams, and each diagram can be constructed by starting with the basic "vertices" like the picture just shown and gluing them together in different ways. The vertex in this case is very simple: three lines meeting at a point. We can take three such vertices and glue them together to make a different diagram, but still with one particle coming in and two coming out.


This is called a "loop diagram," for what are hopefully obvious reasons. The lines inside the diagram, which move around the loop rather than entering or exiting at the left and right, correspond to virtual particles (or, even better, quantum fluctuations in the underlying field).

At each vertex, momentum is conserved; the momentum coming in from the left must equal the momentum going out toward the right. In a loop diagram, unlike the single vertex, that leaves us with some ambiguity; different amounts of momentum can move along the lower part of the loop vs. the upper part, as long as they all recombine at the end to give the same answer we started with. Therefore, to calculate the quantum amplitude associated with this diagram, we need to do an integral over all the possible ways the momentum can be split up. That's why loop diagrams are generally more difficult to calculate, and diagrams with many loops are notoriously nasty beasts.

This process never ends; here is a two-loop diagram constructed from five copies of our basic vertex:.


The only reason this procedure might be useful is if each more complicated diagram gives a successively smaller contribution to the overall result, and indeed that can be the case. (It is the case, for example, in quantum electrodynamics, which is why we can calculate things to exquisite accuracy in that theory.) Remember that our original vertex came associated with a number; that number is just the coupling constant for our theory, which tells us how strongly the particle is interacting (in this case, with itself). In our more complicated diagrams, the vertex appears multiple times, and the resulting quantum amplitude is proportional to the coupling constant raised to the power of the number of vertices. So, if the coupling constant is less than one, that number gets smaller and smaller as the diagrams become more and more complicated. In practice, you can often get very accurate results from just the simplest Feynman diagrams. (In electrodynamics, that's because the fine structure constant is a small number.) When that happens, we say the theory is "perturbative," because we're really doing perturbation theory-- starting with the idea that particles usually just travel along without interacting, then adding simple interactions, then successively more complicated ones. When the coupling constant is greater than one, the theory is "strongly coupled" or non-perturbative, and we have to be more clever.