quantum field theory comes from starting with a theory of fields, and applying the policies 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 policies 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 a person with a lot of public name recognition. John Preskill wrote a terrific 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 important to modern physics and owes a lot of its present form to Wilson's work. (If you want something more technical, you can do worse than Joe Polchinski's lectures.).

So: quantum field theory comes from starting with a theory of fields, and applying the policies of quantum mechanics. A field is simply a mathematical object that is defined by its value at every point in space and time. (Rather than 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 electricity field) or any other structure. The Higgs boson is a particle related to 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 share 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 almost 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 likelihood) that there is one particle, two particles, whatever. And one such state can progress into another state; e.g., a particle can decay, as when a neutron decays to a proton, electron, and an anti-neutrino. The particles related to 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 tip that this process can happen; the policies of quantum field theory give explicit instructions for associating every such diagram with a number, which we can use to determine the likelihood that this process actually takes place. (Admittedly, it will never happen that boson decays into two bosons of exactly the same type; that would breach energy conservation. But one heavy particle can decay into different, lighter particles. We are just keeping things basic 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, regretfully, doesn't give us the total answer to our question of how typically one particle converts into two; it can be thought of as the first (and hopefully largest) term in a limitless 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 really basic: three lines fulfilling 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 evident 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 needs to 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. As a result, to determine the quantum amplitude related to this diagram, we need to do an important over all the possible ways the momentum can be split up. That's why loop diagrams are normally more difficult to determine, and diagrams with lots of loops are notoriously unpleasant 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 beneficial is if each more challenging diagram gives a successively smaller contribution to the overall result, and undoubtedly that can be the case. (It is the case, for example, in quantum electrodynamics, which is why we can determine things to elegant accuracy in that theory.) Bear in mind that our original vertex came related to a number; that number is just the coupling consistent for our theory, which tells us how strongly the particle is interacting (in this case, with itself). In our more challenging diagrams, the vertex appears numerous times, and the resulting quantum amplitude is proportional to the coupling consistent raised to the power of the number of vertices. So, if the coupling consistent is less than one, that number gets smaller and smaller as the diagrams become a growing number of challenging. In practice, you can typically get really accurate results from just the simplest Feynman diagrams. (In electrodynamics, that's due to the fact that the fine structure consistent is a small number.) When that happens, we claim the theory is "perturbative," due to the fact that we're really doing perturbation theory-- starting with the idea that particles normally just travel along without interacting, then adding basic interactions, then successively more challenging ones. When the coupling consistent is greater than one, the theory is "strongly coupled" or non-perturbative, and we have to be more creative.