Last week we looked at particle scattering.
We looked at a B and C particle approaching one another, interacting and moving on. We found that the single interaction term gave zero. So we moved on to the interaction squared term.
< bc | gABC gABC | bc >
When we expanded the fields in terms of creation and annihilation operators, we found that there was only one combination that was significant.
< bc | gABC gABC | bc >
= < bc | (gA-B+C+) (gA+B-C-) | bc >
= g²
The right hand interaction term, (gA+B-C-), annihilated the B and C particles, and created an A particle in their place. The left hand interaction term, (gA-B+C+), then reversed this by annihilating the A particle and recreating the B and C particles.
In a more realistic theory, such as Quantum Electro Dynamics (QED), this would represent the interaction of two charged particles via the exchange of a virtual photon. Once you’ve done this calculation once, say for two electrons, you find that similar configurations, such as two muons, are almost identical. It becomes possible to construct a set of rules for any possible set of interactions. These are called “Feynman Rules”. They bypass the whole need to expand fields in terms of creation and annihilation operators, rearranging them to make terms vanish, and keeping track of any new terms that arise because of this rearrangement.
(The Feynman rules for QED can be found here. https://particle.phys.uvic.ca/~jalbert/424/lecture16r.pdf)
Let’s look at the Feynman way of doing the two examples we’ve already done. First, an A particle decaying into a B and C particle. The probability we want to calculate is:
< bc | gABC | a >
Feynman now tells us to draw a diagram with one line for each initial particle and one line for each final particle.
The type of line usually denotes the type of particle. Photons often get a wavy line and things like electrons usually get a solid line with an arrow. I’m using dashed lines for A particles and solid for B and C. In a real theory, we would also label each particle with it’s own momentum. We would also associate a mathematical object appropriate to each particle. For things like electrons it would be a four component Dirac spinor. For photons it would be a polarisation vector. However, in our toy theory we ignore all of that and just draw the lines.
Connect the incoming and outgoing lines with an interaction vertex. In our model there is only one type of interaction, gABC. It creates a single type of vertex where all three fields connect with one another. So there’s really only one way to connect the three fields.
For each vertex, associated a factor of g, the coupling constant.
Now multiply all the spinors, vectors and constants that we’ve created. In our case, as our model is so simple, we only have the vertex coupling constant, g. The result is our answer.
< bc | gABC | a > = g
Now, isn’t that a lot more fun than the way we did it last time!
This is the important thing about Feynman diagrams. They’re not just pictures. Each element provides a factor in a formula that tells you how to calculate a probability. And it’s a much easier way to do QFT. This is what Julian Schwinger meant when he said that “Feynman brought quantum field theory to the masses.”
Let’s try our second example, BC to BC scattering. First we draw our incoming and outgoing particles. As before, because we’re ignoring details, we don’t associate momenta or any dynamic objects with these lines.
Next, connect the lines with interaction vertices. The only interaction we have is the three field gABC interaction. The rules for our model is that each vertex introduces a factor of g.
Now connect them together and multiply all the factors we’ve accumulated, in our case it’s just the two g’s.
< bc | gABC gABC | bc > = g²
The internal “A” line would also get a factor in most theories. In our case it’s just a constant so I’ve ignored it. The A particle plays a curious role here. It’s called a “virtual” particle.
In this case the A particle appeared because we only had one type of interaction between particles. Every vertex has to have all three particles in it, even if some of the particles are missing in the initial and final states. But these virtual particles are never observed. Note also that, whatever the energy and momentum of the initial state particles, the A particle must carry these to the final state. Now all of the incoming and outgoing particles must obey the rules of special relativity that relate energy, momentum and mass. But the virtual A particle does not!
This is why many physicists really don’t like calling it a particle at all. See for example Matt Strassler’s description of them.
Whether you choose to call it a particle, either virtual or not, the maths remains the same. And once again, remarkably, this bizarre mathematical model always gives the right answers.
But this barely scratches the surface of what QFT can do. We’ve looked at first order (particle decay) processes, and second order (particle scattering process). You can keep on going. Look at this Feynman diagram.
Here, the virtual A particle decays into a B and C particle and then recombines again. We now have 4 vertices, so 4 factors of g. This works provided g is small. When g is small you can keep on adding loops to your heart’s content. The increasing factors of g become smaller and smaller and just provide an answer to a higher degree of precision. When g isn’t small though, as is the case with the strong nuclear force, this doesn’t work
There’s also a hidden calculation nightmare here. The virtual A particle has limits on the energy and momentum it can carry. It must conserve the original and final B and C values. However, the intermediate B and C particles can carry opposite momenta without any limit. These “loop” diagrams can create infinities that have to be massaged out of the theory again using the techniques of renormalization. Which, frankly, I don’t understand.
And that brings me to the end of this series of posts on QFT. We’ve looked at how Quantum Mechanics defines states, operators and probabilities. We’ve seen how to extend this model in Quantum Field Theory to include multiple objects. We’ve shown how creation and annihilation operators allow us to move between these states. We’ve defined quantum fields to be the sum of all possible creation and annihilation operators for a particular type of object, and have defined interactions as products of those fields. And finally we’ve shown how those interactions can be used to calculate particle lifetimes and scattering amplitudes.
Along the way, we’ve seen how states evolve through time thanks to the Schrodinger Equation. We’ve seen how spin affects particle statistics and produces the periodic table. We’ve diverted into group theory and gauge theory to see how the interaction terms, effectively the forces of nature, arise in particle physics.
If you’ve been following all this, then congratulations. You now know as much about QFT as I do.
Platitude of the Day 