Last week we looked at “Creation and Annihilation Operators” (CAOs).
We saw how they acted on Probability Distributions to change the probability of the number of objects. We also saw how to build fields from CAOs.
We’re now ready to explore a toy particle physics theory. In this toy world there are only three types of particles: A, B and C. The only thing we concern ourselves about is whether they exist or don’t exist. We don’t concern ourselves with their mass, energy, spin, charge, or a whole host of other properties.
Each particle type has a field associated with it. Each field consists of a single creation and annihilation operator.
A = A+ + A-
B = B+ + B-
C = C+ + C-
The interaction of the three particles is governed by an “interaction potential” gABC. I need to explain what this is.
The interaction potential, gABC, is the thing that governs interactions between the three types of particles. In the Standard Model (SM) you can’t just have any old interaction. Gauge Theory, the theory that examines the effects of symmetries on the fields, creates specific types of interactions, and these interactions are the only ones allowed in the SM.
As an example, we begin with matter fields, like the one I showed last week for the electron. There’s a simple rotation that you can perform on that field that keeps important properties the same. This is called a U(1) transformation. I’ll discuss this more in the comments. (It also defines a conserved quantity, called electric charge.) When you perform this U(1) transformation by different amounts at different places, you find that you have to introduce a new, massless, field, called a “gauge field”. It becomes immediately obvious that the new field obeys Maxwell’s equations and is clearly the electromagnetic potential. As well as introducing the gauge field, the transformation also introduces an interaction term between the matter field and the gauge field.
Personally, I find this one of the most stunning aspects of the SM. The fact that we can explain the existence of electromagnetism on the basis of something as simple as a rotation, and then go on to explore its interactions to any level of precision we require, is an absolutely beautiful result. To actually see Maxwell’s equations, the bedrock of every course on electromagnetism taught over the last 150 years, emerge as if by magic, is a wonder to behold. First time I saw it I was on a high for days afterwards.
All the interaction terms in the SM arise this way. In our toy theory I’ve just invented the interaction potential. But it’s very similar to the one you get from U(1) and electromagnetism.
“g” is a “coupling constant” that tells us how strong the interaction is. In electromagnetism, it’s the charge of the electron. We’re going to have to discover this experimentally.
ABC is just the product of the three particle fields. It tells us what an interaction between the three types of particles looks like.
But I still have to explain what the “product of the three particle fields” means. Lets take the simpler product BC first. To calculate this we just treat the CAOs in the fields as if they were numbers.
BC = (B+ + B-)(C+ + C-)
And just apply the distributive law of multiplication as we would with any numbers.
BC = B+ (C+ + C-) + B- (C+ + C-)
= (B+C+) + (B+C-) + (B-C+) + (B-C-)
As we’ve done all along with operators, B+C+ means apply C+ first to whatever is to its right, then apply B+ to the result.
e.g. B+C+ |0>
= B+ |c>
= |bc>
To get the full interaction potential we have to multiply BC by A.
ABC
= (A+ + A-) (B+C+ + B+C- + B-C+ + B-C-)
= (A+) (B+C+ + B+C- + B-C+ + B-C-)
+ (A-) (B+C+ + B+C- + B-C+ + B-C-)
= A+B+C+
+A+B+C-
+A+B-C+
+A+B-C-
+A-B+C+
+A-B+C-
+A-B-C+
+A-B-C-
You basically end up with all possible combinations of creation and annihilation operators.
If you think this looks horrendous, then you’re quite right. The real Standard Model is much, much worse. In fact it’s infinitely worse, literally. However, as you’ll see, in any given situation, most terms just give zero.
For example, suppose we have a state with a single “A” particle in it. Call this |a>. Now suppose we want to calculate how the interaction potential affects this state. We want
gABC |a>
To calculate this we simply expand out the product ABC just as we did before, apply each term to |a>, and then add the results together. The first term in the ABC expansion gives this.
A+B+C+ |a>
Now apply each operator, one by one, to the state on its right, e.g. the first operator gives
C+|a> = |ac>
Apllying the full term
A+B+C+ |a>
= A+B+ |ac>
= A+ |abc>
= |aabc>
The next term is interesting.
A+B+C- |a>
There is a C annihilation operator acting on a state with no C particles in it. Remembering our rules from last week, this gives zero.
A+B+C- |a>
= A+B+ . 0
= 0
The same is true with the next term.
A+B-C+ |a>
= A+B- |ac>
= A+ . 0
= 0
In fact, any of the terms with a B- or C- in it will give zero when acting on |a>. There are only two terms, out of the initial eight terms, that give a non-zero result.
A+B+C+ |a> = |aabc>
A-B+C+ |a> = |bc>
This is going to be extremely useful to us next week, when we’re going to perform our first Quantum Field Theory calculation. We’re going to calculate the probability that an A particle will decay into a B and C particle.
Platitude of the Day 