Gamma Matrices and the Clifford Algebra

Transcript

While coming up with the Dirac equation, Paul Dirac noticed that some coefficients in his equation have to be matrices for the whole thing to work. But not any matrices, those matrices used in the Dirac equation have to fulfill certain rules, which are collected in the so-called

Clifford Algebra anticommutation rules. An anticommutator between A and B, denoted with curly brackets, is a shorthand notation for AB+BA. So the anticommutator between two of Dirac‘s matrices has to be equal to ...

something. There are four gamma matrices, gamma^0, gamma^1, gamma^2 and gamma^3, which can be summarized into gamma^mu. This looks like a four-vector, however remember that when we choose, say, mu=2, then the resulting element is a matrix, not a scalar number.

In fact, they are 4 by 4 matrices. To be more general, for D spacetime dimensions, they are 2^\D/2/ by 2^\D/2/ matrices, where the part in red is the floor function. So for our 3 spatial and 1 temporal dimensions, D=4, and 2^4/2 is equal to 4.

But if we consider only one space dimension and one time dimension, which is usually the case for the strings in string theory, the gamma matrices are 2 by 2 matrices. So inside our anticommutator, we put the gamma matrices.

Since we don’t want to write many different relations, for instance gamma^1 with gamma^2, gamma^3 with gamma^0, and so on, we choose a placeholder for both. So we write gamma^mu and gamma^nu. And what‘s on the right-hand side? In order for his equations to work, Paul Dirac

saw that on the right side, there has to be 2 times the Minkowski metric times the unit matrix. Since the metric looks like this, the anticommutator is very often zero, in particular, exactly when mu and nu are different. So gamma^0 with gamma^3? Zero.

Gamma^2 with gamma^1? Zero. And gamma^3 with gamma^3? This is two times minus one times the identity matrix. So minus two times the identity matrix. The useful properties are the following: First,

if you take two different gamma matrices and switch their place, you pick up a minus sign. This follows from the anticommutators like this one here. Second, if you square a gamma matrix, depending on if it‘s the zero one or another, you get plus or minus the identity matrix.

This follows from the anticommutators here. There we have it, using this anticommutation relation you will be able to solve most of the problems you encounter. Aaactually, there is one more gamma matrix, called gamma^5.

You can calculate it either as a product of all other gamma matrices, or via the Levi-Civita epsilon symbol. The interesting thing about this matrix is, that its anticommutators with all other gamma matrices are zero! Did you notice? We did not say anything about how the matrices really look like.

We only know that they are 4x4 matrices, but apart from that, we know nothing. This is the beauty — and the curse — of representations. But that‘s enough stuff for a different

video. So that‘s pretty much it for this time, thanks for watching!