# The Gamma Matrices, Spinors, Anti-Commutation, and all that Jazz

• shirosato
In summary, the conversation involves a person struggling to understand fermions in quantum field theory (QFT). They mention reading about spinors, Clifford algebras, and representations of the Lorentz group, but feel that something fundamental is missing in their understanding. They have several questions about spinors, their relationship to Clifford algebras and the Lorentz group, and the deeper meaning of certain mathematical concepts like \gamma_{5}. Additionally, they mention the history of the Dirac equation and its connection to spin and the Lorentz group. The person expresses a desire to study and understand these concepts more deeply, as they are essential to understanding the laws of the universe.
shirosato
So since I learning QFT a while ago, I've always struggled to understand fermions. I can do computations, but I feel at some level, something fundamental is missing in my understanding. The spinors encountered in QFT develop a lot from "objects that transform under the fundamental representation of SU(2) (Pauli matrices)". I've read quite a bit but its pretty jumbled up and I'm at the point where I just want some clear answers if possible, rigorous or not.

I don't have a good math background so explanations with isomorphisms of SO(1,3) = SL(C,2) + SL(C,2), complexification, etc, make sense intuitively, but not enough to be satisfactory. At some point, I feel I'm just going to have to sit down for a few months and hammer the stuff in my brain once and for all, but then again, that would hinder my day-to-day visible productivity (oh, grad school).

Here's a list, feel free to pick and choose if you'd like. Any help is appreciated.

Questions:

- What is a simple and clear definition of a spinor?

- Are the spinors we deal with in physics a particularly special case of the more general spinor? What makes it special?

- Why do we care about Clifford algebras? Are they related to spinors or spacetime geometry in general?

- What is the straightforward relationship between the Gamma Matrices, Clifford Algebras, Representations of Clifford algebras, representations of the Lorentz Group, and anti-commutation?

- What is the deeper meaning of the existence/definition of $$\gamma_{5}$$? In the chiral basis in 4-d, its easy to see how it is the 'chirality operator' acting on a bispinor, but the way books explain it seems glossy. In even dimensions, by definition, it anti-commutes with all the other gamma matrices, but in odd, it is proportional to the identity (so not useful as a chirality operator?).Extra:

- Why do primers tend to make it seem obvious that there are no chiral fermions in 5-D? To me, it seems purely representation theoretic (though I don't understand the details), but they usually go down the path of Fourier expansion, compactification, where I would guess its just a simple, very general: no (0, 1/2) OR (1/2, 0) representation in 5-D.

- What do the four degrees of freedom of a Weyl spinor physically represent? Since the spin is constrained to lie parallel onto the direction of propagation (left or right handed), there is only one spin state. What exactly is oscillating?

- I think a lot of the problem goes back to the way the Dirac equation is somewhat glossed over when you initially learn it, while it holds a LOT of information and subtleties, that if I understand correctly, Dirac himself did not even know all of at the time. From what I learned, he was motivated to make it first-order in time to make it compatible with his "transformation theory" of QM, and by SR, had to also make it first order in its spatial derivatives. He was not trying to add spin (correct me if I'm wrong). To construct a Lorentz invariant wave equation w/ first-order derivatives, one had to insert matrices (the Gamma matrices, representing the 4-D (D= number of matrices?) Clifford algebra) making $$\Psi$$ a four-component object. It was then discovered to be a 4-component bi-spinor, with its left and right-chiral Weyl spinor components manifest in the chiral/Weyl representation. This is a result of the Dirac (1/2,1/2) representation of the Lorentz algebra being reducible to some SU(2)L + SU(2)R representation (not sure about any of this).

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http://math.ucr.edu/home/baez/week93.html" (towards second half of this entry)

I also would love to understand this all much better. A beautiful web of mathematics with deep connections to the fundamental laws of our universe...great stuff and certainly worth to sit down for some time and studying it

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## 1. What are the Gamma Matrices?

The Gamma Matrices are a set of four matrices in a four-dimensional space, commonly used in physics and mathematics to represent the Dirac algebra. They are denoted by the Greek letter gamma (γ) and are defined by their anti-commutation relations.

## 2. What are spinors?

Spinors are mathematical objects that are used to describe the spin of quantum particles. They are represented by vectors or matrices, and their components transform differently under rotations in space compared to other types of vectors. In physics, spinors are used to describe the intrinsic angular momentum of particles, such as electrons.

## 3. What is anti-commutation?

Anti-commutation is a mathematical operation that involves the multiplication of two quantities with the reversal of their order. In the context of gamma matrices and spinors, the anti-commutation relations are defined as {γμ,γν} = 2ημν, where η is the Minkowski metric. This relation is essential in the Dirac algebra and has significant implications in quantum field theory.

## 4. How are gamma matrices and spinors related?

Gamma matrices and spinors are closely related in the sense that spinors are used to represent the action of the gamma matrices on a spinor space. In other words, a spinor is an eigenvector of the gamma matrices, and the eigenvalues correspond to the spin states of a particle. The relationship between gamma matrices and spinors is crucial in quantum mechanics and quantum field theory.

## 5. What is the significance of "all that jazz" in the title?

"All that jazz" is a colloquial expression that is often used to refer to something complicated, confusing, or difficult to understand. In this context, it refers to the complex and abstract nature of the concepts involved in the study of gamma matrices and spinors. The title is meant to convey the idea that these concepts may be challenging to grasp, but they are fundamental in various fields of physics and mathematics.

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