What is the relationship between the Dirac algebra and the Clifford algebra?

Question

While I'm still trying to understand the issues raised on my previous question, I decided to first address the Clifford algebra used on formulating the famous Dirac equation. In this context, what is found in physics books is the following. There are four $n\times n$ matrices (let's take $n=4$ for simplicity) $\gamma^{\mu}$, $\mu = 0,1,2,3$ satisfying: $$\{\gamma^{\mu},\gamma^{\nu}\} := \gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu} = 2g_{\mu\nu}I$$ where $g = \mbox{diag}(1,-1,-1,-1)$ is the Minkowski metric. These relations should define a Clifford algebra. Moreover, a copy of $\mathbb{C}^{4}$ on which the Dirac matrices $\gamma^{\mu}$ act is called a spinor space, and its elements are called (Dirac) spinors.

As I mentioned in my previous question, the mathematical definition (I know) of a Clifford algebra is the following.

Definition: Let $V$ be a $\mathbb{K}$-vector space, $\varphi: V \times V \to \mathbb{K}$ a symmetric bilinear map and $\Phi: V \to \mathbb{K}$ the quadratic form associated to $\varphi$. A Clifford algebra $\mathcal{Cl}(V, \Phi)$ associated to $V$ and $\Phi$ is a $\mathbb{K}$-associative algebra with unit together with a linear map $i_{\Phi}:V \to \mathcal{Cl}(V,\Phi)$ such that:

(a) $(i_{\Phi}(v))^{2}=\Phi(v)\cdot 1$, $\forall v \in V$,

(b) (Universal Property) For every $\mathbb{K}$-algebra $A$ and every linear map $f: V \to A$ such that $(f(v))^{2}=\Phi(v)\cdot 1_{A}$ ($\forall v \in V$), there exists a unique $\mathbb{K}$-homomorphism $\bar{f}: \mathcal{Cl}(V,\Phi)\to A$ such that $f = \bar{f}\circ i_{\Phi}$.

Property (a) can be rephrased in an equivalent form: $$i_{\Phi}(v)i_{\Phi}(w) + i_{\Phi}(w)i_{\Phi}(v) = 2\varphi(v,w)\cdot 1$$

I'm trying to relate the physicist approach to the above definition. When it comes to Clifford algebras, there are a lot of information out in the internet and it is really difficult to focus on what's important if you have no background on the subject. According to Wikipedia, the Dirac algebra should be $\mathcal{Cl}_{4}(\mathbb{C})$ or $\mathcal{Cl}_{1,3}(\mathbb{C})$ which, frankly, I don't know exactly what it means.

My guess is to take $V = \mathbb{R}^{4}$ the Minkowski space and $\varphi$ the Minkowski inner product: $$\varphi(x,y) = x_{0}y_{0}-x_{1}y_{1}-x_{2}y_{2}-x_{3}y_{3}$$ If $\gamma^{\mu}$, $\mu = 0,1,2,3$ are complex $4\times 4$ known matrices, we can define $i_{\Phi}$ by sending each element $e_{\mu}$ of the canonical basis of $\mathbb{R}^{4}$ to its associated $\gamma^{\mu}$ and extend $i_{\Phi}$ by linearity. However:

(1) If all this reasoning is correct, I don't know how to prove that the universal property is satisfied.

(2) Wikipedia says this construction should be $\mathcal{Cl}_{4}(\mathbb{C})$ and $\mathcal{Cl}_{1,3}(\mathbb{C})$ and I'm taking $\mathbb{R}^{4}$ instead of $\mathbb{C}$, so I don't know what is the connection between these approaches.

Can someone help me with these problems?

Answer 1

Here's a partial answer.

Heads up: that Wikipedia article seems to be written by physicists for physicists, so IMHO it's not the best source to learn these things. Greub's "Multilinear algebra" has a chapter on Clifford algebras. I believe my favorite one used to be R. Shaw's "Linear algebra and group representations", though.

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Clifford algebras can be constructed as quotients of the tensor algebra, just as how it's done with the exterior algebra. Let $V$ be a vector space over $k$ endowed with an orthogonal bilinear form $B:V\times V\to k$ of signature $(p,q)$. Then, the Clifford algebra associated to it is the quotient $$ Cl_{p,q}(k)=Cl(V,B) = \left(\bigoplus_n T^nV\right) / (v\otimes w + w\otimes v=2B(v,w)1), $$ where $T^nV=V^{\otimes n}$ with $T^0V:=k$. The quotient is by the ideal generated by the written relations and $1\in T^0V$ is the unit of this ring.

This definition is equivalent to the one using quadratic forms, so I'd suggest you pick one way of doing it and stick with it. To see this you need to understand the equivalence between bilinear forms and quadratic forms.

Now, in physical applications (especially when dealing with relativistic effects) you usually want to consider a bilinear form with signature $(3,1)$ (or $(1,3)$?), i.e., the one used in the Minkowski spacetime.

Having said all this let me say a couple of things about your questions.

1. You can show this quotient solves the universal property the same way you show the tensor algebra satisfies its universal property. This might be written in Greub's.

2. To really understand what's going on you have to study the classification theory of Clifford algebras and their irreducible representations. From the mathematics side the gamma matrices (Dirac matrices) come from a representation of $Cl(V,B)$. I speculate that physicists complexify this algebra in order to access certain representations that are more sensible from a physical point of view. These notes seem to give a good overview of the whole situation from the mathematical perspective.