41.2 Dirac \gamma matrices and Clif(3, 1)

41.2 Dirac \gamma matrices and Clif(3, 1)

Concept links · terms present in this machine draft; source roles are unverified: vector space · linear map · Lie algebra · Pauli matrices · complexification

In our discussion of the fermionic version of the harmonic oscillator, we defined the Cliford algebra $\mathrm{Cliff}(r,s)$ and found that elements quadratic in its generators gave a basis for the Lie algebra of $\mathfrak{so}(r,s)=\mathfrak{spin}(r,s)$ . Exponentiating these gave an explicit construction of the group $Spin(r,s)$ . We can apply that general theory to the case of Clif(3, 1) and this will give us the representations $\left(\frac{1}{2},0\right)$ and $(0,\frac{1}{2})$

If we complexify our ${\mathbf{R}}^4$ , then its Cliford algebra becomes the algebra of 4 by 4 complex matrices

$$\mathrm{Cliff}(3,1)\otimes \mathbf{C}=\mathrm{Cliff}(4,\mathbf{C})=M(4,\mathbf{C})$$

We will represent elements of $\mathrm{Cliff}(3,1)$ as such 4 by 4 matrices, but should keep in mind that we are working in the complexification of the Cliford algebra that corresponds to the Lorentz group, so there is some sort of condition on the matrices that needs to be kept track of to identify $\mathrm{Cliff}(3,1)\subset M(4,\mathbf{C})$ . There are several diferent choices of how to explicitly represent these matrices, and for diferent purposes, diferent ones are most convenient. The one we will begin with and mostly use is sometimes called the chiral or Weyl representation, and is the most convenient for discussing massless charged particles. We will try and follow the conventions used for this representation in [100]. Note that these 4 by 4 matrices act not on four dimensional space-time, but on spinors. It is a special feature of 4 dimensions that these two diferent representations of the Lorentz group have the same dimension.

Writing 4 by 4 matrices in 2 by 2 block form and using the Pauli matrices ${\sigma }_j$ we assign the following matrices to Cliford algebra generators

$${\gamma }_0=-i\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\gamma }_1=-i\left(\begin{array}{cc}0& {\sigma }_1\\ -{\sigma }_1& 0\end{array}\right),{\gamma }_2=-i\left(\begin{array}{cc}0& {\sigma }_2\\ -{\sigma }_2& 0\end{array}\right),{\gamma }_3=-i\left(\begin{array}{cc}0& {\sigma }_3\\ -{\sigma }_3& 0\end{array}\right)$$

One can easily check that these satisfy the Cliford algebra relations for generators of $\mathrm{Cliff}(3,1)$ : they anticommute with each other and

$${\gamma }_0^2=-1,{\gamma }_1^2={\gamma }_2^2={\gamma }_3^2=1$$

The quadratic Cliford algebra elements $-\frac{1}{2}{\gamma }_j{\gamma }_k$ for $j<k$ satisfy the commutation relations of so(3, 1). These are explicitly

$$-\frac{1}{2}{\gamma }_1{\gamma }_2=-\frac{i}{2}\left(\begin{array}{cc}{\sigma }_3& 0\\ 0& {\sigma }_3\end{array}\right),-\frac{1}{2}{\gamma }_1{\gamma }_3=\frac{i}{2}\left(\begin{array}{cc}{\sigma }_2& 0\\ 0& {\sigma }_2\end{array}\right),-\frac{1}{2}{\gamma }_2{\gamma }_3=-\frac{i}{2}\left(\begin{array}{cc}{\sigma }_1& 0\\ 0& {\sigma }_1\end{array}\right)$$

and

$$-\frac{1}{2}{\gamma }_0{\gamma }_1=\frac{1}{2}\left(\begin{array}{cc}-{\sigma }_1& 0\\ 0& {\sigma }_1\end{array}\right),-\frac{1}{2}{\gamma }_0{\gamma }_2=\frac{1}{2}\left(\begin{array}{cc}-{\sigma }_2& 0\\ 0& {\sigma }_2\end{array}\right),-\frac{1}{2}{\gamma }_0{\gamma }_3=\frac{1}{2}\left(\begin{array}{cc}-{\sigma }_3& 0\\ 0& {\sigma }_3\end{array}\right)$$

They provide a representation $({\pi }^{\prime },{\mathbf{C}}^4)$ of the Lie algebra $\mathfrak{so}(3,1)$ with

$${\pi }^{\prime }(l_1)=-\frac{1}{2}{\gamma }_2{\gamma }_3,{\pi }^{\prime }(l_2)=\frac{1}{2}{\gamma }_1{\gamma }_3,{\pi }^{\prime }(l_3)=-\frac{1}{2}{\gamma }_1{\gamma }_2$$

and

$${\pi }^{\prime }(k_1)=-\frac{1}{2}{\gamma }_0{\gamma }_1,{\pi }^{\prime }(k_2)=-\frac{1}{2}{\gamma }_0{\gamma }_2,{\pi }^{\prime }(k_3)=-\frac{1}{2}{\gamma }_0{\gamma }_3$$

Note that the ${\pi }^{\prime }(l_j)$ are skew-adjoint, since this representation of the $\mathfrak{so}(3)\subset$ $\mathfrak{so}(3,1)$ sub-algebra is unitary. The ${\pi }^{\prime }(k_j)$ are self-adjoint and this representation ${\pi }^{\prime }$ of $\mathfrak{so}(3,1)$ is not unitary.

On the two commuting $\mathfrak{sl}(2,\mathbf{C})$ subalgebras of $\mathfrak{so}(3,1)\otimes \mathbf{C}$ with bases (see section 40.2)

$$A_j=\frac{1}{2}(l_j+i{k}_j),B_j=\frac{1}{2}(l_j-i{k}_j)$$

this representation is

$${\pi }^{\prime }(A_1)=-\frac{i}{2}\left(\begin{array}{cc}{\sigma }_1& 0\\ 0& 0\end{array}\right),{\pi }^{\prime }(A_2)=-\frac{i}{2}\left(\begin{array}{cc}{\sigma }_2& 0\\ 0& 0\end{array}\right),{\pi }^{\prime }(A_3)=-\frac{i}{2}\left(\begin{array}{cc}{\sigma }_3& 0\\ 0& 0\end{array}\right)$$

and

$${\pi }^{\prime }(B_1)=-\frac{i}{2}\left(\begin{array}{cc}0& 0\\ 0& {\sigma }_1\end{array}\right),{\pi }^{\prime }(B_2)=-\frac{i}{2}\left(\begin{array}{cc}0& 0\\ 0& {\sigma }_2\end{array}\right),{\pi }^{\prime }(B_3)=-\frac{i}{2}\left(\begin{array}{cc}0& 0\\ 0& {\sigma }_3\end{array}\right)$$

We see explicitly that the action of the quadratic elements of the Cliford algebra on the spinor representation ${\mathbf{C}}^4$ is reducible, decomposing as the direct sum $S_L\oplus S_R^{\ast }$ of two inequivalent representations on ${\mathbf{C}}^2$

$$\Psi =\left(\genfrac{}{}{0ex}{}{{\psi }_L}{{\psi }_R^{\ast }}\right)$$

with complex conjugation (interchange of $A_j$ and $B_j)$ relating the $\mathfrak{sl}(2,\mathbf{C})$ actions on the components. The $A_j$ act just on $S_L$ , the $B_j$ just on $S_R^{\ast }$ . An alternative standard notation to the two-component van der Waerden notation is to use the four components of ${\mathbf{C}}^4$ with the action of the $\gamma$ matrices. The relation between the two notations is given by

$${\Psi }_A\leftrightarrow \left(\genfrac{}{}{0ex}{}{{\psi }_B}{{\varphi }^{\dot{B}}}\right)$$

where the index $A$ on the left takes values 1, 2, 3, 4 and the indices $B,\dot{B}$ on the right each take values 1, 2.

Note that identifying Minkowski space with elements of the Cliford algebra by

$$(x_0,x_1,x_2,x_3)\to /x=x_0{\gamma }_0+x_1{\gamma }_1+x_2{\gamma }_2+x_3{\gamma }_3$$

identifies Minkowski space with certain 4 by 4 matrices. This again gives the identification used earlier of Minkowski space with linear maps from $S_R^{\ast }$ to $S_L$ since the upper right two by two block of the matrix will be given by

$$-i\left(\begin{array}{cc}{x}_0+x_3& x_1-i{x}_2\\ x_1+i{x}_2& x_0-x_3\end{array}\right)$$

and takes $S_R^{\ast }$ to $S_L$

An important element of the Cliford algebra is constructed by multiplying all of the basis elements together. Physicists traditionally multiply this by $i$ to make it self-adjoint and define

$${\gamma }_5=i{\gamma }_0{\gamma }_1{\gamma }_2{\gamma }_3=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)$$

This can be used to produce projection operators from the Dirac spinors onto the left and right-handed Weyl spinors

$$\frac{1}{2}(1-{\gamma }_5)\Psi ={\psi }_L,\frac{1}{2}(1+{\gamma }_5)\Psi ={\psi }_R^{\ast }$$

There are two other commonly used representations of the Cliford algebra relations, related to the one above by a change of basis. The Dirac representation is useful to describe massive charged particles, especially in the non-relativistic limit. Generators are given by

$${\gamma }_0^D=-i\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),{\gamma }_1^D=-i\left(\begin{array}{cc}0& {\sigma }_1\\ -{\sigma }_1& 0\end{array}\right)$$ $${\gamma }_2^D=-i\left(\begin{array}{cc}0& {\sigma }_2\\ -{\sigma }_2& 0\end{array}\right),{\gamma }_3^D=-i\left(\begin{array}{cc}0& {\sigma }_3\\ -{\sigma }_3& 0\end{array}\right)$$

and the projection operators for Weyl spinors are no longer diagonal, since

$${\gamma }_5^D=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$$

A third representation, the Majorana representation, is given by (now no longer writing in 2 by 2 block form, but as 4 by 4 matrices)

$${\gamma }_0^M=\left(\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 1& 0& 0& 0\end{array}\right),{\gamma }_1^M=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right)$$ $${\gamma }_2^M=\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& -1& 0\\ 0& -1& 0& 0\\ 1& 0& 0& 0\end{array}\right),{\gamma }_3^M=\left(\begin{array}{cccc}0& -1& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& -1& 0\end{array}\right)$$

with

$${\gamma }_5^M=i\left(\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right)$$

The importance of the Majorana representation is that it shows the interesting possibility of having (in signature $(3,1))$ a spinor representation on a real vector space ${\mathbf{R}}^4.$ , since one sees that the Cliford algebra matrices can be chosen to be real. One has

$${\gamma }_0{\gamma }_1{\gamma }_2{\gamma }_3=\left(\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right)$$

and

$${\left({\gamma }_0{\gamma }_1{\gamma }_2{\gamma }_3\right)}^2=-1$$

The Majorana spinor representation is on $S_M={\mathbf{R}}^4$ , with ${\gamma }_0{\gamma }_1{\gamma }_2{\gamma }_3$ a real operator on this space with square −1, so it provides a complex structure on $S_M$ . Recall that a complex structure on a real vector space gives a splitting of the complexification of the real vector space into a sum of two complex vector spaces, related by complex conjugation. In this case this corresponds to

$$S_M\otimes \mathbf{C}=S_L\oplus S_R^{\ast }$$

the fact that complexifying Majorana spinors gives the two kinds of Weyl spinors.

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