Remark

Above we have used the following notation:

• $F_{\omega} \in \Omega^2(M,\mathrm{Ad}(P))$ is the curvature of $\omega$.

• $\mathrm{Ad}(P) := P \times_{\mathrm{Ad}} \mathfrak{g}$ is the adjoint bundle.

• $\| \psi \|_{\kappa}^2 := \psi \wedge_{\kappa} \star \psi \in \Omega^n(M)$. In general, if $U,V,W$ are vector spaces, $\varphi \in \Omega^p(M,V)$, $\psi\in\Omega^q(M,W)$ and $f: V \times W \to U$ is a linear map, we have $\varphi \wedge_{f} \psi \in \Omega^{p+q}(M,U)$.

• $\star$ is the Hodge-star operator determined by the metric on $M$.

Remark

The Euler-Lagrange equations determined by the above action together with the Bianchi identity are called Yang-Mills equations:

where $\mathrm{D}^\omega$ denotes the covariant derivative.

Remark

Let $g: P \to P$ be a gauge transformation.

• If $\omega$ is a connection on $P$, then $g^{*}\omega$ is another connection on $P$.

• One can identify $g$ with a smooth map $\tilde g: P \to G$, namely by $g=r_{\tilde g}$, i.e. $g(p) = p \cdot \tilde g(p)$ for all $p\in P$.

• The pullback along a gauge transformation restricts to an automorphism of $\Omega^k_{\rho}(P,V)$. In terms of the associated map $\tilde g$, we have

  $$g^{*}\psi = \rho(\tilde g^{-1},\psi)\text{.}$$

Remark

One can either keep a connection $\omega$ fixed and consider

as a matter field in an “external gauge field”, or consider the combined action functional

Remark

We also need some facts about spin structures.

• Let $M$ be a spacetime with pseudo-Riemannian metric of signature $(p,q)$. We denote by $FM$ be the principal $SO(p,q)$-bundle over $M$ of orthonormal frames, the frame bundle.

• A spin structure on $M$ is a principal $Spin(p,q)$-bundle $SM$ over $M$ together with a bundle morphism $\lambda: SM \to FM$ such that $\lambda(X\cdot\varphi) = \lambda(X)\cdot\Lambda(\varphi)$ for all $X\in SM$ and all $\varphi\in Spin(p,q)$.

• Let $\theta \in \Omega^1(FM,\mathfrak{so}(p,q))$ be the Levi-Cevita connection on $FM$. Then,

  $$\Theta := \mathrm{d}\Lambda^{-1}(\lambda^{*}\theta) \in \Omega^1(SM,\mathfrak{spin}(p,q))$$

  is a connection on $SM$.

Remark

Finally, we recall the definition of the Dirac operator.

• Let $\rho: C(p,q) \times V \to V$ be a representation, with $V \subset \C(p,q)$. Note that in the above realization of the group $Spin(p,q)$, the representation $\rho$ restricts to a representation of $Spin(p,q)$. The spinor bundle is the vector bundle $V M := SM \times_{\rho} V$.

• Clifford multiplication is a map

  $$TM \otimes V M \to V M: u \otimes s \mapsto u \cdot s\text{.}$$

  It is defined as follows. We write $s=(X,v)$ with $X\in SM$ and $v\in V$. We consider the orthonormal frame $\alpha_X := \lambda(X): TM \to \R^n$. Then, $\alpha_X(u) \in \C(p,q)$ and

  $$u \cdot s := (X,\alpha_X(u)\cdot v)\text{.}$$

  One can show using above-listed properties of the Clifford algebra that this definition does not depend on the choice of the representative $(X,v)$.

• The Dirac operator is

  $$D: \Gamma(M,V M) \to \Gamma(M,V M) : \psi \mapsto \sum_{i} e_i \cdot \mathrm{D}^{\Theta}\psi (e_i)$$

  where $e_i\in TM$ runs over a local orthonormal basis.

Remark

In the physical literature, the picture is slightly different: The representation space $V$ of a spinor is not a subspace of the Clifford algebra, but rather $\C^n$. One can think about this as a further association of $\C^n$ to the Clifford bundle $VM$ using a representation of $\C(p,q)$ on $\C^n$. Below we describe this in the case of the electron, i.e. $(p,q)=(1,3)$ and $V= \Sigma^{+} \oplus \Sigma^{-}$. Another difference is here that instead of $Spin(1,3)$, physicists often use the (non-canonically) isomorphic group $SL(2,\C)$.

• One starts with the following representation $\gamma: \C(1,3) \to \mathrm{Gl}(4,\C)$. Consider the $\R$-linear map

  $$\gamma: \R^4 \to \mathrm{Gl}(4,\C): v \mapsto \begin{pmatrix} 0 & v' \\ v'' & 0 \end{pmatrix}\text{,}$$

  where

  $$v' := \begin{pmatrix} v_0 + v_3 & v_1 - iv_2 \\ v_1 + iv_2 & v_0 - v_3 \\ \end{pmatrix} \quad\text{ and }\quad v'' = \begin{pmatrix} v_0-v_3 & -v_1 + iv_2 \\ -v_1-iv_2 & v_0+v_3 \\ \end{pmatrix}\text{.}$$

  It satisfies $\gamma(v) \cdot \gamma(v)= - \left \langle v,v \right \rangle I_4$. The Clifford algebra has a universal property that implies that $\gamma$ extends uniquely to a representation of $\C(1,3)$. The images of the standard basis vectors $e_0,...,e_3$ are often called $\gamma$-matrices, $\gamma_k := \gamma(e_k)$.

• The restriction of the representation $\gamma$ to $Spin(1,3)$ is a representation $\rho: Spin(1,3) \times \C^4 \to \C^4$. It splits into a direct sum of two representations equivalent to $\Sigma^+$ and $\Sigma^-$. Using $\rho\circ\alpha = \alpha$, one checks using the above definition of the group homomorphism $\Lambda: Spin(p,q) \to SO(p,q)$ that

  $$\gamma(\Lambda(\varphi)v) = \rho(\varphi)\gamma(v)\rho(\varphi)^{-1}\text{.}$$

• The above mentioned identification between $Spin(1,3)$ and $SL(2,\C)$ is

  $$Spin(1,3) \to SL(2,\C) : v_1 \cdot...\cdot v_{2r} \mapsto v_1'v_2''v_3'\cdot...\cdot v_{2r}''\text{.}$$

  Under this isomorphism, $\rho$ becomes

  $$\rho: SL(2,\C) \to \mathrm{Gl}(4,\C) : A \mapsto \begin{pmatrix}A & 0 \\ 0 & A^{*-1} \end{pmatrix}\text{.}$$

  Under this identification, the splitting of $\rho$ into a direct sum yields the defining representation, often called $D^{(1/2,0)}$ and its conjugate, often called $D^{(0,1/2)}$.

• Finally, the bilinear form $H$ becomes

  $$H(v,w) := v^{\mathrm{tr}}\gamma_0\bar w.$$

Remark

If $M=\R^{1,3}$ one can take the trivial spin structure $SM=M \times SL(2,\C)$. It has a canonical global section, so that a spinor $\psi$ can be identified with a map $\psi: M \to \C^4$. The Dirac operator is now $D \psi = \gamma^{i}\partial_i \psi$, where $\gamma^i := \eta^{ik}\gamma_k$. Now, the Dirac equation is

Let’s go back to Dirac’s orginial motivation. Dirac was looking for a first order differential equation

for functions $\psi: \R^{4} \to \C^n$, whose solutions are automatically solutions of the Klein-Gordon equation

where $\triangle= \partial^k\partial_k$. If $\psi$ is a solution to the first equation,

This is the Klein-Gordon equation, if $\alpha^{i}\alpha^{j}=\eta^{ij}$. This can only be satisfied for matrices, so that better $\alpha^{i}\alpha^{j}=\eta^{ij}I_n$. Since $\eta^{ij}I_n$ is a symmetric matrix, this can be written as

The smallest matrices satisfying this relation are the above “gamma matrices”. If $m=0$, there is a solution in dimension two, the “Pauli matrices”.

Remark

For a general spacetime $M$, and unlike in the previous remark, $D^2 \neq \triangle$. Rather, $D^2$ is given by the Lichnerowiz formula which has in fact been proved first by Schrödinger. So, Dirac’s motivation actually fails for “curved spacetimes”.

Remark

(Bundle Splicing)

• Consider principal $G_k$-bundles $P_k$ over $M$, for $k=1,2$. The fibre product $P_1 \times_M P_2$ is a principal $(G_1 \times G_2)$-bundle over $M$, denoted $P_1 \circ P_2$.

• If $\omega_1$ and $\omega_2$ are connections on $P_1$ and $P_2$, respectively, then

  $$\omega_1 \circ \omega_2 := \mathrm{pr}_1^{*}\omega_1 \oplus \mathrm{pr}_2^{*}\omega_2 \in \Omega^1(P_1 \circ P_2,\mathfrak{g}_1 \oplus \mathfrak{g}_2)$$

  is a connection on $P_1 \circ P_2$.

• Suppose $V$ is a vector space and $\rho_k: G_k \to \mathrm{Gl}(V)$ are representations, such that $\rho_1(g_1) \circ \rho_2(g_2) = \rho_2(g_2) \circ \rho_1(g_1)$ for all $g_1\in G_1$ and $g_2 \in G_2$. Then, $\rho_1 \times \rho_2$ is a representation of $G_1 \times G_2$ on $V$.

Remark

Let $M$ be a spacetime with spin structure $SM$ and let $P$ be a Yang-Mills theory over $M$ with gauge group $G$. For $\rho_{SM}: Spin(p,q) \to \mathrm{Gl}(V)$ a representation with $V \subset \C(p,q)$, and $\rho_P: G \to \mathrm{Gl}(V)$ a commuting representation, the associated bundle $VP := (SM \circ P) \times_{(\rho_{SM} \times \rho_P)} V$ still has a Clifford multiplication. For $\omega$ a connection on $P$, one can define a Dirac operator