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The Dirac Electron

Idea

The purpose of this page is to explain - using appropriate mathematical terminology - Dirac’s theory of coupling particles with spin to a Yang-Mills gauge field.

We proceed in three steps: first we recall relevant facts about the gauge field itself, then we discuss charged particles in gauge fields, and finally we add spin.

Yang-Mills Theory

(see: the main article about Yang-Mills theory)

Under a spacetime we understand a smooth, oriented, pseudo-Riemannian manifold.

Definition

A Yang-Mills theory over a spacetime $M$ is:

• A Lie group $G$, called the gauge group, together with an $\mathrm{Ad}$-invariant scalar product $\kappa :𝔤\times 𝔤\to R$ on its Lie algebra $𝔤$.

• A $G$-principal bundle $P$ over $M$.

A gauge field is a connection $\omega \in {\Omega }^1(P,𝔤)$ on $P$. The action functional is

$$S_{\mathrm{YM}}(\omega ):=\frac{1}{2}{\int }_M\parallel F_{\omega }{\parallel }_{\kappa }^2.$$S_{YM}(\omega) := \frac{1}{2} \int_M \| F_{\omega} \|_\kappa^2.

Remark

Above we have used the following notation:

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

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

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

• $\star$ 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:

$${\mathrm{D}}^{\omega }\star F_{\omega }=0\text{and}{\mathrm{D}}^{\omega }{F}_{\omega }=0,$$\mathrm{D}^{\omega}\star F_{\omega} = 0 \quad\text{ and }\quad \mathrm{D}^{\omega}F_{\omega}=0,

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

Definition

A gauge transformation is a smooth bundle morphism $g:P\to P$.

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 ${\Omega }_{\rho }^k(P,V)$. In terms of the associated map $\tilde{g}$, we have

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

Theorem

The Yang-Mills action functional $S_{\mathrm{YM}}$ is gauge-invariant, i.e.

$$S_{\mathrm{YM}}(g^{*}\omega )=S_{\mathrm{YM}}(\omega )$$S_{YM}(g^{*}\omega) = S_{YM}(\omega)

for all gauge transformations $g:P\to P$.

Proof

We have $g^{*}\omega ={\mathrm{Ad}}_{\tilde{g}}^{-1}(\omega )-{\tilde{g}}^{*}\overline{\theta }$ and $g^{*}\Omega ={\mathrm{Ad}}_{\tilde{g}}^{-1}(\Omega )$. Under the isomorphism ${\Omega }^k(P,\mathrm{Ad})\cong {\Omega }^2(M,\mathrm{Ad}(P))$ this corresponds to $F_{g^{*}\omega }={\mathrm{Ad}}_g(F_{\omega })$. Since the bilinear form $\kappa$ is $\mathrm{Ad}$-invariant by assumption,

$$\parallel F_{g^{*}\omega }\parallel ={\mathrm{Ad}}_g(F_{\omega }){\wedge }_{\kappa }{\mathrm{Ad}}_g(F_{\omega })=F_{\omega }{\wedge }_{\kappa }{F}_{\omega }=\parallel F_{\omega }\parallel .$$\| F_{g^{*}\omega} \| = \mathrm{Ad}_{g} (F_{\omega}) \wedge_{\kappa} \mathrm{Ad}_{g} (F_{\omega}) = F_{\omega} \wedge_{\kappa} F_{\omega} = \| F_{\omega}\|.

Example

Let $M$ be a spacetime. A classical electromagnetic field theory over $M$ is a Yang-Mills Theory over $M$ with gauge group $G=U(1)$. In more detail:

• for an electromagnetic field theory given by a $U(1)$-bundle $P$ over $M$, we have $\mathrm{Ad}(P)\cong P\times R$, so that ${\Omega }^k(M,\mathrm{Ad}(P))\cong {\Omega }^k(M)$ and ${\Omega }_{\mathrm{Ad}}^k(P,𝔤)\cong {\Omega }_{\mathrm{Ad}}^k(P)$. In particular, $F_{\omega }\in {\Omega }^2(M)$.

• Since $U(1)$ is abelian, $\mathrm{d}(\mathrm{Ad})=0$ and so ${\mathrm{D}}^{\omega }=\mathrm{d}$ on ${\Omega }_{\mathrm{Ad}}^k(P)$.

• Thus, the Yang-Mills equations reduce to Maxwell’s equations for an electromagnetic field on $M$:

  $$\mathrm{d}\star F_{\omega }=0\text{and}\mathrm{d}{F}_{\omega }=0\text{.}$$\mathrm{d}\star F_{\omega} = 0 \quad\text{ and }\quad \mathrm{d}F_{\omega}=0 \text{.}

General Matter Fields

Definition

Let $G$ be a gauge group. A matter type for $G$ is a tuple $(V,h,\rho ,f)$ consisting of:

• a finite-dimensional real vector space $V$ called the internal state space.

• a scalar product? $h:V\times V\to R$.

• a representation $\rho :G\times V\to V$ that is isometric with respect to $h$ i.e. $h(\rho (g)(v),\rho (g)(w))=h(v,w)$.

• a smooth function $f:V\to R$ that is $\rho$-invariant, i.e. $f(\rho (g)(v))=f(v)$.

Definition

Let $P$ be a principal $G$-bundle over $M$, and let $𝒯=(V,h,\rho ,f)$ be a matter type for $G$. A field for $P$ of type $𝒯$ is a smooth section $\varphi :M\to P{\times }_{\rho }V$. Its action functional is

$$S_{𝒯}(\omega ,\varphi ):={\int }_M\parallel {\mathrm{D}}^{\omega }(\varphi ){\parallel }_h^2+\star (f\circ \varphi )\text{.}$$S_{\mathcal{T}}(\omega,\phi) := \int_M \;\|\, \mathrm{D}^{\omega}(\phi)\, \|^2_{h} + \star (f \circ \phi) \text{.}

Theorem

The action functional $S_{𝒯}$ is gauge invariant, i.e.

$$S_{𝒯}(g^{*}\omega ,g^{*}\varphi )=S_{𝒯}(\omega ,\varphi )$$S_{\mathcal{T}}(g^{*}\omega,g^{*}\phi) = S_{\mathcal{T}}(\omega,\phi)

for all gauge transformations $g:P\to P$.

Proof

One calculates that $g^{*}\omega ={\mathrm{Ad}}_{\tilde{g}}^{-1}(\omega )-{\tilde{g}}^{*}\overline{\theta }$, where $\tilde{g}:P\to G$ is the smooth map associated to $g$ via $g(p)=p\cdot \tilde{g}(p)$. Further, $g^{*}\psi =\rho ({\tilde{g}}^{-1})(\psi )$. A computation shows

$$\mathrm{d}(\rho ({\tilde{g}}^{-1})(\psi ))=\rho ({\tilde{g}}^{-1})(\mathrm{d}\psi )+{\tilde{g}}^{*}\overline{\theta }{\wedge }_{\mathrm{d}\rho }\rho ({\tilde{g}}^{-1})(\psi )\text{,}$$\mathrm{d}(\rho(\tilde g^{-1})(\psi)) = \rho(\tilde g^{-1})(\mathrm{d}\psi) + \tilde g^{*}\bar\theta\wedge_{\mathrm{d}\rho} \rho(\tilde g^{-1})(\psi)\text{,}

where $\mathrm{d}\rho :𝔤\otimes V\to V$. Now we compute

${\mathrm{d}}^{g^{*}\omega }(g^{*}\psi )$

$=\mathrm{d}\rho ({\tilde{g}}^{-1},\psi )+g^{*}\omega {\wedge }_{\mathrm{d}\rho }\rho ({\tilde{g}}^{-1},\psi )$

$=\rho ({\tilde{g}}^{-1},\mathrm{d}\psi )+{\tilde{g}}^{*}\overline{\theta }{\wedge }_{\mathrm{d}\rho }\rho ({\tilde{g}}^{-1},\psi )+({\mathrm{Ad}}_{\tilde{g}}^{-1}(\omega )-{\tilde{g}}^{*}\overline{\theta }){\wedge }_{\mathrm{d}\rho }\rho ({\tilde{g}}^{-1},\psi )$

$=\rho ({\tilde{g}}^{-1},\mathrm{d}\psi )+\rho ({\tilde{g}}^{-1},\omega {\wedge }_{\mathrm{d}\rho }\psi )$

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

Since $h$ is invariant, the invariance of the first term follows. The invariance of the second term is clear.

Remark

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

$$S_{\omega }(\varphi ):=S_{𝒯}(\omega ,\varphi )$$S_{\omega}(\phi) := S_{\mathcal{T}}(\omega,\phi)

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

$$S(\omega ,\varphi ):=S_{YM}(\omega )+S_{𝒯}(\omega ,\varphi )\text{.}$$S(\omega,\phi) := S_{Y\!M}(\omega) + S_{\mathcal{T}}(\omega,\phi)\text{.}

Example

(Scalar particle in an external, trivial gauge field)

We consider $G=\left\{e\right\}$, $P=M$, so that necessarily $\omega =0$. A scalar field is field for $M$ of matter type $(R,h,id,f)$ where $f(x):=-\frac{1}{2}{m}^2{x}^2$ and $h(x,y)=xy$. The action functional is

$$S(0,\varphi )=\frac{1}{2}{\int }_M\parallel \mathrm{d}\varphi {\parallel }_h^2+\star m^2{\varphi }^2\text{.}$$S(0,\phi) =\frac{1}{2} \int_M \| \mathrm{d}\phi \|_h^2 +\star m^{2}\phi^2\text{.}

The Euler-Lagrange equation is the Klein-Gordon equation?

$$(▵+m^2)\varphi =0\text{,}$$(\triangle + m^2)\phi = 0\text{,}

where $▵:=\delta \circ \mathrm{d}:{\Omega }^k(M)\to {\Omega }^k(M)$ is the Laplace operator and $\delta :=\star \mathrm{d}\star$ is the exterior coderivative.

Example

(Charged particle in an electromagnetic field, e.g. a ${\pi }^{-}$-meson)

Let $P$ be a $U(1)$-principal bundle over $M$. A field of charge $n\in Z$ is a field for $P$ of matter type $(C,h,{\rho }_n,f)$, where ${\rho }_n:U(1)\times C\to C$ is defined by ${\rho }_n(z,z\prime ):=z^{n}z\prime$ and $f(z):=-\frac{1}{2}{m}^2\mid z{\mid }^2$. The action functional is

$$S(\omega ,\varphi )=\frac{1}{2}{\int }_M\parallel {\mathrm{D}}^{\omega }\varphi {\parallel }^2+\star m^2\parallel \varphi {\parallel }^2\text{.}$$S(\omega,\phi) =\frac{1}{2} \int_M \| \mathrm{D}^{\omega}\phi \|^2 +\star m^{2} \| \phi \| ^2\text{.}

The Euler-Lagrange equation is covariant Klein-Gordon equation

$$({▵}^{\omega }+m^2)\varphi =0\text{,}$$(\triangle^{\omega} +m^2) \phi = 0\text{,}

where ${▵}^{\omega }:={\delta }^{\omega }\circ {\mathrm{D}}^{\omega }$ is the covariant Laplace operator and ${\delta }^{\omega }:=\star {\mathrm{D}}^{\omega }\star$ is the exterior covariant coderivative.

Matter Fields with Spin

The Klein-Gordon equation?s found above are – unlike the Schrödinger equation – of second order on time. Dirac’s motivation was to find a first order equation which upon iteration yields the Klein-Gordon equation. We first discuss free spinors (where free means that they are not coupled to an electromagnetic field, but still feel the “gravity” of the spacetime manifold), and then add the coupling.

Free Spinors

Example

We recall some facts about Clifford algebras and the spin group.

• We denote by $C(p,q)$ the Clifford algebra on $R^{p,q}$, i.e. the quotient of the tensor algebra of $R^{p+q}$ by the ideal generated by $v\otimes w+w\otimes v+2⟨v,w⟩\cdot 1$, where $⟨-,-⟩$ is the Minkowski scalar product of signature $(p,q)$.

• The map $v\mapsto -v$ extends to an anti-automorphism $\alpha :C(p,q)\to C(p,q)$, whose eigenspace decomposition yields the usual $Z_2$-grading on $C(p,q)$.

• We have $\dim C(p,q)=2^{p+q}$.

• The Clifford algebra inherits a bilinear form $H(v,w):=(v^{\mathrm{tr}}w{)}_0$, where $({)}^{\mathrm{tr}}$ is the anti-automorphism of the tensor algebra that reverts the order of tensor products, and $({)}_0$ denotes the degree 0 part.

• We denote by $\mathrm{SO}(p,q)$ the group of linear maps $R^{p+q}\to R^{p+q}$ that preserve the product $⟨-,-⟩$. We define

  $$\mathrm{Spin}(p,q):=\{v_1\cdot ...\cdot v_{2r}\in C(p,q)\mid v_i\in R^{p+q},\parallel v_i\parallel =1,r\in N\}\text{.}$$Spin(p,q) := \left \lbrace v_{1}\cdot ... \cdot v_{2r}\in C(p,q)\mid v_i \in \R^{p+q}, \| v_i \|=1,r \in \N \right \rbrace \text{.}

  Then, we define a group homomorphism $\Lambda :\mathrm{Spin}(p,q)\to \mathrm{SO}(p,q)$ by $\Lambda (\phi )v=\alpha (\phi )v{\phi }^{-1}$. This gives a central extension

  $$1\to Z_2\to \mathrm{Spin}(p,q)\to \mathrm{SO}(p,q)\to 1\text{.}$$1 \to \Z_2 \to Spin(p,q) \to SO(p,q) \to 1\text{.}

• We denote by $C(p,q):=C(p,q){\otimes }_{R}C$ the complexification of the Clifford algebra. The bilinear form $H$ on $C(p,q)$ extends to a sesquilinear form $H$ on $C(p,q)$ defined by $H(v,w)=(v^{\mathrm{tr}}\overline{w}{)}_0$.

• Multiplication in $C(p,q)$ restricts to an action of $spinp,q$ on $C(p,q)$. One can decompose $C(p,q)$ into $k$ copies of a subrepresentation $\Sigma$:

  $$C(p,q)=\Sigma \oplus ...\oplus \Sigma \text{.}$$\C(p,q) = \Sigma \oplus ... \oplus \Sigma \text{.}

• The representation $\Sigma$ is isometric with respect to $H$. If $p+q$ is odd, $k=2^{(p+q-1)/2}$, and $\Sigma$ is irreducible. If $p+q$ is even, $k=2^{(p+q)/2}$, and $\Sigma ={\Sigma }^{+}\oplus {\Sigma }^{-}$ with ${\Sigma }^{\pm }$ irreducible and ${\dim }_C{\Sigma }^{\pm }=2^{(p+q-2)/2}$.

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 $\mathrm{FM}$ be the principal $\mathrm{SO}(p,q)$-bundle over $M$ of orthonormal frames, the frame bundle.

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

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

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

  is a connection on $\mathrm{SM}$.

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 $\mathrm{Spin}(p,q)$, the representation $\rho$ restricts to a representation of $\mathrm{Spin}(p,q)$. The spinor bundle is the vector bundle $VM:=\mathrm{SM}{\times }_{\rho }V$.

• Clifford multiplication is a map

  $$\mathrm{TM}\otimes VM\to VM:u\otimes s\mapsto u\cdot s\text{.}$$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 $X\in \mathrm{SM}$ and $v\in V$. We consider the orthonormal frame ${\alpha }_X:=\lambda (X):\mathrm{TM}\to R^n$. Then, ${\alpha }_X(u)\in C(p,q)$ and

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

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

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

Definition

Let $M$ be a spacetime with spin structure $\mathrm{SM}$, and considered as a $\mathrm{Spin}(p,q)$ as a Yang-Mills theory over $M$. A free spinor is a field for $\mathrm{SM}$ of type $(V,h,\rho ,f)$, where $V\subset C(p,q)$, the scalar product $h$ is

$$h(v,w):=\frac{1}{2}(H(v,w)+H(w,v))\text{,}$$h(v,w):=\frac{1}{2}(H(v,w) + H(w,v))\text{,}

and $\rho$ is the restriction of the multiplication in $C(p,q)$ to $\mathrm{Spin}(p,q)$. The action functional is

$$S(\psi ):={\int }_{M}D\psi {\wedge }_h\star \psi +\star f\circ \psi \text{.}$$S(\psi) := \int_M D \psi \wedge_{h} \star \psi + \star f \circ \psi\text{.}

Definition

The Euler-Lagrange equation determined by the action functional $S(\psi )$ is the Dirac equation

$$D\psi +\mathrm{i}m\psi =0\text{.}$$D\psi + \mathrm{i}m\psi = 0\text{.}

Definition

(Weyl spinors)

We assume spacetime to have even dimension. Weyl spinor?s have $V={\Sigma }^{\pm }$, with the sign corresponding to left/right-handed spinors. Thus, ${\dim }_C(V)=2$. Further $f=0$ (they are massless). In the standard model, neutrinos are left-handed Weyl spinors.

Definition

(Dirac spinors)

We assume spacetime to have signature $(1,3)$. Dirac spinor?s have $V={\Sigma }^{+}\oplus {\Sigma }^{-}$, so that ${\dim }_C(V)=4$. The function $f$ is taken to be $f(v)=-\mathrm{mh}(v,v)$. In the standard model, electrons are Dirac spinors.

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 $\mathrm{VM}$ 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 $\mathrm{Spin}(1,3)$, physicists often use the (non-canonically) isomorphic group $\mathrm{SL}(2,C)$.

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

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

  where

  $$v\prime :=\left(\begin{array}{cc}{v}_0+v_3& v_1-{\mathrm{iv}}_2\\ v_1+{\mathrm{iv}}_2& v_0-v_3\\ \end{array}\right)\text{and}v″=\left(\begin{array}{cc}{v}_0-v_3& -v_1+{\mathrm{iv}}_2\\ -v_1-{\mathrm{iv}}_2& v_0+v_3\\ \end{array}\right)\text{.}$$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 $\gamma (v)\cdot \gamma (v)=-⟨v,v⟩I_4$. 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 $\mathrm{Spin}(1,3)$ is a representation $\rho :\mathrm{Spin}(1,3)\times C^4\to C^4$. 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 $\Lambda :\mathrm{Spin}(p,q)\to \mathrm{SO}(p,q)$ that

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

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

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

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

  $$H(v,w):=v^{\mathrm{tr}}{\gamma }_0\overline{w}.$$H(v,w) := v^{\mathrm{tr}}\gamma_0\bar w.

Remark

If $M=R^{1,3}$ one can take the trivial spin structure $\mathrm{SM}=M\times \mathrm{SL}(2,C)$. 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 ${\gamma }^i:={\eta }^{\mathrm{ik}}{\gamma }_k$. Now, the Dirac equation is

$${\gamma }^i{\partial }_i\psi +\mathrm{i}m\psi =0\text{.}$$\gamma^i\partial_i \psi + \mathrm{i} m\psi = 0\text{.}

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

$${\alpha }^k{\partial }_k\psi +\mathrm{im}\psi =0$$\alpha^k\partial_k\psi +im\psi=0

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

$$(▵+m^2)\psi =0\text{,}$$(\triangle + m^2)\psi = 0\text{,}

where $▵={\partial }^k{\partial }_k$. If $\psi$ is a solution to the first equation,

$${\alpha }^j{\partial }_j{\alpha }^k{\partial }_k\psi ={\alpha }^j{\partial }_j(-\mathrm{im}\psi )=-m^2\psi \text{.}$$\alpha^{j}\partial_j\alpha^k\partial_k \psi = \alpha^{j}\partial_j(-im\psi) = - m^2\psi\text{.}

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

$$\frac{1}{2}({\alpha }^i{\alpha }^j+{\alpha }^j{\alpha }^i)={\eta }^{\mathrm{ij}}\text{.}$$\frac{1}{2}(\alpha^{i}\alpha^j + \alpha^j\alpha^i)= \eta^{ij}\text{.}

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, $D^2\ne ▵$. 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”.

Charged Spinors

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

  $${\omega }_1\circ {\omega }_2:={\mathrm{pr}}_1^{*}{\omega }_1\oplus {\mathrm{pr}}_2^{*}{\omega }_2\in {\Omega }^1(P_1\circ P_2,{𝔤}_1\oplus {𝔤}_2)$$\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 $\mathrm{SM}$ and let $P$ be a Yang-Mills theory over $M$ with gauge group $G$. For ${\rho }_{\mathrm{SM}}:\mathrm{Spin}(p,q)\to \mathrm{Gl}(V)$ a representation with $V\subset C(p,q)$, and ${\rho }_P:G\to \mathrm{Gl}(V)$ a commuting representation, the associated bundle $\mathrm{VP}:=(\mathrm{SM}\circ P){\times }_{({\rho }_{\mathrm{SM}}\times {\rho }_P)}V$ still has a Clifford multiplication. For $\omega$ a connection on $P$, one can define a Dirac operator

$$D^{\omega }:\Gamma (M,\Sigma M\circ P)\to \Gamma (M,\Sigma M\circ P):\psi \mapsto \sum _i{e}_i\cdot {\mathrm{D}}^{\Theta \circ \omega }(\psi )(e_i)\text{.}$$D^{\omega}: \Gamma(M,\Sigma M \circ P) \to \Gamma(M,\Sigma M \circ P):\psi \mapsto \sum_{i} e_i \cdot \mathrm{D}^{\Theta \circ \omega}(\psi)(e_i)\text{.}

Definition

Let $M$ be a spacetime with spin structure, let $P$ be a Yang-Mills theory with gauge group $G$ over $M$, and let ${\rho }_P$ be a representation of $G$ on $V$ commuting with ${\rho }_{\mathrm{SM}}$. A charged spinor is a field for $\mathrm{SM}\circ P$ of type $(V,h,{\rho }_{\mathrm{SM}}\times {\rho }_P,f)$, where $V\subset C(p,q)$ and $H$ is given as before. Its action functional is

$$S(\omega ,\psi ):={\int }_M{D}^{\omega }\psi {\wedge }_h\star \psi +\star f\circ \psi \text{.}$$S(\omega, \psi) := \int_M D^{\omega} \psi \wedge_{h} \star \psi + \star f \circ \psi\text{.}

Definition

(Spinor in an electromagnetic field)

Here, ${\rho }_{\mathrm{SM}}:\mathrm{Spin}(p,q)\to \mathrm{Gl}(V)$ is some representation, and ${\rho }_P:U(1)\to \mathrm{Gl}(V)$ is given by complex multiplication with $z^n$, where $n\in Z$ is the charge of the spinor. Obviously ${\rho }_{\mathrm{SM}}$ and ${\rho }_P$ commute. The Euler-Lagrange equation is

$$D^{\omega }\psi +\mathrm{im}\psi =0\text{.}$$D^{\omega}\psi + im\psi = 0\text{.}

If $M=R^{1,3}$ one can take $\mathrm{SM}=M\times \mathrm{SL}(2,C)$. The canonical global section identifies $\psi$ with a smooth function $\psi :R^{3,1}\to C^4$ and the connection $\omega$ with a 1-form with components $A_i$. Then,

$$D^{\omega }\psi ={\gamma }^i({\partial }_i+A_i)\psi \text{.}$$D^{\omega}\psi = \gamma^{i}(\partial_{i} + A_i)\psi\text{.}

This gives the “Dirac equation?” one usually finds in a textbook.

Related concepts

• fermionic path integral

• minimal coupling

standard model of particle physics and cosmology

theory:	Einstein-	Yang-Mills-	Dirac-	Higgs
	gravity	electroweak and strong nuclear force	fermionic matter	scalar field
field content:	vielbein field $e$	principal connection $\nabla$	spinor $\psi$	scalar field $H$
Lagrangian:	scalar curvature density	field strength squared	Dirac operator component density	field strength squared + potential density
$L=$	$R(e)\mathrm{vol}(e)+$	$⟨F_{\nabla }\wedge {\star }_e{F}_{\nabla }⟩+$	$(\psi ,D_{(e,\nabla )}\psi )\mathrm{vol}(e)+$	$\nabla \overline{H}\wedge {\star }_e\nabla H+(\lambda {\mid H\mid }^4-{\mu }^2{\mid H\mid }^2)\mathrm{vol}(e)$

References

Useful literature on this topic is:

• Christian Bär, Introduction to Spin Geometry, Oberwolfach Reports 53 (2006), p. 3135-3136.

• D. Bleecker, Gauge Theory and Variational Principles, Addison-Weasley, 1981.

• H. Blaine Lawson Jr. , Marie-Louise Michelson, Spin geometry, Princeton Univ. Press, 1989.

• G. L. Naber, Topology, Geometry and Gauge Dields, Springer, 1999.