\def\Cl{\mathop{\rm Cl}}
\def\CL{\mathop{{\bf C\/}l}}
\def\R{{\bf R}}
\def\SO{\mathop{\rm SO}}
\def\Spin{\mathop{\rm Spin}}
\def\Z{{\bf Z}}
\def\C{{\bf C}}
\def\so{{\rm so}}
\def\spin{{\rm spin}}
\def\GL{{\rm GL}}

Konrad Waldorf.

Classical Gauge Theory and Matter Fields II.

No elementary particle can be described in the framework of the previous lecture.
Some non-elementary particles, like $\pi^-$-meson, can be described in these terms.

Dirac's starting point was that Schr\"odinger's equation is first order,
whereas Klein-Gordon's equation $(\Delta^\omega+m^2)\phi=0$ is second order.

We consider Clifford algebra $\Cl(p,q)$ of $\R^{p,q}$.
This algebra has two involutive anti-automorphisms:
$\alpha$ is induced by $v\mapsto-v$ on~$\R^{p,q}$
and $\cdot^t\colon v_1\otimes\cdots\otimes v_r\to v_r\otimes\cdots\otimes v_1$.
The Clifford algebra has a real valued metric $H(v,w)=(v^tw)_0$.

Spin.  We have $\SO(p,q)$ and $\Spin(p,q)=\{v_1\cdots v_{2r}\mid v_i\in\R^{p,q}\land\|v_i\|=1\}$.
$\Lambda\colon\Spin(p,q)\to\SO(p,q)$ is given by $\Lambda(\phi)(v)=\alpha(\phi)v\phi^{-1}$.
We have an exact sequence $1\to\Z/(2)\to\Spin(p,q)\to\SO(p,q)\to1$.
$\Cl(p,q)\otimes_\R\C$ decomposes into $k$ copies of a subrepresentation~$\Sigma$.
If $p+q$ is odd, then $\Sigma$ is irreducible, $k=2^{(p+q-1)/2}$,
if $p+q$ is even, then $k=2^{(p+q)/2}$ and $\Sigma$ decomposes into
sum of two irreducible representations $\Sigma^+\oplus\Sigma^-$.
For example, if $p+q=4$, then $\dim\Sigma^\pm=2$.

Spin structure.  Suppose $M$ is a spacetime with signature~$(p,q)$.
$FM$ (frame bundle of~$M$) is an $\SO(p,q)$-bundle.
A spin structure on~$M$ is a $\Spin(p,q)$-bundle~$SM$
with $\lambda\colon SM\to FM$ such that $\lambda(X\phi)=\lambda(X)\Lambda(\phi)$.
If $\theta\in\Omega^1(FM,\so(p,q))$ is the Levi-Civita connection on~$FM$,
then $\Theta=d\Lambda^{-1}(\lambda^*\theta)\in\Omega^1(SM,\spin(p,q))$.

Dirac operator.  Let $\rho\colon\Spin(p,q)\to\GL(V)$ be a representation of Spin-group
on a subrepresentation $V\subset\CL(p,q)$ of~$\CL(p,q)$.
The spinor bundle $\Sigma M$ is defined as $SM\times_\rho V$.
We have Clifford multiplication $TM\otimes\Sigma M\to\Sigma M$.
Now we can define the Dirac operator $D\colon\Gamma(M,\Sigma M)\to\Gamma(M,\Sigma M)$:
$\psi\mapsto\sum_ie_i\cdot D^\Theta\psi(e_i)$.

Spinors.  A spinor is a field for~$SM$ of matter type~$(V,h,\rho,f)$ where
$V$ is the same as above, $\rho$ is the restruction of Clifford multiplication
to~$\Spin(p,q)$, $h(v,w)=(H(v,w)+H(w,v))/2$.  Fields are sections of $\Sigma M$ as usual,
but we have a different action functional $S(\psi)=\int_M D\psi\wedge_h*\psi+*(f\psi)$.

Example.  (Weyl spinor.)  $f=0$.  $V=\Sigma^\pm$.  $\dim_\C V=2$.
In the standard model we have ``neutrinos''.

Example.  (Dirac spinor.)  $V=\Sigma$, $f(v)=mh(v,v)/2$.  $\dim_\C V=4$.
Standard model: ``electrons''.
Euler-Lagrange equations: $D\psi+im\psi=0$.  (Dirac equation.)
Particles that satisfy Dirac equation also satisfy Klein-Gordon equation.

\bye