[Click here for a PDF of this notes with full details.]

### DISCLAIMER: Rough notes from class, with some additional side notes.

These are notes for the UofT course PHY2403H, Quantum Field Theory, taught by Prof. Erich Poppitz, fall 2018.

## Overview.

See the PDF above for full notes for the first part of this particular lecture. We covered

- Normalization:

\begin{equation*}

u^{r \dagger} u^{s}

= 2 p_0 \delta^{r s}.

\end{equation*} -
Products of \( p \cdot \sigma, p \cdot \overline{\sigma} \)

\begin{equation*}

(p \cdot \sigma) (p \cdot \overline{\sigma})

=

(p \cdot \overline{\sigma}) (p \cdot \sigma)

= m^2.

\end{equation*} - Adjoint orthogonality conditions for \( u \)

\begin{equation*}

\overline{u}^r(\Bp) u^{s}(\Bp) = 2 m \delta^{r s}.

\end{equation*} -
Solutions in the \( e^{i p \cdot x} \) “direction”

\begin{equation}\label{eqn:qftLecture21:99}

v^s(p)

=

\begin{bmatrix}

\sqrt{p \cdot \sigma} \eta^s \\

-\sqrt{p \cdot \overline{\sigma}} \eta^s \\

\end{bmatrix},

\end{equation}

where \( \eta^1 = (1,0)^\T, \eta^2 = (0,1)^\T \). -
\(v\) normalization

\begin{equation*}

\begin{aligned}

\overline{v}^r(p) v^s(p) &= – 2 m \delta^{rs} \\

v^{r \dagger}(p) v^s(p) &= 2 p^0 \delta^{rs}.

\end{aligned}

\end{equation*} -
Dirac adjoint orthogonality conditions.

\begin{equation*}

\begin{aligned}

\overline{u}^r(p) v^s(p) &= 0 \\

\overline{v}^r(p) u^s(p) &= 0.

\end{aligned}

\end{equation*} -
Dagger orthogonality conditions.

\begin{equation*}

\begin{aligned}

v^{r \dagger}(-\Bp) u^s(\Bp) &= 0 \\

u^{r\dagger}(\Bp) v^s(-\Bp) &= 0.

\end{aligned}

\end{equation*} -
Tensor product.
Given a pair of vectors

\begin{equation*}

x =

\begin{bmatrix}

x_1 \\

\vdots \\

x_n \\

\end{bmatrix},

\qquad

y =

\begin{bmatrix}

y_1 \\

\vdots \\

y_n \\

\end{bmatrix},

\end{equation*}

the tensor product is the matrix of all elements \( x_i y_j \)\begin{equation*}

x \otimes y^\T =

\begin{bmatrix}

x_1 \\

\vdots \\

x_n \\

\end{bmatrix}

\otimes

\begin{bmatrix}

y_1 \cdots y_n

\end{bmatrix}

=

\begin{bmatrix}

x_1 y_1 & x_1 y_2 & \cdots & x_1 y_n \\

x_2 y_1 & x_2 y_2 & \cdots & x_2 y_n \\

x_3 y_1 & \ddots & & \\

\vdots & & & \\

x_n y_1 & \cdots & & x_n y_n

\end{bmatrix}.

\end{equation*} -
Direct product relations.

\begin{equation*}

\begin{aligned}

\sum_{s = 1}^2 u^s(p) \otimes \overline{u}^s(p) &= \gamma \cdot p + m \\

\sum_{s = 1}^2 v^s(p) \otimes \overline{v}^s(p) &= \gamma \cdot p – m \\

\end{aligned}

\end{equation*}