Notation and conventions

Units

This site uses natural units throughout: $\hbar = c = \epsilon_0 = 1$. In these units:

• Energy, mass, and momentum all have dimensions of [mass]
• Length and time have dimensions of [mass]$^{-1}$
• The fine structure constant $\alpha = e^2/(4\pi) \approx 1/137$ is dimensionless
• The electron mass $m \approx 0.511$ MeV sets the scale

To convert back to SI: restore factors of $\hbar$ and $c$ using dimensional analysis. For example, the Compton wavelength $\lambda_C = 1/m$ in natural units becomes $\lambda_C = \hbar/(mc)$ in SI.

Metric signature

We use the mostly-minus convention, following Peskin & Schroeder:

$$\eta_{\mu\nu} = \text{diag}(+1, -1, -1, -1)$$

So $p^2 = p_\mu p^\mu = E^2 - |\mathbf{p}|^2 = m^2$ for an on-shell particle (positive for timelike momenta). Some references (Weinberg, Srednicki) use the mostly-plus convention $\text{diag}(-1,+1,+1,+1)$, which flips all signs.

Four-vectors

Greek indices $\mu, \nu = 0,1,2,3$ run over all spacetime components. Latin indices $i, j = 1,2,3$ run over spatial components only.

$$x^\mu = (t, \mathbf{x}), \quad p^\mu = (E, \mathbf{p}), \quad A^\mu = (\phi, \mathbf{A})$$

Raising and lowering: $p_\mu = \eta_{\mu\nu}p^\nu$, so $p_0 = E$ and $p_i = -p^i$.

Gamma matrices

The Dirac gamma matrices satisfy the Clifford algebra:

$$\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu}$$

We use the Dirac representation:

$$\gamma^0 = \begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix}, \quad \gamma^i = \begin{pmatrix} 0 & \sigma^i \\ -\sigma^i & 0 \end{pmatrix}$$

Slash notation: $\not{p} = \gamma^\mu p_\mu$.

The fifth gamma matrix: $\gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3$.

The Dirac conjugate: $\bar{\psi} = \psi^\dagger\gamma^0$.

Trace identities

Frequently used in amplitude calculations:

$$\text{Tr}[\mathbf{1}] = 4$$

$$\text{Tr}[\gamma^\mu\gamma^\nu] = 4\eta^{\mu\nu}$$

$$\text{Tr}[\text{odd number of } \gamma\text{'s}] = 0$$

$$\text{Tr}[\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma] = 4(\eta^{\mu\nu}\eta^{\rho\sigma} - \eta^{\mu\rho}\eta^{\nu\sigma} + \eta^{\mu\sigma}\eta^{\nu\rho})$$

Traces involving $\gamma^5$:

$$\text{Tr}[\gamma^5] = 0, \quad \text{Tr}[\gamma^5\gamma^\mu\gamma^\nu] = 0$$

$$\text{Tr}[\gamma^5\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma] = -4i\epsilon^{\mu\nu\rho\sigma}$$

where $\epsilon^{0123} = +1$ is the totally antisymmetric Levi-Civita tensor.

Mandelstam variables

For a $2 \to 2$ scattering process with momenta $p_1 + p_2 \to p_3 + p_4$:

$$s = (p_1 + p_2)^2, \quad t = (p_1 - p_3)^2, \quad u = (p_1 - p_4)^2$$

They satisfy $s + t + u = \sum_i m_i^2$.

Spinor conventions

Dirac spinors $u^s(p)$ (electrons) and $v^s(p)$ (positrons) satisfy:

$$(\not{p} - m)u^s(p) = 0, \quad (\not{p} + m)v^s(p) = 0$$

Normalization: $\bar{u}^s(p)u^{s'}(p) = 2m\,\delta^{ss'}$, $\bar{v}^s(p)v^{s'}(p) = -2m\,\delta^{ss'}$.

Completeness relations (used in spin sums):

$$\sum_s u^s(p)\bar{u}^s(p) = \not{p} + m, \quad \sum_s v^s(p)\bar{v}^s(p) = \not{p} - m$$