The master formulae in QFT:

\[\int \mathcal{D}\phi^*\mathcal{D}\phi\exp\left\lbrace i \int d^4x (\phi^* M \phi + JM) \right\rbrace = \frac{\mathcal{N}}{\det{M}} \exp(iJM^{-1}J)\]

where $\mathcal{N}$ is some normalization factor.

For fermions,

\[\int \mathcal{D}\psi\mathcal{D}\overline{\psi}\exp\left\lbrace i \int d^4x (\overline{\psi} M \psi) \right\rbrace = \mathcal{N}\det{M}.\]

Some conventions. Metric:

\[\begin{align*} g_{\mu\nu}&=\text{diag}\lbrace1,-1,-1,-1\rbrace,\\ \epsilon_{0123} &= 1, \\ \end{align*}\]

Fourier transform:

\[\begin{align*} f(x) &= \int \frac{d^nk}{(2\pi)^n} \, e^{-ip\cdot x}\widetilde{f}(p),\\ \widetilde{f}(p)&=\int dx \, e^{ip\cdot x}f(x), \end{align*}\]

and $\delta$-function:

\[\delta^{(n)}(k) = \frac{1}{(2\pi)^n}\int d^nx \, e^{i k x}.\]

The gauge Lie group $G$ has an underlying Lie algebra $\mathfrak{g}$, whose generator satisfies

\[[T^a,T^b] = i f^{abc} T^c\]

We require the generators in the fundamental representations to satisfy normalization condition

\[\text{Tr }{T^a T^b} = \frac{1}{2} \delta^{ab}.\]

Gauge field is a $\mathfrak{g}$-valued field,

\[A_\mu = A_\mu^a T^a, \, T^a \in \mathfrak{g}.\]

The $\mathfrak{g}$-valued field strength is

\[F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu -i[A_\mu, A_\nu],\]

which in the language of differential forms is

\[\boxed{ F = dA-iA \wedge A }.\]

Our convention for covariant derivative acting on a field in the fundamental representation is

\[D_{\mu}{\psi} \equiv \partial_\mu \psi -i A_\mu \psi,\]

while acting on a field in the adjoint representation is

\[D_{\mu}{\phi} \equiv \partial_\mu \phi -i [A_\mu,\phi],\]

and

\[\begin{align} [\mathcal{D}_\mu,\mathcal{D}_\nu]\psi &= -i F_{\mu\nu}\psi,\\ [\mathcal{D}_\mu, \mathcal{D}_\nu]\phi &= -i [F_{\mu\nu},\phi], \end{align}\]

where again $\psi,\phi$ are fields in the fundamental and adjoint representation respectively.

The gauge transformation for various fields are

  • gauge field: $A_\mu \to \Omega (A_\mu + i \partial_\mu) \Omega ^{\dagger},\, \Omega=e^{i \omega^i T^i} \in SU(N)$
  • fundamental scalar field: $\phi \to \Omega \phi$
  • fundamental spinor field: $\psi \to \Omega \psi$
  • adjoint scalar field: $\phi \to \Omega \phi \Omega^\dagger$

The non-abelian, gauge dependent magnetic field (chromo-magnetic field) is

\[B_i \equiv -\frac{1}{2} \epsilon_{ijk}F_{jk}\]

and the non-abelian electric field

\[E_i \equiv F_{0i}\]

The Weyl (chiral) basis of gamma matrices is

\[\gamma^\mu = \begin{pmatrix} 0 & \sigma^\mu \\ \bar\sigma^\mu & 0 \end{pmatrix},\quad \bar\sigma^\mu \equiv (\mathbb{1},\sigma^i).\]

Going to the Euclidean spacetime:

\[t\to -i\tau\]

Some notations concerning SUSY.

\[\begin{align} \eta_{\mu\nu} &= \text{diag}{1,-1,-1,-1}, \\ \sigma^\mu_{a \dot{a}} &= (\mathbb{1},\sigma^i) \quad \text{numerically}, \\ \overline{\sigma}^{\mu \dot{a} a} &= (\mathbb{1},-\sigma^i)\quad \text{numerically}, \\ {\left(\sigma^{\mu\nu}\right)_{a}}^{b} &\equiv \frac{i}{4} {\left( \sigma^{[\mu}\overline{\sigma}^{\nu]}\right)_{a}}^{b}, \\ {\left(\overline{\sigma}^{\mu\nu}\right)^{\dot{a}}}_{\dot{b}} &\equiv \frac{i}{4} {\left(\bar\sigma^{[\mu}\sigma^{\nu]}\right)^{\dot{a}}}_{\dot{b}}. \end{align}\]

Under Lorentz transformation,

\[\psi_a \mapsto {\exp\left( -\frac{i}{2}\omega_{\mu\nu}\sigma^{\mu\nu}\right)_{a}}^{b}\; \psi_b, \quad \bar\psi^{\dot{a}} \mapsto {\exp\left( -\frac{i}{2}\omega_{\mu\nu}\bar\sigma^{\mu\nu}\right)^{\dot{a}}}_{\dot{b}} \bar{\psi}^{\dot{b}}.\]

The contraction works in a peculiar way. The metric is

\[\epsilon_{ab} = \sigma^2(i\to 1) = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \quad \epsilon^{ab} = -\epsilon_{ab}, \quad \epsilon^{\dot{a}\dot{b}} = -\epsilon_{\dot{a}\dot{b}}.\]

The metric acts from left on a field, and acts from right to a derivative,

\[\epsilon_{ab} \psi^b = \psi_a, \quad \partial_a \epsilon^{ab} = \partial^b.\]

The gamma matrices:

\[\begin{align} \gamma^\mu &= \begin{pmatrix} 0 & \sigma^\mu \\ \bar\sigma^\mu & 0 \end{pmatrix}, \\ \gamma^5 &= i\gamma^0\gamma^1\gamma^2\gamma^3. \end{align}\] \[\begin{align} \chi^a \chi^b &= - \frac{1}{2} \epsilon^{ab} \chi^2, \quad \chi_a \chi_b = \frac{1}{2} \epsilon_{ab} \chi^2, \\ \bar\chi^{\dot{a}} \bar\chi^{\dot{b}} &= \frac{1}{2} \epsilon^{\dot{a} \dot{b}} \bar\chi^2, \quad \bar\chi_{\dot{a}} \bar\chi_{\dot{b}} = -\frac{1}{2} \epsilon_{\dot{a} \dot{b}} \bar\chi^2. \end{align}\]

\((\theta \sigma^\mu \bar\theta)(\theta \sigma^\nu \bar\theta) = \frac{1}{2} \eta^{\mu\nu} \theta^2 \bar\theta^2.\) \(\begin{align} A^\mu &= \frac{1}{2} A_{a\dot{b}} \sigma^{\mu\dot{b} a}, \\ A_{a\dot{b}} &= A^\mu \sigma_{\mu a\dot{b}}, \\ F_{a}^{b} &= F_{\mu\nu} {(\sigma^{\mu\nu})_{a}}^{b} \end{align}\)

Note that the complex conjugate reverses the order of Grassmann operators, and

\[\boxed{(\partial_a)^\ast = -\bar\partial_{\dot{a}}}.\]

The chiral field:

\[\begin{align} x_L &\equiv x^\mu - i\theta \sigma^\mu \bar\theta, \quad x_R \equiv x^\mu + i\theta \sigma^\mu \bar\theta, \\ D_a &= \partial_a -i (\sigma^\mu \bar\theta)_a \partial_\mu, \quad \bar{D}_{\dot{a}} = (D_a)^\ast = -\bar\partial_{\dot{a}} + i (\theta \sigma^\mu )_{\dot{a}} \partial_\mu. \end{align}\]

The supergauge transformation:

\[\begin{align} \Phi &\rightarrow e^{i q\Lambda(y,\theta)} \Phi, \\ V(x,\theta,\bar\theta) &\rightarrow V(x,\theta,\bar\theta) - i (\Lambda - \bar\Lambda), \end{align}\]

where the charge $q$ is usually absorbed in the definition of the gauge field $A$. The gauge invariant combination is

\(\bar\Phi e^{V} \Phi.\) The Wess-Zumino supergauge greatly reduces the number of component fields of a real superfield:

\(V(x,\theta,\bar\theta) = -2 \theta \sigma^\mu \bar\theta A_\mu -2i\bar\theta^2 (\theta\lambda) + 2i \theta^2 \bar\theta\bar\lambda + \theta^2 \bar\theta^2 D.\) The super-generalization of field strength is a left-chiral superfield, \(W_a \equiv \frac{1}{8} \bar{D}^2 D_a V,\) to derive it, go to $\lbrace y^\mu, \theta, \theta\rbrace$ coordinates and note that

\[\bar{D}_ {\dot{a}} \to -\bar\partial_ {\dot{a}}\]

and

\[\bar\partial^2 \bar\theta^2 = -4.\]