\(c = 1\)

\(\hbar = 1\)

\(\epsilon_{0} = 1\)

\(\mu_{0} = 1\)

\( \overrightarrow{\nabla} \cdot \overrightarrow{E} = \frac{\rho}{\epsilon_{0}} \hspace{2cm} \overrightarrow{\nabla} \times \overrightarrow{E} = - \frac{\partial \overrightarrow{B}}{\partial t} \)

\( \overrightarrow{\nabla} \cdot \overrightarrow{B} = 0 \hspace{2.25cm} \overrightarrow{\nabla} \times \overrightarrow{B} = \mu_{0} \overrightarrow{J} + \frac{1}{c^{2}} \frac{\partial \overrightarrow{E}}{\partial t} \)

\( \overrightarrow{\nabla} \cdot \overrightarrow{E} = \rho \hspace{2.4cm} \overrightarrow{\nabla} \times \overrightarrow{E} = - \frac{1}{c} \frac{\partial \overrightarrow{B}}{\partial t} \)

\( \overrightarrow{\nabla} \cdot \overrightarrow{B} = 0 \hspace{2.45cm} \overrightarrow{\nabla} \times \overrightarrow{B} = \frac{1}{c} ( \overrightarrow{J} + \frac{\partial \overrightarrow{E}}{\partial t} ) \)

\( \overrightarrow{\nabla} \cdot \overrightarrow{E} = \rho \hspace{2.4cm} \overrightarrow{\nabla} \times \overrightarrow{E} = - \frac{\partial \overrightarrow{B}}{\partial t} \)

\( \overrightarrow{\nabla} \cdot \overrightarrow{B} = 0 \hspace{2.45cm} \overrightarrow{\nabla} \times \overrightarrow{B} = ( \overrightarrow{J} + \frac{\partial \overrightarrow{E}}{\partial t} ) \)

Fine Structure Constant: \( \alpha = \frac{e^{2}}{4\pi\hbar c} \approx 137 \Rightarrow \frac{e^{2}}{4\pi\hbar} \Rightarrow \frac{e^{2}}{4\pi} \)

\(\gamma = \frac{1}{\sqrt{1 - \beta^2}}\)

\(\beta = \frac{v}{c}\)

\(\bar{t} = \gamma (t - \frac{vx}{c^2})\)

\(\bar{x} = \gamma (x - vt)\)

\(\bar{y} = y\)

\(\bar{z} = z\)

\(t = \gamma (\bar{t} + \frac{v\bar{x}}{c^2})\)

\(x = \gamma (\bar{x} + v\bar{t})\)

\(y = \bar{y}\)

\(z = \bar{z}\)

Vector: \(\overrightarrow{x}, \overrightarrow{p}\)

Components: \(\overrightarrow{x}\) = \(x^{i}, i = 1, 2, 3 \)

Four Vectors: Italics, Greek indices.

Vector:

Components: \(x^{\mu}\), \(\mu = 0, 1, 2, 3\)

Time-like Component: \(x^{0}\)

Current Density: j = (\(\rho, \overrightarrow{j}\))

Vector potential: A = (\(V, \overrightarrow{A}\))

\( \frac{\partial \phi}{\partial \bar{x}^{\mu}} = \underset{\nu}{\Sigma} (\frac{\partial x^{\nu}}{\partial \bar{x}^{\mu}}) \frac{\partial \phi}{\partial x^{\nu}} \) (Covariant Transformation)

Covariant: \( \partial_{\bar{\mu}} = \frac{\partial \phi}{\partial \bar{x}^{\mu}} = (\frac{\partial x^{\nu}}{\partial \bar{x}^{\mu}}) \frac{\partial \phi}{\partial x^{\nu}} \) (Equation 4)

\( \Lambda_{\phantom{\nu}\mu}^{\nu} = (\frac{\partial {x}^{\nu}}{\partial \bar{x}^{\mu}}) \)

\( \bar{x}_{\mu} = \Lambda_{\mu}^{\phantom{\mu}\nu}x_{\nu} \) (Eqn. 2)

\(x \cdot x = x^{0}x^{0} - \overrightarrow{x} \cdot \overrightarrow{x} \)

\(x \cdot x = (x^{0})^{2} - (x^{1})^{2} - (x^{2})^{2} - (x^{3})^{2} \)

\( \Rightarrow \partial^{2} = \frac{\partial^{2}}{\partial t^{2}} - \nabla^{2} = \frac{\partial^{2}}{\partial t^{2}} - \frac{\partial^{2}}{\partial x^{2}} - \frac{\partial^{2}}{\partial y^{2}} - \frac{\partial^{2}}{\partial z^{2}} \)

\( a_{\mu} = \eta_{\mu\nu} a^{\nu} \)

\( a \cdot b = \eta_{\mu\nu} a^{\mu} b^{\nu} = a_{\mu}b^{\mu} = a^{\mu}b_{\mu} = \eta^{\mu\nu}a_{\mu}b_{\nu} \)

(2) \( \epsilon^{ijkl} = -1\) , for

(3) \( \epsilon^{ijkl} = 0\) , for anything else (e.g. 0012 \( \Longrightarrow \epsilon^{0012} = 0\) )

\( \tilde{f}(k) = \tilde{f}(\omega,\overrightarrow{k}) = \int \mathrm{d}^{3}x \hspace{0.1cm} \mathrm{d}t \hspace{0.1cm} \mathrm{e}^{(\omega t - \mathrm{i} \overrightarrow{k} \cdot \overrightarrow{x})} \hspace{0.1cm} f(t, \overrightarrow{x}) \)

\( \hat{L}x(t) = \int \mathrm{d}u \hspace{0.1cm} \hat{L} \hspace{0.1cm} G(t,u) \hspace{0.1cm} f(u) \)

\( \hspace{1.2cm} = \int \mathrm{d}u \hspace{0.1cm} \delta(t - u) \hspace{0.1cm} f(u) \)

\( \hspace{1.2cm} = f(t) \)

Given: \( \hat{L} G(t,u) = \delta(t - u) \)

\( \Rightarrow G(\overrightarrow{x},\overrightarrow{u}) = - \frac{1}{4\pi \lvert \overrightarrow{x} - \overrightarrow{u} \rvert }\)

\( \Rightarrow \phi(x, t_{x}) = \int \mathrm{d}y \hspace{0.1cm} G^{+}(x,t_{x},y,t_{y}) \hspace{0.1cm} \phi(y,t_{y}) \)