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basic_tools:symbols

# Symbols

• Derivatives with respect to the four-vector $x^{\mu}=(ct,\vec{x})$ are denoted by

\begin{eqnarray} \partial_{\mu}\equiv {\partial\over \partial x^{\mu}} =\left({1\over c}{\partial\over\partial t},\vec{\nabla}\right). \end{eqnarray}

• Space-time indices are labelled by Greek letters ($\mu,\nu,\ldots=0,1,2,3$)
• Latin indices are used for spatial directions ($i,j,\ldots=1,2,3$).
• Moreover, $\sigma^{\mu}=(\mathbf{1},\sigma^{i})$ where $\sigma^{i}$ are the Pauli matrices $$\sigma^{1}=\left( \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right), \quad \sigma^{2}=\left( \begin{array}{rr} 0 & -i \\ i & 0 \end{array} \right), \quad \sigma^{3}=\left( \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array} \right).$$

Math Symbols \begin{align} &\mathbb{N} = \{0, 1, 2, 3, \ldots\} \\ &\mathbb{Z} = \{0, \pm 1, \pm 2, \pm 3, \ldots \} \\ &\mathbb{Q} = \rm Rational \; Numbers \it \\ &\mathbb{R} = \rm Real \; Numbers \it \\ &\mathbb{C} = \rm Complex \; Numbers \it \\ &\mathbb{Z}_n = \mathbb{Z} \; \mod \; n \\ &\Rightarrow \rm \; is \; read \; implies"\\ &\rm iff \; is \; read \; if\; and \; only \; if" \\ &\forall \rm \; is \; read \; for \; every" \\ &\exists \rm \; is \; read \; there \; exists" \\ &\in \; \rm is \; read \; in" \\ &\ni \; \rm is \; read \; such \; that" \\ &\dot{=} \; \rm is \; represented \; by" \\ &\subset \; \rm is \; subset\; of" \\ &\equiv \; \rm is\;defined\; as" \\ &= \quad \text{exactly equal} \notag \\ &\propto \quad \text{equality except perhaps for a factor with dimension} \notag\\ &\sim \quad \text{equality except perhaps for a factor without dimensions} \notag \\ &\approx \quad \text{equality except perhaps for a factor close to 1} \end{align} 