Definitions

• Definition of a group $(G,\bullet )$ which is a set of elements $G=\{a,b,c,\cdots\}$ and a composition or operation $\bullet$ which has the following properties

  • Closure: $\forall a,b \in G, c=a\bullet b\in G$
  • Associativity: $\forall a,b,c \in G, a\bullet(b\bullet c)=(a\bullet b)\bullet c$
  • Identity: $\exist e\in G$ such that $e\bullet a=a\bullet e\forall a \in G$
  • Inverse: $\forall a \in G \exist a^{-1} \in G$ such that $a^{-1}\bullet a= a\bullet a^{-1}=e$

• $O(n)=\{ M\in \mathbb{GL}(n,\R):MM^T=M^TM=I \}$ the group of orthogonal $n\times n$ matrices

• $SO(n)=\{M\in O(n):\det(M)=1\}$ the special orthogonal group which are the rotation matrices

• Minkowski metric and inverse:

  $$ \eta_{\mu\upsilon}=\eta^{\mu\upsilon}=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} $$

• Inner product of two 4-vectors

  $$ \overset{↝}{\bold A}\cdot \overset{↝}{\bold B}=\eta_{\mu\upsilon}A_\mu B_\upsilon=A_0B_0- A_iB_i=A_0B_0-\vec A\cdot \vec B $$

• Common 4-vectors

  $$ U^\mu=\frac{\text dX^\mu}{\text d\tau}=\left ( c\frac{\text dt}{\text d\tau},\frac{\text d\vec x}{\text d\tau} \right )=\gamma(c,\vec v) $$

  where $\eta_{\mu\upsilon}U^\mu U ^\upsilon=c^2$

  $$ P^\mu=m\frac{\text dX^\mu}{\text d\tau}=m U^\mu $$

  where $P_\mu P^\mu=(mc)^2$

  $$ k^\mu=\left ( \frac \omega c ,\vec k \right ) $$

  where $k_\mu k^\mu=0$

• Contravariant and covariant indices $A-\mu=\eta_{\mu\upsilon} A^\upsilon$ and $A^\mu=\eta^{\mu\upsilon} A_\upsilon$

Lorentz

• Lorentz invariants

  $$ \overset{↝}{\bold A}\cdot \overset{↝}{\bold B}=\eta_{\mu\upsilon}A^\mu B^\upsilon=A_\mu B^\mu=A^\mu B_\mu $$

• Lorentz transformation

  $$ A'^\mu=\Lambda ^\mu_\upsilon A^\upsilon \quad ;\quad A_\alpha'=[(\Lambda^T)^{-1}]\alpha^\beta A\beta=(\Lambda^{-1})^\beta_\alpha A_\beta $$

• Doppler effect and aberration

  $$ \begin{aligned} p'&=\gamma p \left ( 1-\frac vc \cos\theta \right ) \\ \cos\theta'&=\frac{\cos\theta-\frac vc}{1-\frac vc\cos\theta} \\ \sin\theta'&=\frac{\sin\theta}{\gamma(1-\frac vc \cos\theta)} \end{aligned} $$

Collisions

• Conservation of 4-momentum in a 2 - 2 scatter

  $$ \overset{↝}{\bold P}_1+\overset{↝}{\bold P}_2=\overset{↝}{\bold P}_A+\overset{↝}{\bold P}_B $$

  which implies that the energy is conserved $E_1+E_2=E_A+E_B$ as is the 3-momentum $\vec p_1+\vec p_2=\vec p_A+\vec p_B$

• The on-shell conditions imply that $\overset{↝}{\bold P}_i^2=(m_i c)^2$ for $i=1,2,A,B$ and in particular $\overset{↝}{\bold P}_i^2=0$ for massless particles

• Invariant mass

  $$ (m_\text{inv}c)^2=\left ( \frac 1c \sum_i E_i \right )^2- \left ( \sum_i \vec p_i \right )^2 $$

  and at threshold

  $$ m_\text{inv}c=\frac 12 \sum_i E_i=\sum_i\sqrt{|\vec p_i|^2+(m_ic)^2}\ge c\sum_i m_i $$