View source for Total 4-momentum

=Total 4-momentum=
As was [[DV_Calculations_of_4-momentum_components#4-Momentum_Invariants | shown earlier]] the scalar product of a 4-Momentum vector with itself ,
<center><math>{\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2=s</math></center> ,

and the length of a 4-Momentum vector composed of 4-Momentum vectors, 

<center><math>{\mathbf P^2}=({\mathbf P_1}+{\mathbf P_2})^2=(E_1+E_2)^2-(\vec p_1 +\vec p_2 )^2=(m_1+m_2)^2=s</math></center>,

are invariant quantities.

It was [[DV_Calculations_of_4-momentum_components#Equal_masses | further shown ]] that 

<center><math>{\mathbf P^*}^2={\mathbf P}^2</math></center>


<center>''where'' <math>{\mathbf P^*}=({\mathbf P_1^*}+{\mathbf P_2^*})^2</math> ''represents the 4-Momentum Vector in the CM frame''</center>


<center> ''and'' <math>{\mathbf P}=({\mathbf P_1}+{\mathbf P_2})^2</math> ''represents the 4-Momentum Vector in the initial Lab frame''</center>

which can be expanded to 

<center><math>{\mathbf P^*}^2={\mathbf P^{'*}}^2={\mathbf P}^2={\mathbf P^'}^2</math></center>


<center>''where'' <math>{\mathbf P^'}=({\mathbf P_1^'}+{\mathbf P_2^'})^2</math> ''represents the 4-Momentum Vector in the final Lab frame''</center>


<center>''and'' <math>{\mathbf P^{'*}}=({\mathbf P_1^{'*}}+{\mathbf P_2^{'*}})^2</math> ''represents the 4-Momentum Vector in the final CM frame''</center>