Difference between revisions of "Variables Used in Elastic Scattering"

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=Lorentz Invariant Quantities=
 
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</math></center>
 
 
 
<center>''where'' <math>{\mathbf P^'}</math> ''represents the 4-Momentum Vector in the final Lab frame''</center>
 
 
=Mandelstam Representation=
 
 
[[File:Mandelstam.png | 400 px]]
 

Latest revision as of 19:10, 1 June 2017