Difference between revisions of "Variables Used in Elastic Scattering"

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<center><math>{\mathbf P^*}^2={\mathbf P}^2</math></center>
 
<center><math>{\mathbf P^*}^2={\mathbf P}^2</math></center>
  
<center>''where'' <math>{\mathbf P^*}</math> ''represents the 4-Momentum Vector in the CM frame''</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}</math> ''represents the 4-Momentum Vector in the initial Lab 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  
 
which can be expanded to  

Revision as of 19:28, 31 January 2016

Lorentz Invariant Quantities

As was shown earlier the scalar product of a 4-Momentum vector with itself ,

[math]{\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2=s[/math]

,

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

[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]

,

are invariant quantities.

It was further shown that

[math]{\mathbf P^*}^2={\mathbf P}^2[/math]
where [math]{\mathbf P^*}==({\mathbf P_1^*}+{\mathbf P_2^*})^2[/math] represents the 4-Momentum Vector in the CM frame
and [math]{\mathbf P}==({\mathbf P_1}+{\mathbf P_2})^2[/math] represents the 4-Momentum Vector in the initial Lab frame

which can be expanded to

[math]{\mathbf P^*}^2={\mathbf P}^2={\mathbf P^'}^2[/math]
where [math]{\mathbf P^'}[/math] represents the 4-Momentum Vector in the final Lab frame

Mandelstam Representation

Mandelstam.png