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| − | =Total 4-momentum=
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| − | As was [[DV_Calculations_of_4-momentum_components#4-Momentum_Invariants | shown earlier]] the scalar product of a 4-Momentum vector with itself ,
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| − | <center><math>{\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2</math></center> ,
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| − | and the length of a 4-Momentum vector composed of 4-Momentum vectors,
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| − | <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>,
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| − | are invariant quantities.
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| − | It was [[DV_Calculations_of_4-momentum_components#Equal_masses | further shown ]] that
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| − | <center><math>{\mathbf P^*}^2={\mathbf P}^2</math></center>
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| − | <center>''where'' <math>{\mathbf P^*}=({\mathbf P_1^*}+{\mathbf P_2^*})^2</math> ''represents the 4-Momentum Vector in the CM frame''</center>
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| − | <center> ''and'' <math>{\mathbf P}=({\mathbf P_1}+{\mathbf P_2})^2</math> ''represents the 4-Momentum Vector in the initial Lab frame''</center>
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| − | which can be expanded to
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| − | <center><math>{\mathbf P^*}^2={\mathbf P^{'*}}^2={\mathbf P}^2={\mathbf P^'}^2</math></center>
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| − | <center>''where'' <math>{\mathbf P^'}=({\mathbf P_1^'}+{\mathbf P_2^'})^2</math> ''represents the 4-Momentum Vector in the final Lab frame''</center>
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| − |
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| − | <center>''and'' <math>{\mathbf P^{'*}}=({\mathbf P_1^{'*}}+{\mathbf P_2^{'*}})^2</math> ''represents the 4-Momentum Vector in the final CM frame''</center>
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