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| ==New 4-Momentum Quantities== | | ==New 4-Momentum Quantities== |
| Working in just the Lab frame, we can form new | | Working in just the Lab frame, we can form new 4-Momentum Vectors comprised of 4-Momenta in this frame, with |
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| <center><math>{\mathbf P_1}- {\mathbf P_1^'}= \left( \begin{matrix}E_1-E_1'\\ p_{1(x)}-p_{1(x)}^' \\ p_{1(y)}-p_{1(y)}^' \\ p_{1(z)}-p_{1(z)}^'\end{matrix} \right)={\mathbf P_a}</math></center> | | <center><math>{\mathbf P_1}- {\mathbf P_1^'}= \left( \begin{matrix}E_1-E_1'\\ p_{1(x)}-p_{1(x)}^' \\ p_{1(y)}-p_{1(y)}^' \\ p_{1(z)}-p_{1(z)}^'\end{matrix} \right)={\mathbf P_a}</math></center> |
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| <center><math>{\mathbf P_2}- {\mathbf P_2^'}= \left( \begin{matrix}E_2-E_2'\\ p_{2(x)}-p_{2(x)}^' \\ p_{2(y)}-p_{2(y)}^' \\ p_{2(z)}-p_{2(z)}^'\end{matrix} \right)={\mathbf P_d}</math></center> | | <center><math>{\mathbf P_2}- {\mathbf P_2^'}= \left( \begin{matrix}E_2-E_2'\\ p_{2(x)}-p_{2(x)}^' \\ p_{2(y)}-p_{2(y)}^' \\ p_{2(z)}-p_{2(z)}^'\end{matrix} \right)={\mathbf P_d}</math></center> |
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| <center><math>{\mathbf P_1}- {\mathbf P_1^'}= \left( \begin{matrix}E_1-E_1'\\ p_{1(x)}-p_{1(x)}^' \\ p_{1(y)}-p_{1(y)}^' \\ p_{1(z)}-p_{1(z)}^'\end{matrix} \right)={\mathbf P_a}</math></center>
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| <center><math>{\mathbf P_1}- {\mathbf P_1^'}= \left( \begin{matrix}E_1-E_1'\\ p_{1(x)}-p_{1(x)}^' \\ p_{1(y)}-p_{1(y)}^' \\ p_{1(z)}-p_{1(z)}^'\end{matrix} \right)={\mathbf P_a}</math></center>
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| =Mandelstam Representation= | | =Mandelstam Representation= |
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| [[File:Mandelstam.png | 400 px]] | | [[File:Mandelstam.png | 400 px]] |
Revision as of 19:52, 31 January 2016
Lorentz Invariant Quantities
Total 4-Momentums
As was shown earlier the scalar product of a 4-Momentum vector with itself ,
[math]\displaystyle{ {\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]\displaystyle{ {\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]\displaystyle{ {\mathbf P^*}^2={\mathbf P}^2 }[/math]
where [math]\displaystyle{ {\mathbf P^*}=({\mathbf P_1^*}+{\mathbf P_2^*})^2 }[/math] represents the 4-Momentum Vector in the CM frame
and [math]\displaystyle{ {\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]\displaystyle{ {\mathbf P^*}^2={\mathbf P}^2={\mathbf P^'}^2 }[/math]
where [math]\displaystyle{ {\mathbf P^'}=({\mathbf P_1^'}+{\mathbf P_2^'})^2 }[/math] represents the 4-Momentum Vector in the final Lab frame
New 4-Momentum Quantities
Working in just the Lab frame, we can form new 4-Momentum Vectors comprised of 4-Momenta in this frame, with
[math]\displaystyle{ {\mathbf P_1}- {\mathbf P_1^'}= \left( \begin{matrix}E_1-E_1'\\ p_{1(x)}-p_{1(x)}^' \\ p_{1(y)}-p_{1(y)}^' \\ p_{1(z)}-p_{1(z)}^'\end{matrix} \right)={\mathbf P_a} }[/math]
[math]\displaystyle{ {\mathbf P_1}- {\mathbf P_2^'}= \left( \begin{matrix}E_1-E_2'\\ p_{1(x)}-p_{2(x)}^' \\ p_{1(y)}-p_{2(y)}^' \\ p_{1(z)}-p_{2(z)}^'\end{matrix} \right)={\mathbf P_b} }[/math]
[math]\displaystyle{ {\mathbf P_2}- {\mathbf P_1^'}= \left( \begin{matrix}E_2-E_1'\\ p_{2(x)}-p_{1(x)}^' \\ p_{2(y)}-p_{1(y)}^' \\ p_{2(z)}-p_{1(z)}^'\end{matrix} \right)={\mathbf P_c} }[/math]
[math]\displaystyle{ {\mathbf P_2}- {\mathbf P_2^'}= \left( \begin{matrix}E_2-E_2'\\ p_{2(x)}-p_{2(x)}^' \\ p_{2(y)}-p_{2(y)}^' \\ p_{2(z)}-p_{2(z)}^'\end{matrix} \right)={\mathbf P_d} }[/math]
Mandelstam Representation
