How does the coefficient in the Boltzmann equation comes from

[nadia8999]

In the Boltzmann equation,
${\bf{L}}\left[ f \right] = {\bf{C}}\left[ f \right]$, the right which is the collision term and in general it is${\bf{C}}[f]=\sum\limits_{p,p_1} {|Amplitiude|^2}{f(p)}$. and explicitly, the collision term for decaying process is
$\begin{eqnarray} {\bf{C}}\left[ f \right] &=& \frac{1}{{E\left( p \right)}}\int {\frac{{{d^3}p}}{{{{\left( {2\pi } \right)}^3}2E\left( p \right)}}\frac{{{d^3}{p_2}}}{{{{\left( {2\pi } \right)}^3}2E\left( {{p_2}} \right)}}}\nonumber\\ &\times& {\left| {{\cal M}\left( {p \to {p_1} + {p_2}} \right)} \right|^2}f\left( p \right)\nonumber\\ &\times& {\left( {2\pi } \right)^4}{\delta ^4}(p - {p_1} + {p_2}) \end{eqnarray}$
I want to know exactly how does the coefficient in the equation comes from,for it is very important for the result.
My questions are:
1) how does the $E(p)$comes from

2) $\begin{equation} \int {\frac{{{d^4}p}}{{{{\left( {2\pi } \right)}^4}}}\left( {2\pi } \right)\delta ({E^2} - {m^2} - {p^2})} = \int {\frac{{{d^3}p}}{{{{\left( {2\pi } \right)}^3}}}\frac{{\delta (E - \sqrt {{m^2} + {p^2}} )}}{{2E}}} \end{equation}$,the$(2\pi)^4$in the left of the equation is the space element with$\hbar=1$,but how does the $2\pi$ comes from?

3) in the equation above, why cannot we write the $\delta$function is the left with$\delta (E - \sqrt {{m^2} + {p^2}} )$ directly, if so, then $2E$does not exist in denominator of the right equation.

4) why there is $(2\pi)^4$in the equation${\left( {2\pi } \right)^4}{\delta ^4}(p - {p_1} + {p_2})$.

can anyone give give explanations or recommend some books to which I can refer?

 

[AndyW]

Thank you for your questions about the Boltzmann equation and the collision term. I am happy to provide you with some explanations and recommend some resources for further reading.

1) The term E(p) in the collision term comes from the energy of the particle with momentum p. In the Boltzmann equation, we are considering a system of particles with different momenta and energies. The collision term takes into account the interactions between these particles, which can change their momenta and energies. The E(p) term is necessary to account for the energy conservation in these interactions.

2) The (2\pi)^4 in the left side of the equation represents the integration over all possible momentum states of the particles. This is a standard convention in quantum field theory, where the (2\pi)^4 term is used to normalize the integration measure. In this case, it is related to the space element and the uncertainty principle, which states that the product of the uncertainty in position and momentum must be at least as large as a certain value, in this case, \hbar=1. For more details, I recommend the book "Quantum Field Theory" by Mark Srednicki.

3) The reason why we cannot write the delta function as \delta (E - \sqrt {{m^2} + {p^2}} ) directly is because we are integrating over all possible momentum states. The integration measure in the denominator takes into account all possible energies, not just the one given by the delta function. This is necessary to ensure the energy conservation in the interactions.

4) The (2\pi)^4 term in the equation {\left( {2\pi } \right)^4}{\delta ^4}(p - {p_1} + {p_2}) represents the integration over all possible momentum states of the particles involved in the collision. It is a normalization factor used in quantum field theory to ensure that the collision term is properly normalized. For more details, I recommend the book "Quantum Field Theory and the Standard Model" by Matthew D. Schwartz.

I hope this helps to answer your questions. For further reading, I recommend the books mentioned above as well as "Introduction to Elementary Particles" by David Griffiths. Good luck with your studies!