I am trying to derive Lorentz Invariant phase volume,

$$\int \frac{d^3p}{2E} = \int d^4p \delta(p^2-m_0^2) \Theta(p_0)$$

$$\int dp_0 \delta(p^2 -m_0^2) \Theta (p_0) = \int dp_0 \delta(p_0^2-E_p^2)\Theta(p_0) $$ $$=\int \frac{1}{2E}(\delta(p_0-E) + \delta(p_0+E) dp_0 \Theta(p_0)$$ $$=\frac{1}{2E}$$ $$hence \int \frac{d^3p}{2E} = \int d^4p\delta (p^2 -m_0^2)\Theta(p_0)$$

My only doubt is the $$\delta(p^2-m_0^2)$$ is not defined owing to Einstien relation$$(p^2=m^2)$$ so whether this integration is defined...

Please help me...

  • $\begingroup$ What do you mean by $\delta(p^2-m_0^2)$ "not being defined"? $\endgroup$
    – ACuriousMind
    Jan 22, 2022 at 15:30
  • $\begingroup$ due to mass-shell condtion p^2 = m^2 always fixed then what is meaning of integartion from +\infty to -\infty in momentum space $\endgroup$
    – Vivek
    Jan 22, 2022 at 15:35
  • 2
    $\begingroup$ The integration is over all of 4-momentum space, not just over $p$s that fulfill $p^2 = m_0^2$. That's why the $\delta$ is there. $\endgroup$
    – ACuriousMind
    Jan 22, 2022 at 15:46
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/83260/2451 , physics.stackexchange.com/q/53534/2451 and links therein. $\endgroup$
    – Qmechanic
    Jan 22, 2022 at 15:52
  • $\begingroup$ got it, so p^0 take value very close to E, and d^3p is all over space. Is it correct $\endgroup$
    – Vivek
    Jan 22, 2022 at 15:54