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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...

Qmechanic
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Vivek
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  • What do you mean by $\delta(p^2-m_0^2)$ "not being defined"? – ACuriousMind Jan 22 '22 at 15:30
  • due to mass-shell condtion p^2 = m^2 always fixed then what is meaning of integartion from +\infty to -\infty in momentum space – Vivek Jan 22 '22 at 15:35
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    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. – ACuriousMind Jan 22 '22 at 15:46
  • Possible duplicates: https://physics.stackexchange.com/q/83260/2451 , https://physics.stackexchange.com/q/53534/2451 and links therein. – Qmechanic Jan 22 '22 at 15:52
  • got it, so p^0 take value very close to E, and d^3p is all over space. Is it correct – Vivek Jan 22 '22 at 15:54
  • Thank you so much Your answer helped I am new to QED. I was just wandering whether K_0 or p_0 as variable can have value other than E_k (off-shell) – Vivek Jan 22 '22 at 16:40

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