- #51
- 24,488
- 15,057
Sure, you can use the Dirac lemma, when you formulate the propagator as an p^0 integral along the real axis and define the integrand by shifting the poles infinitesimally up and/or down away from the real axis. E.g., for the free time-ordered propagator, needed in vacuum-qft perturbation theory
\Delta(p)=\frac{1}{p^2-m^2+\mathrm{i} 0^+}.
This you can reformulate as
\Delta(p)=\mathrm{PV} \frac{1}{p^2-m^2} - \mathrm{i} \pi \delta(p^2-m^2).
The proof is simpler for the statement
\frac{1}{p^0-z+\mathrm{i} 0^+}=\mathrm{PV}\frac{1}{p_0-z}-\mathrm{i} \pi \delta(p^0-z)
and using the residue theorm on an arbitrary holomorphic test function.
\Delta(p)=\frac{1}{p^2-m^2+\mathrm{i} 0^+}.
This you can reformulate as
\Delta(p)=\mathrm{PV} \frac{1}{p^2-m^2} - \mathrm{i} \pi \delta(p^2-m^2).
The proof is simpler for the statement
\frac{1}{p^0-z+\mathrm{i} 0^+}=\mathrm{PV}\frac{1}{p_0-z}-\mathrm{i} \pi \delta(p^0-z)
and using the residue theorm on an arbitrary holomorphic test function.
