Dirac Principle Value Identity applied to Propagators

[maverick280857]

Hi,

How is

$$\frac{1}{\displaystyle{\not}{P}-m+i\epsilon}-\frac{1}{\displaystyle{\not}{P}-m-i\epsilon} = \frac{2\pi}{i}(\displaystyle{\not}{P}+m)\delta(P^2-m^2)$$

? This is equation (4-91) of Itzykson and Zuber (page 189). I know that

$$\frac{1}{x\mp i\epsilon} = \mathcal{P}\left(\frac{1}{x}\right) \pm i\pi\delta(x)$$

But this doesn't seem to give the right hand side of the first equation above. What am I missing?

Thanks in advance!

 

[maverick280857]

Hi,

How is

$$\frac{1}{\displaystyle{\not}{P}-m+i\epsilon}-\frac{1}{\displaystyle{\not}{P}-m-i\epsilon} = \frac{2\pi}{i}(\displaystyle{\not}{P}+m)\delta(P^2-m^2)$$

? This is equation (4-91) of Itzykson and Zuber (page 189). I know that

$$\frac{1}{x\mp i\epsilon} = \mathcal{P}\left(\frac{1}{x}\right) \pm i\pi\delta(x)$$

But this doesn't seem to give the right hand side of the first equation above. What am I missing?

Thanks in advance!

How does the $(\displaystyle{\not}{P} + m)$ appear?

 

[Bill_K]

It's because P/ - m is a matrix, and so first you have to write 1/(P/ - m) as (P/ + m)/(P² - m²).

So in detail,

1/(P/ - m + iε) - 1/(P/ - m + iε) = (P/ + m)[1/(P² - m² + iε) - 1/(P² - m² - iε)]
= (P/ + m)[-iπ δ(P² - m²) -iπ δ(P² - m²)] = (P/ + m)(-2iπ δ(P² - m²))

 

[maverick280857]

It's because P/ - m is a matrix, and so first you have to write 1/(P/ - m) as (P/ + m)/(P² - m²).

So in detail,

1/(P/ - m + iε) - 1/(P/ - m + iε) = (P/ + m)[1/(P² - m² + iε) - 1/(P² - m² - iε)]
= (P/ + m)[-iπ δ(P² - m²) -iπ δ(P² - m²)] = (P/ + m)(-2iπ δ(P² - m²))

Thanks BillK, that cleared it up!