- #1

Pouramat

- 28

- 1

- Homework Statement
- My question comes from the textbook by Peskin & Schroeder,

If $$S_F(x-y)$$ is Green’s function of Dirac operator, how we should verify

$$ (i {\partial}_{\mu} \gamma^{\mu} -m)S_F (x-y)= i \delta^{(4)} (x-y) . $$

!!Didn’t know how to write slashed partial!!

all of $$\partial _x$$ in my solution are slashed but I did not know how to write it.

- Relevant Equations
- Using $$S_F(x-y)$$ definition:

\begin{align}

S_F(x-y) &= < 0|T \psi (x) \bar\psi (y) |0> \\

& = \theta(x^0-y^0) <0|\psi (x) \bar\psi (y) |0>- \theta(y^0-x^0) <0|\bar\psi (y) \psi (x) |0>

\end{align}

I started from eq(3.113) and (3.114) of Peskin and merge them with upper relation for $S_F$, as following:

\begin{align}

S_F(x-y) &=

\theta(x^0-y^0)(i \partial_x +m) D(x-y) -\theta(y^0-x^0)(i \partial_x -m) D(y-x) \\

&= \theta(x^0-y^0)(i \partial_x +m) < 0| \phi(x) \phi(y)|0 > -\theta(y^0-x^0)(i \partial_x -m) < 0| \phi(y) \phi(x)|0 >

\end{align}

Now we can calculate Green's Function of Dirac operator using this form of $S_F$

\begin{align}

(i \partial_x -m) S_F =& [(i \partial_x -m) \theta(x^0-y^0)][(i \partial_x +m) < 0| \phi(x) \phi(y)|0 >]\\

&+\theta(x^0-y^0)[(\partial^2-m^2) <0| \phi(x) \phi(y)|0>] \\

&-[(i \partial_x -m) \theta(y^0-x^0)][(i \partial_x-m) <0| \phi(y) \phi(x)|0 >] \\

&- \theta(y^0-x^0)[(i \partial_x -m)(i \partial_x -m)< 0| \phi(y) \phi(x)|0 >]

\end{align}

All of the terms are fine except the last line.The 1st and 3rd terms simplify as following The 2nd term is zero using klein-Gordon equation

The 1st term :

\begin{equation}

[(i \partial_x -m) \theta(x^0-y^0)][(i \partial_x +m) < 0| \phi(x) \phi(y)|0 >] = [-\partial_0 \theta(x^0-y^0)][<0| \pi(x) \phi(y)|0>]

\end{equation}

The 3nd term:

\begin{equation}

[(i \partial_x -m) \theta(y^0-x^0)][(i \partial_x +m) < 0| \phi(y) \phi(x)|0 >] = [-\partial_0 \theta(y^0-x^0)][< 0| \phi(x) \pi(y)|0 >]

\end{equation}

if the 4th term like the 2nd term was Klein-Gordon equation the problem gets solved, but it isn't.

\begin{align}

S_F(x-y) &=

\theta(x^0-y^0)(i \partial_x +m) D(x-y) -\theta(y^0-x^0)(i \partial_x -m) D(y-x) \\

&= \theta(x^0-y^0)(i \partial_x +m) < 0| \phi(x) \phi(y)|0 > -\theta(y^0-x^0)(i \partial_x -m) < 0| \phi(y) \phi(x)|0 >

\end{align}

Now we can calculate Green's Function of Dirac operator using this form of $S_F$

\begin{align}

(i \partial_x -m) S_F =& [(i \partial_x -m) \theta(x^0-y^0)][(i \partial_x +m) < 0| \phi(x) \phi(y)|0 >]\\

&+\theta(x^0-y^0)[(\partial^2-m^2) <0| \phi(x) \phi(y)|0>] \\

&-[(i \partial_x -m) \theta(y^0-x^0)][(i \partial_x-m) <0| \phi(y) \phi(x)|0 >] \\

&- \theta(y^0-x^0)[(i \partial_x -m)(i \partial_x -m)< 0| \phi(y) \phi(x)|0 >]

\end{align}

All of the terms are fine except the last line.The 1st and 3rd terms simplify as following The 2nd term is zero using klein-Gordon equation

The 1st term :

\begin{equation}

[(i \partial_x -m) \theta(x^0-y^0)][(i \partial_x +m) < 0| \phi(x) \phi(y)|0 >] = [-\partial_0 \theta(x^0-y^0)][<0| \pi(x) \phi(y)|0>]

\end{equation}

The 3nd term:

\begin{equation}

[(i \partial_x -m) \theta(y^0-x^0)][(i \partial_x +m) < 0| \phi(y) \phi(x)|0 >] = [-\partial_0 \theta(y^0-x^0)][< 0| \phi(x) \pi(y)|0 >]

\end{equation}

if the 4th term like the 2nd term was Klein-Gordon equation the problem gets solved, but it isn't.

Last edited: