Green’s function of Dirac operator

Pouramat
Messages
27
Reaction score
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.
 
Last edited:
Thread 'Need help understanding this figure on energy levels'
This figure is from "Introduction to Quantum Mechanics" by Griffiths (3rd edition). It is available to download. It is from page 142. I am hoping the usual people on this site will give me a hand understanding what is going on in the figure. After the equation (4.50) it says "It is customary to introduce the principal quantum number, ##n##, which simply orders the allowed energies, starting with 1 for the ground state. (see the figure)" I still don't understand the figure :( Here is...
Thread 'Understanding how to "tack on" the time wiggle factor'
The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself. Part (a) is quite easy. We get $$\sigma_1 = 2\lambda, \mathbf{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \sigma_2 = \lambda, \mathbf{v}_2 = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{pmatrix} \sigma_3 = -\lambda, \mathbf{v}_3 = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \\ 0 \end{pmatrix} $$ There are two ways...
Back
Top