The Dirac Equation: Understanding Spinors and Approximations

[park]

TL;DR

dirac equation and it's solution

I'm studying about dirac equation and it's solution.
When we starts with the equation (2.75), I can understand that it is possible to set 2 kinds of spinor.

But my question is...
1. After the assumption (2.100), how can we set the equation like (2.101)

2. I can't get (2.113) from (2.111) using (2.112)... Approximation and operator made me so crazy!
Please help me...

 

[nrqed]

First step: what is $(\vec{p} \cdot \sigma) (\vec{p} \cdot \vec{\sigma})$ equal to?

 

[park]

First step: what is $(\vec{p} \cdot \sigma) (\vec{p} \cdot \vec{\sigma})$ equal to?

Equal to $\boldsymbol{p}^2$ !

 

[nrqed]

Equal to $\boldsymbol{p}^2$ !

Right!

Now, the key point is that they work up to order $p^4$, i.e. they drop all terms of higher order.

So notice that the following term on the left of (2.111) is

$$ (T+e \phi) (-p^2/(8m^2c^2)= -p^4/(16m^3c^2) ~\text{plus terms of order } p^4 \text{ and higher}.$$

This term cancels exactly the term $- (\vec{p} \cdot \vec{\sigma})^2\, p^2/(16m^3c^2)$ that appears on the right side. This leaves Eq, (2.113), which is valid up to order $p^4$.