Need help reviewing Mathematica expression which came from Physics

Question

Please help me check the conversion of these two expressions in physics to their respective Mathematica expression. Where sigma is the Pauli matrices in standard form, E/p0 is energy, m is mass and p is the momentum.

For eq(5.29) I got:

w = ((e + m)/(2*m))^(1/2) {{1 + p3/(e + m), (p1 - (I *p2))/(e + m), 0,
 0}, {(p1 + (I* p2))/(e + m), 1 - p3/(e + m), 0, 0}, {0, 0, 
1 - p3/(e + m), -((p1 - (I* p2))/(e + m))}, {0, 
0, -((p1 + (I* p2))/(e + m)), 1 + p3/(e + m)}}

And for eq(5.30)

P = (m^(-1))*((\[Gamma]0*p0) +(\[Gamma]1*p1) + (\[Gamma]2*
   p2) + (\[Gamma]3*p3));

Where,

\[Gamma]0 = {{0, 0, 1, 0}, {0, 0, 0, 1}, {1, 0, 0, 0}, {0, 1, 0, 0}};
\[Gamma]1 = {{0, 0, 0, 1}, {0, 0, 1, 0}, {0, -1, 0, 0}, {-1, 0, 0, 0}};
\[Gamma]2 = {{0, 0, 0, -I}, {0, 0, I, 0}, {0, I, 0, 0}, {-I, 0, 0, 0}};
\[Gamma]3 = {{0, 0, 1, 0}, {0, 0, 0, -1}, {-1, 0, 0, 0}, {0, 1, 0, 0}};

Answer 1

\[Sigma]p=Sum[PauliMatrix[i]*Symbol["p"<>ToString[i]],{i,1,3}]
Id2=IdentityMatrix[2];
O2=IdentityMatrix[2]*0;

wExp=(IdentityMatrix[4]+ArrayFlatten[{{\[Sigma]p/(e+m),O2},{O2,-\[Sigma]p/(e+m)}}])*Sqrt[(e+m)/(2m)];
%//MatrixForm

results in

while

\[Sigma]phat=Sum[PauliMatrix[i]*Symbol["phat"<>ToString[i]],{i,1,3}];
PExp=(ArrayFlatten[{{O2,Id2},{Id2,O2}}]*p0+ArrayFlatten[{{O2,\[Sigma]phat},{-\[Sigma]phat,O2}}]*p)/m;
%//MatrixForm

results in

The expression for w given by OP coincides with wExp while P and PExp differ since no explicit definitions for the expressions $\hat{p}$ and $p$ in Eq. (5.30) where given in the question. I would invite OP to fill in the blanks/expressions for $p$ and $\hat{p}$.