Problem #1 :

Clear["Global`*"]

g=DiagonalMatrix[{1,-1,-1,-1}];
g //MatrixForm

p1={e1,0,0,+pz};
p2={e2,0,0,-pz};
p12=p1+p2;

p2.g.p2

p12.g.p12

eqs={p1.g.p1==m12,p2.g.p2==m22,p12.g.p12==s}

sol=Solve[eqs,{e1,e2,pz}] //Simplify;
sol[[4]]

tmp1= pz /.sol[[4]]

tmp2=(4 s)(tmp1^2) //Expand

del[a_,b_,c_]=(a^2)+(b^2)+(c^2)-2(a b + b c + c a);

del[m12,m22,s]==tmp2 //Simplify

Problem #2 :

a) CMS System

Clear["Global`*"]

g=DiagonalMatrix[{1,-1,-1,-1}];
g //MatrixForm

(* BEFORE *)
p1={e1,0,0,+pz};
p2={e2,0,0,-pz};
p12=p1+p2;
e1=e2=e; (* set energies equal *)

sBefore= p12.g.p12

(* AFTER *)
p3={m,0,0,0};
p4={m,0,0,0};
p5={m,0,0,0};
p6={m,0,0,0};
pAFTER=(p3+p4+p5+p6)
e1=e2=e; (* set energies equal *)

sAfter= pAFTER.g.pAFTER

eq= sBefore== sAfter

Solve[eq,e]

a) Fixed Target System

Clear["Global`*"]

g=DiagonalMatrix[{1,-1,-1,-1}];
g //MatrixForm

(* BEFORE *)
p1={e,0,0,+pz};
p2={m,0,0,0};
p12=p1+p2;

sBefore= p12.g.p12

eq1= p1.g.p1==(m^2)

sol= Solve[eq1,pz]

sBefore= sBefore /.sol[[2]] //Simplify

(* AFTER *)
p3={m,0,0,0};
p4={m,0,0,0};
p5={m,0,0,0};
p6={m,0,0,0};
pAFTER=(p3+p4+p5+p6)
e1=e2=e; (* set energies equal *)

sAfter= pAFTER.g.pAFTER

eq= sBefore== sAfter

Solve[eq,e]

Problem #3 :

a) CMS System

Clear["Global`*"]

g=DiagonalMatrix[{1,-1,-1,-1}];
g //MatrixForm

(* BEFORE *)
p1={e1,0,0,+pz};
p2={e2,0,0,-pz};
p12=p1+p2;
e1=e2=e; (* set energies equal *)

sBefore= p12.g.p12

(* AFTER *)
p3={mX,0,0,0};

sAfter= p3.g.p3

eq= sBefore== sAfter

msol=Solve[eq,e]

e /.msol[[2]] /.{m->1,mX->125} //N

a) Fixed Target System

Clear["Global`*"]

g=DiagonalMatrix[{1,-1,-1,-1}];
g //MatrixForm

(* BEFORE *)
p1={e,0,0,+pz};
p2={m,0,0,0};
p12=p1+p2;

sBefore= p12.g.p12

eq1= p1.g.p1==(m^2)

sol= Solve[eq1,pz]

sBefore= sBefore /.sol[[2]] //Simplify

(* AFTER *)
p3={mX,0,0,0}

sAfter=p3.g.p3

eq= sBefore== sAfter

msol= Solve[eq,e] //Simplify

e /.msol /.{m->1,mX->125} //N

Problem #5 :

a) z-Boost:

Clear["Global`*"]

boostz=γ{{1,0,0,β},{0,1,0,0},{0,0,1,0},{β,0,0,1}};
boostz //MatrixForm

(* CMS *)
p1={e,0,0,+pz};
p2={e,0,0,-pz};

boostz.p2

eq1= boostz.p2=={m,0,0,0} //Thread

eq1[[4]]

bsol= Solve[eq1[[4]],β][[1]]

Check the boost:

boostz.p2 /.bsol  //Together

pzSol=Solve[(e^2)-(pz^2)==(m^2),pz][[1]]

boostz.p2 /.bsol  /.pzSol /.{γ->e/m} //Together

Boost p1

tmp1= boostz.p1

tmp2= tmp1[[1]] /.bsol /.pzSol  /.{γ->e/m} //Together

tmp2 /.{e->2000,m->1}

tmp2 /.{(m^2)->0}  /.{e->2000,m->1}

tmp2 /.{(m^2)->0} /.{m->e/γ}

γ /.{γ->e/m}   /.{e->2000,m->1}

Scattering angle

eq= Tan[θlab] == (1/γ)Tan[((θcm)/2)]

eq /.{γ->2000}

sol= Solve[eq ,θlab][[1]]

θlab /.sol /.{γ->2000,θcm->90 Degree,(C[1])->0} //N

Problem #6 :

Clear["Global`*"]

f={{0,-ex,-ey,-ez},{ex,0,-bz,+by},{ey,bz,0,-bx},{ez,-by,bx,0}};
f //MatrixForm

Transpose[f]+f

boostz={{1,0,0,β},{0,1,0,0},{0,0,1,0},{β,0,0,1}};
boostz //MatrixForm

boostz.f.Transpose[boostz] //MatrixForm

Problem #8 :

Clear["Global`*"]

pmu={e,p Sin[θ],0,p Cos[θ]}

g=DiagonalMatrix[{1,-1,-1,-1}];

eq1= pmu.g.pmu ==m^2 //Simplify

sol=Solve[eq1,p][[2]]

arg= ((pmu[[1]] + pmu[[4]])/(pmu[[1]] - pmu[[4]])) /.sol /.{m-> x e}

y[θ_,x_]=(1/2)Log[arg] /.{e->1}

η[θ_]=-Log[Tan[θ/2]]

tab=Table[
 ParametricPlot[{y[θ,x],η[θ]},{θ,0,Pi},
PlotStyle->{Thick,Hue[x]}],
{x,-1,1,0.1}];

z=3;
Show[tab,PlotRange->{{-z,z},{-z,z}}]