In[]:=
originalLines={{0,k},{1,k}}
Out[]=
{{0,k},{1,k}}
In[]:=
transformedLines=lorentz[β]/@{{0,k},{1,k}}
Out[]=
-
kβ
1-
2
β
,
k
1-
2
β
,
1-kβ
1-
2
β
,
k-β
1-
2
β

In[]:=
arbitraryPoint[line_]:={line〚1,1〛+q(line〚2,1〛-line〚1,1〛),line〚1,2〛+q(line〚2,2〛-line〚1,2〛)}
In[]:=
arbitraryPoint[originalLines]
Out[]=
{q,k}
In[]:=
arbitraryPoint[transformedLines]//FullSimplify
Out[]=

q-kβ
1-
2
β
,
k-qβ
1-
2
β

In[]:=
verticalCrossDensity[line_]:=With[{point=arbitraryPoint[line]},FullSimplify[1/((#/.k0)-(#/.k1))&@(point[[1]]/.Solve[point[[2]]0,q][[1]])]]
In[]:=
horizontalCrossDensity[line_]:=With[{point=arbitraryPoint[line]},FullSimplify[1/((#/.k1)-(#/.k0))&@(point[[2]]/.Solve[point[[1]]0,q][[1]])]]

Momentum

Vertical lines from each vertex
In[]:=
verticalCrossDensity[lorentz[β]/@{{0,k},{1,k}}]
Out[]=
-
β
1-
2
β
“Light-cone” lines going diagonally left and right from each vertex
In[]:=
{verticalCrossDensity[lorentz[β]/@{{0,k},{1/2,k+1/2}}],verticalCrossDensity[lorentz[β]/@{{0,k},{1/2,k-1/2}}]}//FullSimplify
Out[]=

1-β
1+β
,-
1+β
1-
2
β

The total is twice as large as vertical lines as there are two lines going out from each vertex
In[]:=
Total[%]//FullSimplify
Out[]=
-
2β
1-
2
β

Energy

Vertical lines from each vertex
In[]:=
horizontalCrossDensity[lorentz[β]/@{{0,k},{1,k}}]
Out[]=
1
1-
2
β
“Light-cone” lines going diagonally left and right from each vertex
In[]:=
{horizontalCrossDensity[lorentz[β]/@{{0,k},{1/2,k+1/2}}],horizontalCrossDensity[lorentz[β]/@{{0,k},{1/2,k-1/2}}]}//FullSimplify
Out[]=

1-β
1+β
,
1+β
1-
2
β

The total is twice as large as vertical lines as there are two lines going out from each vertex
In[]:=
Total[%]//FullSimplify
Out[]=
2
1-
2
β
In[]:=
{horizontalCrossDensity[lorentz[β]/@{{0,λk},{1/2,λk+α/2}}],horizontalCrossDensity[lorentz[β]/@{{0,λk},{1/2,λk-α/2}}]}//FullSimplify
Out[]=

1-αβ
1-
2
β
λ
,
1+αβ
1-
2
β
λ

In[]:=
{verticalCrossDensity[lorentz[β]/@{{0,k},{1/2,k+α/2}}],verticalCrossDensity[lorentz[β]/@{{0,k},{1/2,k-α/2}}]}//FullSimplify
Out[]=

α-β
1-
2
β
,-
α+β
1-
2
β

In[]:=
e0First[%27]^2-p0First[%28]^2//FullSimplify
Out[]=
p0
2
(α-β)
-e0
2
(-1+αβ)
-1+
2
β
In[]:=
1+αβ
1-
2
β
/.α1
Out[]=
1+β
1-
2
β
In[]:=
%27/.β0
Out[]=
{1,1}
In[]:=
%28/.β0
Out[]=
{α,-α}