Dirac Matrices

Author

Eric W. Weisstein
September 14, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/LinearAlgebra/DiracMatrices.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DiracMatrices.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Definition

sp[1]:={{0,1},{1,0}};
sp[2]:={{0,-I},{I,0}};
sp[3]:={{1,0},{0,-1}};

σ[i_]:=Partition[Flatten[Transpose/@Outer[Times,IdentityMatrix[2],sp[i]]],4]

σ[0]:=IdentityMatrix[4]

ρ[i_]:=Partition[Flatten[Transpose/@Outer[Times,sp[i],IdentityMatrix[2]]],4]

ρ[0]:=IdentityMatrix[4]

e[i_,j_]:=r[i].s[j];

addn[i_]:=Mod[i,3]+1

?Dirac*

?SpecialFunctions`*

Table[ρ[i]//MatrixForm,{i,0,3}]

Table[σ[i]//MatrixForm,{i,0,3}]

Table[e[i,j]//MatrixForm,{i,0,3},{j,0,3}]

Table[MatrixForm[σ[i].σ[j]+σ[j].σ[i]-2KroneckerDelta[i-j]IdentityMatrix[4]],{i,3},{j,3}]

Table[MatrixForm[ρ[i].ρ[j]+ρ[j].ρ[i]-2KroneckerDelta[i-j]IdentityMatrix[4]],{i,3},{j,3}]

Table[MatrixForm[σ[i].ρ[j]-ρ[j].σ[i]],{i,3},{j,3}]

Table[MatrixForm[σ[i].σ[addn[i]]-I σ[addn[addn[i]]]],{i,3}]

Table[MatrixForm[ρ[i].ρ[addn[i]]-I ρ[addn[addn[i]]]],{i,1,3}]

Table[Det[e[i,j]],{i,0,3},{j,0,3}]

Table[MatrixForm[e[i,j].e[i,j]],{i,0,3},{j,0,3}]

Table[MatrixForm[(IdentityMatrix[4]+e[j,i])/2-(IdentityMatrix[4]+e[j,i]).(IdentityMatrix[4]+e[j,i])/4],{i,0,3},{j,3}]

Table[MatrixForm[e[i,j]-Conjugate[Transpose[e[i,j]]]],{i,0,3},{j,0,3}]

Table[Plus@@Table[e[i,j][[k,k]],{k,1,4}],{i,0,3},{j,0,3}]