FeynCalc manual (development version)

FCMatrixProduct

FCMatrixProduct[mat1, mat2, ...] can be used to obtain products of matrices with entries containing noncommutative symbols. Using the usual Dot on such matrices would otherwise destroy the original ordering.

The resulting expression can be then further simplified using DotSimplify.

See also

Overview, DataType, DeclareNonCommutative, UnDeclareNonCommutative.

Examples

Generic matrices

Consider two generic 2×22 \times 2-matrices containing noncommutative heads

DeclareNonCommutative[opA, opB, opC, opD]
mat[1] = {{opA[1], opB[1]}, {opC[1], opD[1]}} 
 
mat[2] = {{opA[2], opB[2]}, {opC[2], opD[2]}}

(  opA(1)  opB(1)  opC(1)  opD(1))\left( \begin{array}{cc} \;\text{opA}(1) & \;\text{opB}(1) \\ \;\text{opC}(1) & \;\text{opD}(1) \\ \end{array} \right)

(  opA(2)  opB(2)  opC(2)  opD(2))\left( \begin{array}{cc} \;\text{opA}(2) & \;\text{opB}(2) \\ \;\text{opC}(2) & \;\text{opD}(2) \\ \end{array} \right)

Using the usual Dot product the elements of the resulting matrix are now multiplied with each other commutatively. Hence, the result is incorrect.

mat[1] . mat[2]

(  opA(1)  opA(2)+opB(1)  opC(2)  opA(1)  opB(2)+opB(1)  opD(2)  opA(2)  opC(1)+opC(2)  opD(1)  opB(2)  opC(1)+opD(1)  opD(2))\left( \begin{array}{cc} \;\text{opA}(1) \;\text{opA}(2)+\text{opB}(1) \;\text{opC}(2) & \;\text{opA}(1) \;\text{opB}(2)+\text{opB}(1) \;\text{opD}(2) \\ \;\text{opA}(2) \;\text{opC}(1)+\text{opC}(2) \;\text{opD}(1) & \;\text{opB}(2) \;\text{opC}(1)+\text{opD}(1) \;\text{opD}(2) \\ \end{array} \right)

With FCMatrixProduct the proper ordering is preserved

FCMatrixProduct[mat[1], mat[2]]

(  opA(1).opA(2)+opB(1).opC(2)  opA(1).opB(2)+opB(1).opD(2)  opC(1).opA(2)+opD(1).opC(2)  opC(1).opB(2)+opD(1).opD(2))\left( \begin{array}{cc} \;\text{opA}(1).\text{opA}(2)+\text{opB}(1).\text{opC}(2) & \;\text{opA}(1).\text{opB}(2)+\text{opB}(1).\text{opD}(2) \\ \;\text{opC}(1).\text{opA}(2)+\text{opD}(1).\text{opC}(2) & \;\text{opC}(1).\text{opB}(2)+\text{opD}(1).\text{opD}(2) \\ \end{array} \right)

We can also multiply more than two matrices at once

mat[3] = {{opA[3], opB[3]}, {opC[3], opD[3]}}

(  opA(3)  opB(3)  opC(3)  opD(3))\left( \begin{array}{cc} \;\text{opA}(3) & \;\text{opB}(3) \\ \;\text{opC}(3) & \;\text{opD}(3) \\ \end{array} \right)

out = FCMatrixProduct[mat[1], mat[2], mat[3]]

(  opB(1).(opC(2).opA(3)+opD(2).opC(3))+opA(1).(opA(2).opA(3)+opB(2).opC(3))  opA(1).(opA(2).opB(3)+opB(2).opD(3))+opB(1).(opC(2).opB(3)+opD(2).opD(3))  opC(1).(opA(2).opA(3)+opB(2).opC(3))+opD(1).(opC(2).opA(3)+opD(2).opC(3))  opC(1).(opA(2).opB(3)+opB(2).opD(3))+opD(1).(opC(2).opB(3)+opD(2).opD(3)))\left( \begin{array}{cc} \;\text{opB}(1).(\text{opC}(2).\text{opA}(3)+\text{opD}(2).\text{opC}(3))+\text{opA}(1).(\text{opA}(2).\text{opA}(3)+\text{opB}(2).\text{opC}(3)) & \;\text{opA}(1).(\text{opA}(2).\text{opB}(3)+\text{opB}(2).\text{opD}(3))+\text{opB}(1).(\text{opC}(2).\text{opB}(3)+\text{opD}(2).\text{opD}(3)) \\ \;\text{opC}(1).(\text{opA}(2).\text{opA}(3)+\text{opB}(2).\text{opC}(3))+\text{opD}(1).(\text{opC}(2).\text{opA}(3)+\text{opD}(2).\text{opC}(3)) & \;\text{opC}(1).(\text{opA}(2).\text{opB}(3)+\text{opB}(2).\text{opD}(3))+\text{opD}(1).(\text{opC}(2).\text{opB}(3)+\text{opD}(2).\text{opD}(3)) \\ \end{array} \right)

Now use DotSimplify to expand noncommutative products

DotSimplify[out]

(  opA(1).opB(2).opC(3)+opB(1).opC(2).opA(3)+opA(1).opA(2).opA(3)+opB(1).opD(2).opC(3)  opA(1).opB(2).opD(3)+opA(1).opA(2).opB(3)+opB(1).opC(2).opB(3)+opB(1).opD(2).opD(3)  opD(1).opC(2).opA(3)+opC(1).opA(2).opA(3)+opC(1).opB(2).opC(3)+opD(1).opD(2).opC(3)  opC(1).opA(2).opB(3)+opC(1).opB(2).opD(3)+opD(1).opC(2).opB(3)+opD(1).opD(2).opD(3))\left( \begin{array}{cc} \;\text{opA}(1).\text{opB}(2).\text{opC}(3)+\text{opB}(1).\text{opC}(2).\text{opA}(3)+\text{opA}(1).\text{opA}(2).\text{opA}(3)+\text{opB}(1).\text{opD}(2).\text{opC}(3) & \;\text{opA}(1).\text{opB}(2).\text{opD}(3)+\text{opA}(1).\text{opA}(2).\text{opB}(3)+\text{opB}(1).\text{opC}(2).\text{opB}(3)+\text{opB}(1).\text{opD}(2).\text{opD}(3) \\ \;\text{opD}(1).\text{opC}(2).\text{opA}(3)+\text{opC}(1).\text{opA}(2).\text{opA}(3)+\text{opC}(1).\text{opB}(2).\text{opC}(3)+\text{opD}(1).\text{opD}(2).\text{opC}(3) & \;\text{opC}(1).\text{opA}(2).\text{opB}(3)+\text{opC}(1).\text{opB}(2).\text{opD}(3)+\text{opD}(1).\text{opC}(2).\text{opB}(3)+\text{opD}(1).\text{opD}(2).\text{opD}(3) \\ \end{array} \right)

Dirac matrices in terms of Pauli matrices

Let us define Dirac matrices in the Dirac basis in terms of Pauli matrices

gamma[0] = {{1, 0}, {0, -1}}; 
 
gamma[i_] := {{0, CSI[i]}, {-CSI[i], 0}};

and express γiγjγi\gamma^i \gamma^j \gamma^i as a 2×22 \times 2-matrix

FCMatrixProduct[gamma[i], gamma[j], gamma[i]] 
 
DotSimplify[%]

(0.(σ‾j.(−σ‾i)+0.0)+σ‾i.(0.(−σ‾i)+(−σ‾j).0)0.(0.σ‾i+σ‾j.0)+σ‾i.((−σ‾j).σ‾i+0.0)0.(0.(−σ‾i)+(−σ‾j).0)+(−σ‾i).(σ‾j.(−σ‾i)+0.0)0.((−σ‾j).σ‾i+0.0)+(−σ‾i).(0.σ‾i+σ‾j.0))\left( \begin{array}{cc} 0.\left(\overline{\sigma }^j.\left(-\overline{\sigma }^i\right)+0.0\right)+\overline{\sigma }^i.\left(0.\left(-\overline{\sigma }^i\right)+\left(-\overline{\sigma }^j\right).0\right) & 0.\left(0.\overline{\sigma }^i+\overline{\sigma }^j.0\right)+\overline{\sigma }^i.\left(\left(-\overline{\sigma }^j\right).\overline{\sigma }^i+0.0\right) \\ 0.\left(0.\left(-\overline{\sigma }^i\right)+\left(-\overline{\sigma }^j\right).0\right)+\left(-\overline{\sigma }^i\right).\left(\overline{\sigma }^j.\left(-\overline{\sigma }^i\right)+0.0\right) & 0.\left(\left(-\overline{\sigma }^j\right).\overline{\sigma }^i+0.0\right)+\left(-\overline{\sigma }^i\right).\left(0.\overline{\sigma }^i+\overline{\sigma }^j.0\right) \\ \end{array} \right)

(0−σ‾i.σ‾j.σ‾iσ‾i.σ‾j.σ‾i0)\left( \begin{array}{cc} 0 & -\overline{\sigma }^i.\overline{\sigma }^j.\overline{\sigma }^i \\ \overline{\sigma }^i.\overline{\sigma }^j.\overline{\sigma }^i & 0 \\ \end{array} \right)