FeynCalc manual (development version)

FCGramMatrix

FCGramMatrix[{p1, p2, ...}] creates a Gram matrix from the given list of momenta.

See also

Overview, FCGramDeterminant.

Examples

FCGramMatrix[{p1, p2}]

(2  p122(p1  p2)2(p1  p2)2  p22)\left( \begin{array}{cc} 2 \;\text{p1}^2 & 2 (\text{p1}\cdot \;\text{p2}) \\ 2 (\text{p1}\cdot \;\text{p2}) & 2 \;\text{p2}^2 \\ \end{array} \right)

FCGramMatrix[{p1, p2, p3}]

(2  p122(p1  p2)2(p1  p3)2(p1  p2)2  p222(p2  p3)2(p1  p3)2(p2  p3)2  p32)\left( \begin{array}{ccc} 2 \;\text{p1}^2 & 2 (\text{p1}\cdot \;\text{p2}) & 2 (\text{p1}\cdot \;\text{p3}) \\ 2 (\text{p1}\cdot \;\text{p2}) & 2 \;\text{p2}^2 & 2 (\text{p2}\cdot \;\text{p3}) \\ 2 (\text{p1}\cdot \;\text{p3}) & 2 (\text{p2}\cdot \;\text{p3}) & 2 \;\text{p3}^2 \\ \end{array} \right)

FCGramMatrix[{p1, p2, p3}, Head -> {CartesianPair, CartesianMomentum},Dimension -> D - 1] 
 
Det[%]

(2  p122(p1  p2)2(p1  p3)2(p1  p2)2  p222(p2  p3)2(p1  p3)2(p2  p3)2  p32)\left( \begin{array}{ccc} 2 \;\text{p1}^2 & 2 (\text{p1}\cdot \;\text{p2}) & 2 (\text{p1}\cdot \;\text{p3}) \\ 2 (\text{p1}\cdot \;\text{p2}) & 2 \;\text{p2}^2 & 2 (\text{p2}\cdot \;\text{p3}) \\ 2 (\text{p1}\cdot \;\text{p3}) & 2 (\text{p2}\cdot \;\text{p3}) & 2 \;\text{p3}^2 \\ \end{array} \right)

8  p32(p1  p2)28  p12(p2  p3)28  p22(p1  p3)2+8  p12  p22  p32+16(p1  p2)(p1  p3)(p2  p3)-8 \;\text{p3}^2 (\text{p1}\cdot \;\text{p2})^2-8 \;\text{p1}^2 (\text{p2}\cdot \;\text{p3})^2-8 \;\text{p2}^2 (\text{p1}\cdot \;\text{p3})^2+8 \;\text{p1}^2 \;\text{p2}^2 \;\text{p3}^2+16 (\text{p1}\cdot \;\text{p2}) (\text{p1}\cdot \;\text{p3}) (\text{p2}\cdot \;\text{p3})

FCGramDeterminant[{p1, p2, p3}, Head -> {CartesianPair, CartesianMomentum}, Dimension -> D - 1]

8  p32(p1  p2)28  p12(p2  p3)28  p22(p1  p3)2+8  p12  p22  p32+16(p1  p2)(p1  p3)(p2  p3)-8 \;\text{p3}^2 (\text{p1}\cdot \;\text{p2})^2-8 \;\text{p1}^2 (\text{p2}\cdot \;\text{p3})^2-8 \;\text{p2}^2 (\text{p1}\cdot \;\text{p3})^2+8 \;\text{p1}^2 \;\text{p2}^2 \;\text{p3}^2+16 (\text{p1}\cdot \;\text{p2}) (\text{p1}\cdot \;\text{p3}) (\text{p2}\cdot \;\text{p3})