FeynCalc manual (development version)

Contract

Contract[expr] contracts pairs of Lorentz or Cartesian indices of metric tensors, vectors and (depending on the value of the option EpsContract) of Levi-Civita tensors in expr.

For contractions of Dirac matrices with each other use DiracSimplify.

Contract[exp1, exp2] contracts (exp1*exp2), where exp1 and exp2 may be larger products of sums of metric tensors and 4-vectors. This can be also useful when evaluating polarization sums, where exp2 should be the product (or expanded sum) of the polarization sums for the vector bosons.

See also

Overview, Pair, CartesianPair, DiracSimplify, MomentumCombine.

Examples

MT[\[Mu], \[Nu]] FV[p, \[Mu]] 
 
Contract[%]

p‾μgˉμν\overline{p}^{\mu } \bar{g}^{\mu \nu }

p‾ν\overline{p}^{\nu }

FV[p, \[Mu]] GA[\[Mu]] 
 
Contract[%]

γˉμp‾μ\bar{\gamma }^{\mu } \overline{p}^{\mu }

γˉ⋅p‾\bar{\gamma }\cdot \overline{p}

The default dimension for a metric tensor is 4.

MT[\[Mu], \[Mu]] 
 
Contract[%]

gˉμμ\bar{g}^{\mu \mu }

44

A quick way to enter DD-dimensional metric tensors is given by MTD.

MTD[\[Mu], \[Nu]]  MTD[\[Mu], \[Nu]] 
 
Contract[%]

(gμν)2(g^{\mu \nu})^2

DD

FV[p, \[Mu]] FV[q, \[Mu]] 
 
Contract[% ]

p‾μq‾μ\overline{p}^{\mu } \overline{q}^{\mu }

p‾⋅q‾\overline{p}\cdot \overline{q}

FV[p - q, \[Mu]] FV[a - b, \[Mu]] 
 
Contract[%]

(a‾−b‾)μ(p‾−q‾)μ\left(\overline{a}-\overline{b}\right)^{\mu } \left(\overline{p}-\overline{q}\right)^{\mu }

a‾⋅p‾−a‾⋅q‾−b‾⋅p‾+b‾⋅q‾\overline{a}\cdot \overline{p}-\overline{a}\cdot \overline{q}-\overline{b}\cdot \overline{p}+\overline{b}\cdot \overline{q}

FVD[p - q, \[Nu]] FVD[a - b, \[Nu]] 
 
Contract[%]

(a−b)ν(p−q)ν(a-b)^{\nu } (p-q)^{\nu }

a⋅p−a⋅q−b⋅p+b⋅qa\cdot p-a\cdot q-b\cdot p+b\cdot q

LC[\[Mu], \[Nu], \[Alpha], \[Sigma]] FV[p, \[Sigma]] 
 
Contract[%]

p‾σϵˉμνασ\overline{p}^{\sigma } \bar{\epsilon }^{\mu \nu \alpha \sigma }

ϵˉαμνp‾\bar{\epsilon }^{\alpha \mu \nu \overline{p}}

LC[\[Mu], \[Nu], \[Alpha], \[Beta]] LC[\[Mu], \[Nu], \[Alpha], \[Sigma]] 
 
Contract[%]

ϵˉμναβϵˉμνασ\bar{\epsilon }^{\mu \nu \alpha \beta } \bar{\epsilon }^{\mu \nu \alpha \sigma }

−6gˉβσ-6 \bar{g}^{\beta \sigma }

LCD[\[Mu], \[Nu], \[Alpha], \[Beta]] LCD[\[Mu], \[Nu], \[Alpha], \[Sigma]] 
 
Contract[%] // Factor2

ϵμναβϵμνασ\overset{\text{}}{\epsilon }^{\mu \nu \alpha \beta } \overset{\text{}}{\epsilon }^{\mu \nu \alpha \sigma }

(1−D)(2−D)(3−D)gβσ(1-D) (2-D) (3-D) g^{\beta \sigma }

Contractions of Cartesian tensors are also possible. They can live in 33, D−1D-1 or D−4D-4 dimensions.

KD[i, j] CV[p, i] 
 
Contract[%]

p‾iδˉij\overline{p}^i \bar{\delta }^{ij}

p‾j\overline{p}^j

CV[p, i] CGA[i] 
 
Contract[%]

γ‾ip‾i\overline{\gamma }^i \overline{p}^i

γ‾⋅p‾\overline{\gamma }\cdot \overline{p}

KD[i, i] 
 
Contract[%]

δˉii\bar{\delta }^{ii}

33

KD[i, j]^2 
 
Contract[%]

(δˉij)2(\bar{\delta}^{ij})^2

33

CV[p - q, j] CV[a - b, j] 
 
Contract[%]

(a‾−b‾)j(p‾−q‾)j\left(\overline{a}-\overline{b}\right)^j \left(\overline{p}-\overline{q}\right)^j

(a‾−b‾)⋅(p‾−q‾)(\overline{a}-\overline{b})\cdot (\overline{p}-\overline{q})

CLC[i, j, k] CV[p, k] 
 
Contract[%]

p‾kϵˉijk\overline{p}^k \bar{\epsilon }^{ijk}

ϵˉijp‾\bar{\epsilon }^{ij\overline{p}}

CLC[i, j, k] CLC[i, j, l] 
 
Contract[%]

ϵˉijkϵˉijl\bar{\epsilon }^{ijk} \bar{\epsilon }^{ijl}

2δˉkl2 \bar{\delta }^{kl}

CLCD[i, j, k] CLCD[i, j, l] 
 
Contract[%] // Factor2

ϵijkϵijl\overset{\text{}}{\epsilon }^{ijk} \overset{\text{}}{\epsilon }^{ijl}

(2−D)(3−D)δkl(2-D) (3-D) \delta ^{kl}