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.
Overview, Pair, CartesianPair, DiracSimplify, MomentumCombine.
MT[\[Mu], \[Nu]] FV[p, \[Mu]]
Contract[%]\overline{p}^{\mu } \bar{g}^{\mu \nu }
\overline{p}^{\nu }
FV[p, \[Mu]] GA[\[Mu]]
Contract[%]\bar{\gamma }^{\mu } \overline{p}^{\mu }
\bar{\gamma }\cdot \overline{p}
The default dimension for a metric tensor is 4.
MT[\[Mu], \[Mu]]
Contract[%]\bar{g}^{\mu \mu }
4
A quick way to enter D-dimensional
metric tensors is given by MTD.
MTD[\[Mu], \[Nu]] MTD[\[Mu], \[Nu]]
Contract[%](g^{\mu \nu})^2
D
FV[p, \[Mu]] FV[q, \[Mu]]
Contract[% ]\overline{p}^{\mu } \overline{q}^{\mu }
\overline{p}\cdot \overline{q}
FV[p - q, \[Mu]] FV[a - b, \[Mu]]
Contract[%]\left(\overline{a}-\overline{b}\right)^{\mu } \left(\overline{p}-\overline{q}\right)^{\mu }
\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)^{\nu } (p-q)^{\nu }
a\cdot p-a\cdot q-b\cdot p+b\cdot q
LC[\[Mu], \[Nu], \[Alpha], \[Sigma]] FV[p, \[Sigma]]
Contract[%]\overline{p}^{\sigma } \bar{\epsilon }^{\mu \nu \alpha \sigma }
\bar{\epsilon }^{\alpha \mu \nu \overline{p}}
LC[\[Mu], \[Nu], \[Alpha], \[Beta]] LC[\[Mu], \[Nu], \[Alpha], \[Sigma]]
Contract[%]\bar{\epsilon }^{\mu \nu \alpha \beta } \bar{\epsilon }^{\mu \nu \alpha \sigma }
-6 \bar{g}^{\beta \sigma }
LCD[\[Mu], \[Nu], \[Alpha], \[Beta]] LCD[\[Mu], \[Nu], \[Alpha], \[Sigma]]
Contract[%] // Factor2\overset{\text{}}{\epsilon }^{\mu \nu \alpha \beta } \overset{\text{}}{\epsilon }^{\mu \nu \alpha \sigma }
(1-D) (2-D) (3-D) g^{\beta \sigma }
Contractions of Cartesian tensors are also possible. They can live in 3, D-1 or D-4 dimensions.
KD[i, j] CV[p, i]
Contract[%]\overline{p}^i \bar{\delta }^{ij}
\overline{p}^j
CV[p, i] CGA[i]
Contract[%]\overline{\gamma }^i \overline{p}^i
\overline{\gamma }\cdot \overline{p}
KD[i, i]
Contract[%]\bar{\delta }^{ii}
3
KD[i, j]^2
Contract[%](\bar{\delta}^{ij})^2
3
CV[p - q, j] CV[a - b, j]
Contract[%]\left(\overline{a}-\overline{b}\right)^j \left(\overline{p}-\overline{q}\right)^j
(\overline{a}-\overline{b})\cdot (\overline{p}-\overline{q})
CLC[i, j, k] CV[p, k]
Contract[%]\overline{p}^k \bar{\epsilon }^{ijk}
\bar{\epsilon }^{ij\overline{p}}
CLC[i, j, k] CLC[i, j, l]
Contract[%]\bar{\epsilon }^{ijk} \bar{\epsilon }^{ijl}
2 \bar{\delta }^{kl}
CLCD[i, j, k] CLCD[i, j, l]
Contract[%] // Factor2\overset{\text{}}{\epsilon }^{ijk} \overset{\text{}}{\epsilon }^{ijl}
(2-D) (3-D) \delta ^{kl}