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[%]
pμgˉμν
pν
FV[p, \[Mu]] GA[\[Mu]]
Contract[%]
γˉμpμ
γˉ⋅p
The default dimension for a metric tensor is 4.
MT[\[Mu], \[Mu]]
Contract[%]
gˉμμ
4
A quick way to enter D-dimensional metric tensors is given by MTD
.
MTD[\[Mu], \[Nu]] MTD[\[Mu], \[Nu]]
Contract[%]
(gμν)2
D
FV[p, \[Mu]] FV[q, \[Mu]]
Contract[% ]
pμqμ
p⋅q
FV[p - q, \[Mu]] FV[a - b, \[Mu]]
Contract[%]
(a−b)μ(p−q)μ
a⋅p−a⋅q−b⋅p+b⋅q
FVD[p - q, \[Nu]] FVD[a - b, \[Nu]]
Contract[%]
(a−b)ν(p−q)ν
a⋅p−a⋅q−b⋅p+b⋅q
LC[\[Mu], \[Nu], \[Alpha], \[Sigma]] FV[p, \[Sigma]]
Contract[%]
pσϵˉμνασ
ϵˉαμνp
LC[\[Mu], \[Nu], \[Alpha], \[Beta]] LC[\[Mu], \[Nu], \[Alpha], \[Sigma]]
Contract[%]
ϵˉμναβϵˉμνασ
−6gˉβσ
LCD[\[Mu], \[Nu], \[Alpha], \[Beta]] LCD[\[Mu], \[Nu], \[Alpha], \[Sigma]]
Contract[%] // Factor2
ϵμναβϵμνασ
(1−D)(2−D)(3−D)gβσ
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[%]
piδˉij
pj
CV[p, i] CGA[i]
Contract[%]
γipi
γ⋅p
δˉii
3
(δˉij)2
3
CV[p - q, j] CV[a - b, j]
Contract[%]
(a−b)j(p−q)j
(a−b)⋅(p−q)
CLC[i, j, k] CV[p, k]
Contract[%]
pkϵˉijk
ϵˉijp
CLC[i, j, k] CLC[i, j, l]
Contract[%]
ϵˉijkϵˉijl
2δˉkl
CLCD[i, j, k] CLCD[i, j, l]
Contract[%] // Factor2
ϵijkϵijl
(2−D)(3−D)δkl