Standard spacetime Kronecker delta \(\delta^a_b\) as STTens 1 1 (SField Rational).

delta3 = fromListT2 $ zip [(singletonInd (Ind3 i),singletonInd (Ind3 i)) | i <- [0..3]] (repeat $ SField 1)

Standard Kronecker delta for the Ind9 index type \(\delta^I_J\) as ATens 0 0 1 1 0 0 (SField Rational).

delta9 = fromListT6 $ zip [(Empty, Empty, singletonInd (Ind9 i),singletonInd (Ind9 i), Empty, Empty) | i <- [0..9]] (repeat $ SField 1)

Standard Kronecker delta for the Ind20 index type \(\delta^A_B\) as ATens 1 1 0 0 0 0 (SField Rational).

delta20 = fromListT6 $ zip [(singletonInd (Ind20 i),singletonInd (Ind20 i), Empty, Empty, Empty, Empty) | i <- [0..20]] (repeat $ SField 1)

Spacetime Kronecker delta as ATens.

Spacetime Minkowski metric \(\eta_{ab}\) as ATens 0 0 0 0 0 2 (SField Rational). The Minkowski metric could also be defined as STTens 0 2 (SField Rational) in similar fashion.

eta =  fromListT2 map (\(x,y,z) -> ((Empty,Append (Ind3 x) $ Append (Ind3 y) Empty),SField z)) [(0,0,-1),(1,1,1),(2,2,1),(3,3,1)]

Inverse spacetime Minkowski metric \(\eta^{ab}\) as ATens 0 0 0 0 2 0 (SField Rational). The inverse Minkowski metric could also be defined as STTens 2 0 (SField Rational) in similar fashion.

invEta = fromListT2 $ map (\(x,y,z) -> ((Append (Ind3 x) $ Append (Ind3 y) Empty,Empty),SField z)) [(0,0,-1),(1,1,1),(2,2,1),(3,3,1)]

Minkowski metric lifted to ATens.

Inverse Minkowski metric lifted to ATens.

The tensor \(\eta_I\) provides an equivalent version of the Minkowski metric that uses an index of type Ind9 to label the 10 different values of the symmetric spacetime index pair.

Covariant spacetime Levi-Civita symbol \(\epsilon_{abcd}\) as type ATTens 0 4 (SField Rational).

Contravariant spacetime Levi-Civita symbol \(\epsilon^{abcd}\) as type STTens4 0 (SField Rational). T

Covariant Levi-Civita symbol lifted to ATens.

Contravariant Levi-Civita symbol lifted to ATens.

Flat area metric tensor. Can be obtained via the interJArea intertwiner \( J_A^{abcd}\) as: \( N_A = J_A^{abcd} \left ( \eta_{ac} \eta_{bd} - \eta_{ad} \eta_{bc} - \epsilon_{abcd} \right ) \).

The tensor \(I^I_{ab} \) maps between covariant Ind9 indices and symmetric pairs of covariant Ind3 indices.

The tensor \(J_I^{ab} \) maps between covariant Ind9 indices and pairs of covariant Ind3 indices.

The tensor \( I^A_{abcd}\) maps between covariant Ind20 indices and blocks of 4 of covariant Ind3 indices.

The tensor \( J_A^{abcd}\) maps between contravariant Ind20 indices and blocks of 4 of contravariant Ind3 indices.

Can be obtained as: \(C^{Am}_{Bn} = -4 \cdot I^A_{nbcd} J_B^{mbcd} \)

interArea = SField (-4 :: Rational) &. contrATens3 (1,1) (contrATens3 (2,2) $ contrATens3 (3,3) $ interIArea &* interJArea

Can be obtained as : \(K^{Im}_{Jn} = -2 \cdot I^I_{nb} J_J^{mb} \)

interMetric = SField (-2 :: Rational) &. contrATens3 (0,0) (interI2 &* interJ2)

Is given by: \( K^m_{Jn} = K^{Im}_{Jn} \eta_I\)

flatInterMetric = contrATens2 (0,1) $ interMetric &* etaAbs

Is given by: \( C^m_{Bn} = C^{Am}_{Bn} N_A \)

flatInter = contrATens1 (0,1) $ interArea &* flatArea

Is given by: \( C_{An}^{Bm} \delta_p^q - \delta_A^B \delta_p^m \delta_n^q \)

Is given by: \( C_{An}^{Bm} \delta_I^J + \delta_A^B K^{Im}_{Jn}\)

Is given by: \( C_{An}^{B(m\vert} 2 J_I^{\vert p) q} - \delta^B_A J_I ^{pm} \delta_n^q \)

Is given by: \( C_{An}^{B(m\vert} J_I^{\vert p q )} \)

Is given by: \( K_{In}^{Jm} \delta_p^q - \delta_I^J \delta_p^m \delta_n^q \)

Is given by: \( K_{In}^{Jm} \delta_K^L + \delta_I^J K^{Km}_{Ln}\)

Is given by: \( K_{In}^{J(m\vert} 2 J_L^{\vert p) q} - \delta^I_J J_L ^{pm} \delta_n^q \)

Is given by: \( K_{In}^{J(m\vert} J_L^{\vert p q )} \)

tensorRank6' generic4Ansatz = 21

tensorRank6' generic5Ansatz = 21*4

tensorRank6' generic5Ansatz = 21*10

tensorRank6' generic8Ansatz = 21*22/2

tensorRank6' generic21Ansatz = 21*21*4

tensorRank6' generic5Ansatz = 84*85/2

tensorRank6' generic5Ansatz = 21*21*10

tensorRank6' generic5Ansatz = 21*21*10*4

tensorRank6' generic5Ansatz = 210*211/2