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docs: metric-lemmas

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jstoobysmith 2024-11-26 09:41:26 +00:00
parent fe0f2c26c7
commit 3cb6fe4982
3 changed files with 18 additions and 2 deletions
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14
HepLean/Lorentz/ComplexTensor/Metrics/Basis.lean
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@ -87,6 +87,8 @@ lemma contrMatrix_basis_expand : {η | μ ν}ᵀ.tensor =
simp only [Fin.isValue, Lorentz.complexContrBasisFin4, Basis.coe_reindex, Function.comp_apply]
rfl
/-- The expansion of the Lorentz contrvariant metric in terms of basis vectors as
a structured tensor tree. -/
lemma contrMatrix_basis_expand_tree : {η | μ ν}ᵀ.tensor =
(TensorTree.add (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 0))) <|
TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 1)))) <|
@ -94,6 +96,7 @@ lemma contrMatrix_basis_expand_tree : {η | μ ν}ᵀ.tensor =
(smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 3))))).tensor :=
contrMatrix_basis_expand
/-- The expansion of the Fermionic left metric in terms of basis vectors. -/
lemma leftMetric_expand : {εL | α β}ᵀ.tensor =
- basisVector ![Color.upL, Color.upL] (fun | 0 => 0 | 1 => 1)
+ basisVector ![Color.upL, Color.upL] (fun | 0 => 1 | 1 => 0) := by
@ -112,12 +115,15 @@ lemma leftMetric_expand : {εL | α β}ᵀ.tensor =
· rfl
· rfl
/-- The expansion of the Fermionic left metric in terms of basis vectors as a structured
tensor tree. -/
lemma leftMetric_expand_tree : {εL | α β}ᵀ.tensor =
(TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.upL, Color.upL]
(fun | 0 => 0 | 1 => 1)))) <|
(tensorNode (basisVector ![Color.upL, Color.upL] (fun | 0 => 1 | 1 => 0)))).tensor :=
leftMetric_expand
/-- The expansion of the Fermionic alt-left metric in terms of basis vectors. -/
lemma altLeftMetric_expand : {εL' | α β}ᵀ.tensor =
basisVector ![Color.downL, Color.downL] (fun | 0 => 0 | 1 => 1)
- basisVector ![Color.downL, Color.downL] (fun | 0 => 1 | 1 => 0) := by
@ -135,6 +141,8 @@ lemma altLeftMetric_expand : {εL' | α β}ᵀ.tensor =
· rfl
· rfl
/-- The expansion of the Fermionic alt-left metric in terms of basis vectors as a
structured tensor tree. -/
lemma altLeftMetric_expand_tree : {εL' | α β}ᵀ.tensor =
(TensorTree.add (tensorNode (basisVector ![Color.downL, Color.downL]
(fun | 0 => 0 | 1 => 1))) <|
@ -142,6 +150,7 @@ lemma altLeftMetric_expand_tree : {εL' | α β}ᵀ.tensor =
(fun | 0 => 1 | 1 => 0))))).tensor :=
altLeftMetric_expand
/-- The expansion of the Fermionic right metric in terms of basis vectors. -/
lemma rightMetric_expand : {εR | α β}ᵀ.tensor =
- basisVector ![Color.upR, Color.upR] (fun | 0 => 0 | 1 => 1)
+ basisVector ![Color.upR, Color.upR] (fun | 0 => 1 | 1 => 0) := by
@ -160,12 +169,15 @@ lemma rightMetric_expand : {εR | α β}ᵀ.tensor =
· rfl
· rfl
/-- The expansion of the Fermionic right metric in terms of basis vectors as a
structured tensor tree. -/
lemma rightMetric_expand_tree : {εR | α β}ᵀ.tensor =
(TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.upR, Color.upR]
(fun | 0 => 0 | 1 => 1)))) <|
(tensorNode (basisVector ![Color.upR, Color.upR] (fun | 0 => 1 | 1 => 0)))).tensor :=
rightMetric_expand
/-- The expansion of the Fermionic alt-right metric in terms of basis vectors. -/
lemma altRightMetric_expand : {εR' | α β}ᵀ.tensor =
basisVector ![Color.downR, Color.downR] (fun | 0 => 0 | 1 => 1)
- basisVector ![Color.downR, Color.downR] (fun | 0 => 1 | 1 => 0) := by
@ -183,6 +195,8 @@ lemma altRightMetric_expand : {εR' | α β}ᵀ.tensor =
· rfl
· rfl
/-- The expansion of the Fermionic alt-right metric in terms of basis vectors as a
structured tensor tree. -/
lemma altRightMetric_expand_tree : {εR' | α β}ᵀ.tensor =
(TensorTree.add (tensorNode (basisVector
![Color.downR, Color.downR] (fun | 0 => 0 | 1 => 1))) <|

4
HepLean/Lorentz/ComplexTensor/Metrics/Lemmas.lean
View file

@ -101,7 +101,7 @@ informal_lemma altRightMetric_contr_rightMetric where
def leftMetricMulRightMap := (Sum.elim ![Color.upL, Color.upL] ![Color.upR, Color.upR]) ∘
finSumFinEquiv.symm
/- Expansion of the product of `εL` and `εR` in terms of a basis. -/
/-- Expansion of the product of `εL` and `εR` in terms of a basis. -/
lemma leftMetric_prod_rightMetric : {εL | α α' ⊗ εR | β β'}ᵀ.tensor
= basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
- basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
@ -134,7 +134,7 @@ lemma leftMetric_prod_rightMetric : {εL | α α' ⊗ εR | β β'}ᵀ.tensor
funext x
fin_cases x <;> rfl
/- Expansion of the product of `εL` and `εR` in terms of a basis, as a tensor tree. -/
/-- Expansion of the product of `εL` and `εR` in terms of a basis, as a tensor tree. -/
lemma leftMetric_prod_rightMetric_tree : {εL | α α' ⊗ εR | β β'}ᵀ.tensor
= (TensorTree.add (tensorNode
(basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1))) <|