151 lines
6 KiB
Text
151 lines
6 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.Lorentz.ComplexTensor.Metrics.Basis
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import HepLean.Lorentz.ComplexTensor.Units.Basic
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/-!
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## Basic lemmas regarding metrics
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-/
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open IndexNotation
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open CategoryTheory
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open MonoidalCategory
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open Matrix
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open MatrixGroups
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open Complex
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open TensorProduct
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open IndexNotation
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open CategoryTheory
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open TensorTree
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open OverColor.Discrete
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noncomputable section
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namespace complexLorentzTensor
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/-!
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## Symmetry properties
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-/
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informal_lemma coMetric_symm where
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math :≈ "The covariant metric is symmetric {η' | μ ν = η' | ν μ}ᵀ"
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deps :≈ [``coMetric]
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informal_lemma contrMetric_symm where
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math :≈ "The contravariant metric is symmetric {η | μ ν = η | ν μ}ᵀ"
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deps :≈ [``contrMetric]
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informal_lemma leftMetric_antisymm where
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math :≈ "The left metric is antisymmetric {εL | α α' = - εL | α' α}ᵀ"
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deps :≈ [``leftMetric]
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informal_lemma rightMetric_antisymm where
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math :≈ "The right metric is antisymmetric {εR | β β' = - εR | β' β}ᵀ"
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deps :≈ [``rightMetric]
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informal_lemma altLeftMetric_antisymm where
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math :≈ "The alt-left metric is antisymmetric {εL' | α α' = - εL' | α' α}ᵀ"
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deps :≈ [``altLeftMetric]
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informal_lemma altRightMetric_antisymm where
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math :≈ "The alt-right metric is antisymmetric {εR' | β β' = - εR' | β' β}ᵀ"
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deps :≈ [``altRightMetric]
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/-!
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## Contractions with each other
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-/
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informal_lemma coMetric_contr_contrMetric where
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math :≈ "The contraction of the covariant metric with the contravariant metric is the unit
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{η' | μ ρ ⊗ η | ρ ν = δ' | μ ν}ᵀ"
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deps :≈ [``coMetric, ``contrMetric, ``coContrUnit]
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informal_lemma contrMetric_contr_coMetric where
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math :≈ "The contraction of the contravariant metric with the covariant metric is the unit
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{η | μ ρ ⊗ η' | ρ ν = δ | μ ν}ᵀ"
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deps :≈ [``contrMetric, ``coMetric, ``contrCoUnit]
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informal_lemma leftMetric_contr_altLeftMetric where
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math :≈ "The contraction of the left metric with the alt-left metric is the unit
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{εL | α β ⊗ εL' | β γ = δL | α γ}ᵀ"
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deps :≈ [``leftMetric, ``altLeftMetric, ``leftAltLeftUnit]
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informal_lemma rightMetric_contr_altRightMetric where
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math :≈ "The contraction of the right metric with the alt-right metric is the unit
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{εR | α β ⊗ εR' | β γ = δR | α γ}ᵀ"
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deps :≈ [``rightMetric, ``altRightMetric, ``rightAltRightUnit]
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informal_lemma altLeftMetric_contr_leftMetric where
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math :≈ "The contraction of the alt-left metric with the left metric is the unit
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{εL' | α β ⊗ εL | β γ = δL' | α γ}ᵀ"
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deps :≈ [``altLeftMetric, ``leftMetric, ``altLeftLeftUnit]
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informal_lemma altRightMetric_contr_rightMetric where
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math :≈ "The contraction of the alt-right metric with the right metric is the unit
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{εR' | α β ⊗ εR | β γ = δR' | α γ}ᵀ"
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deps :≈ [``altRightMetric, ``rightMetric, ``altRightRightUnit]
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/-!
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## Other relations
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-/
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/-- The map to color one gets when multiplying left and right metrics. -/
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def leftMetricMulRightMap := (Sum.elim ![Color.upL, Color.upL] ![Color.upR, Color.upR]) ∘
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finSumFinEquiv.symm
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/-- Expansion of the product of `εL` and `εR` in terms of a basis. -/
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lemma leftMetric_prod_rightMetric : {εL | α α' ⊗ εR | β β'}ᵀ.tensor
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= basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
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- basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
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- basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
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+ basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
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rw [prod_tensor_eq_fst (leftMetric_expand_tree)]
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rw [prod_tensor_eq_snd (rightMetric_expand_tree)]
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rw [prod_add_both]
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rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_prod _ _ _]
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rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_tensor_eq <| prod_smul _ _ _]
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rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_smul _ _ _]
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rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_eq_one _ _ (by simp)]
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rw [add_tensor_eq_fst <| add_tensor_eq_snd <| smul_prod _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
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rw [add_tensor_eq_fst <| add_tensor_eq_fst <| prod_basisVector_tree _ _]
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rw [add_tensor_eq_fst <| add_tensor_eq_snd <| smul_tensor_eq <| prod_basisVector_tree _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| prod_basisVector_tree _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| prod_basisVector_tree _ _]
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rw [← add_assoc]
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simp only [add_tensor, smul_tensor, tensorNode_tensor]
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change _ = basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
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+- basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
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+- basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
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+ basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0)
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congr 1
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congr 1
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congr 1
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all_goals
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congr
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funext x
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fin_cases x <;> rfl
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/-- Expansion of the product of `εL` and `εR` in terms of a basis, as a tensor tree. -/
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lemma leftMetric_prod_rightMetric_tree : {εL | α α' ⊗ εR | β β'}ᵀ.tensor
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= (TensorTree.add (tensorNode
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(basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1))) <|
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TensorTree.add (TensorTree.smul (-1 : ℂ) (tensorNode
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(basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)))) <|
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TensorTree.add (TensorTree.smul (-1 : ℂ) (tensorNode
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(basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)))) <|
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(tensorNode
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(basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0)))).tensor := by
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rw [leftMetric_prod_rightMetric]
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, add_tensor, tensorNode_tensor,
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smul_tensor, neg_smul, one_smul]
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rfl
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end complexLorentzTensor
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