80 lines
3.3 KiB
Text
80 lines
3.3 KiB
Text
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/-
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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.Tensors.ComplexLorentz.Metrics.Basis
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import HepLean.Tensors.ComplexLorentz.Basis
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/-!
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## Metrics and basis expansions
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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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/-- 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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lemma leftMetric_mul_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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lemma leftMetric_mul_rightMetric_tree :
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{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_mul_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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