164 lines
6.4 KiB
Text
164 lines
6.4 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.Tensors.ComplexLorentz.BasisTrees
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/-!
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## Lemmas related to complex Lorentz tensors.
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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 Fermion
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open complexLorentzTensor
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set_option maxRecDepth 20000 in
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lemma contr_rank_2_symm {T1 : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
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{T2 : (Lorentz.complexCo ⊗ Lorentz.complexCo).V} :
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{T1 | μ ν ⊗ T2 | μ ν = T2 | μ ν ⊗ T1 | μ ν}ᵀ := by
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rw [perm_tensor_eq (contr_tensor_eq (contr_tensor_eq (prod_comm _ _ _ _)))]
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rw [perm_tensor_eq (contr_tensor_eq (perm_contr _ _))]
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rw [perm_tensor_eq (perm_contr _ _)]
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rw [perm_perm]
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rw [perm_eq_id]
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· rw [(contr_tensor_eq (contr_swap _ _))]
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rw [perm_contr]
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rw [perm_tensor_eq (contr_swap _ _)]
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rw [perm_perm]
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rw [perm_eq_id]
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· rfl
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· rfl
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· apply OverColor.Hom.ext
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ext x
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exact Fin.elim0 x
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lemma contr_rank_2_symm' {T1 : (Lorentz.complexCo ⊗ Lorentz.complexCo).V}
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{T2 : (Lorentz.complexContr ⊗ Lorentz.complexContr).V} :
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{T1 | μ ν ⊗ T2 | μ ν = T2 | μ ν ⊗ T1 | μ ν}ᵀ := by
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rw [perm_tensor_eq contr_rank_2_symm]
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rw [perm_perm]
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rw [perm_eq_id]
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apply OverColor.Hom.ext
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ext x
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exact Fin.elim0 x
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set_option maxRecDepth 20000 in
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/-- Contracting a rank-2 anti-symmetric tensor with a rank-2 symmetric tensor gives zero. -/
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lemma antiSymm_contr_symm {A : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
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{S : (Lorentz.complexCo ⊗ Lorentz.complexCo).V}
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(hA : {A | μ ν = - (A | ν μ)}ᵀ) (hs : {S | μ ν = S | ν μ}ᵀ) :
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{A | μ ν ⊗ S | μ ν}ᵀ.tensor = 0 := by
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have h1 {M : Type} [AddCommGroup M] [Module ℂ M] {x : M} (h : x = - x) : x = 0 := by
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rw [eq_neg_iff_add_eq_zero, ← two_smul ℂ x] at h
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simpa using h
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refine h1 ?_
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rw [← neg_tensor]
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rw [neg_perm] at hA
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nth_rewrite 1 [contr_tensor_eq (contr_tensor_eq (prod_tensor_eq_fst hA))]
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nth_rewrite 1 [(contr_tensor_eq (contr_tensor_eq (prod_tensor_eq_snd hs)))]
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rw [contr_tensor_eq (contr_tensor_eq (neg_fst_prod _ _))]
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rw [contr_tensor_eq (neg_contr _)]
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rw [neg_contr]
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rw [neg_tensor]
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apply congrArg
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rw [contr_tensor_eq (contr_tensor_eq (prod_perm_left _ _ _ _))]
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rw [contr_tensor_eq (perm_contr _ _)]
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rw [perm_contr]
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rw [perm_tensor_eq (contr_tensor_eq (contr_tensor_eq (prod_perm_right _ _ _ _)))]
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rw [perm_tensor_eq (contr_tensor_eq (perm_contr _ _))]
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rw [perm_tensor_eq (perm_contr _ _)]
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rw [perm_perm]
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nth_rewrite 1 [perm_tensor_eq (contr_contr _ _ _)]
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rw [perm_perm]
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rw [perm_eq_id]
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· rfl
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· rfl
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lemma symm_contr_antiSymm {S : (Lorentz.complexCo ⊗ Lorentz.complexCo).V}
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{A : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
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(hA : {A | μ ν = - (A | ν μ)}ᵀ) (hs : {S | μ ν = S | ν μ}ᵀ) :
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{S | μ ν ⊗ A | μ ν}ᵀ.tensor = 0 := by
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rw [contr_rank_2_symm', perm_tensor, antiSymm_contr_symm hA hs]
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rfl
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lemma antiSymm_add_self {A : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
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(hA : {A | μ ν = - (A | ν μ)}ᵀ) :
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{A | μ ν + A | ν μ}ᵀ.tensor = 0 := by
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rw [← TensorTree.add_neg (twoNodeE complexLorentzTensor Color.up Color.up A)]
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apply TensorTree.add_tensor_eq_snd
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rw [neg_tensor_eq hA, neg_tensor_eq (neg_perm _ _), neg_neg]
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/-!
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## The contraction of Pauli matrices with Pauli matrices
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And related results.
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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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lemma leftMetric_mul_rightMetric : {Fermion.leftMetric | α α' ⊗ Fermion.rightMetric | 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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{Fermion.leftMetric | α α' ⊗ Fermion.rightMetric | 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 Fermion
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end
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