207 lines
8.3 KiB
Text
207 lines
8.3 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.Basic
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import HepLean.Lorentz.ComplexTensor.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 expansion of the Lorentz covariant metric in terms of basis vectors. -/
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lemma coMetric_basis_expand : {η' | μ ν}ᵀ.tensor =
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basisVector ![Color.down, Color.down] (fun _ => 0)
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- basisVector ![Color.down, Color.down] (fun _ => 1)
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- basisVector ![Color.down, Color.down] (fun _ => 2)
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- basisVector ![Color.down, Color.down] (fun _ => 3) := by
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rw [tensorNode_coMetric]
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
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Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Functor.id_obj, Fin.isValue]
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erw [Lorentz.coMetric_apply_one, Lorentz.coMetricVal_expand_tmul]
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simp only [Fin.isValue, map_sub]
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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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erw [pairIsoSep_tmul, basisVector]
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apply congrArg
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funext i
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fin_cases i
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all_goals
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.zero_eta, Fin.isValue, OverColor.mk_hom,
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cons_val_zero, Fin.cases_zero]
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change _ = Lorentz.complexCoBasisFin4 _
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simp only [Fin.isValue, Lorentz.complexCoBasisFin4, Basis.coe_reindex, Function.comp_apply]
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rfl
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/-- Provides the explicit expansion of the co-metric tensor in terms of the basis elements, as
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a tensor tree. -/
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lemma coMetric_basis_expand_tree : {η' | μ ν}ᵀ.tensor =
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(TensorTree.add (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 0))) <|
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TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 1)))) <|
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TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 2)))) <|
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(smul (-1) (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 3))))).tensor :=
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coMetric_basis_expand
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/-- The expansion of the Lorentz contravariant metric in terms of basis vectors. -/
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lemma contrMatrix_basis_expand : {η | μ ν}ᵀ.tensor =
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basisVector ![Color.up, Color.up] (fun _ => 0)
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- basisVector ![Color.up, Color.up] (fun _ => 1)
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- basisVector ![Color.up, Color.up] (fun _ => 2)
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- basisVector ![Color.up, Color.up] (fun _ => 3) := by
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rw [tensorNode_contrMetric]
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
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Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V]
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erw [Lorentz.contrMetric_apply_one, Lorentz.contrMetricVal_expand_tmul]
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simp only [Fin.isValue, map_sub]
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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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erw [pairIsoSep_tmul, basisVector]
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apply congrArg
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funext i
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fin_cases i
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all_goals
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.zero_eta, Fin.isValue, OverColor.mk_hom,
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cons_val_zero, Fin.cases_zero]
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change _ = Lorentz.complexContrBasisFin4 _
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simp only [Fin.isValue, Lorentz.complexContrBasisFin4, Basis.coe_reindex, Function.comp_apply]
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rfl
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/-- The expansion of the Lorentz contravariant metric in terms of basis vectors as
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a structured tensor tree. -/
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lemma contrMatrix_basis_expand_tree : {η | μ ν}ᵀ.tensor =
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(TensorTree.add (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 0))) <|
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TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 1)))) <|
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TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 2)))) <|
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(smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 3))))).tensor :=
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contrMatrix_basis_expand
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/-- The expansion of the Fermionic left metric in terms of basis vectors. -/
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lemma leftMetric_expand : {εL | α β}ᵀ.tensor =
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- basisVector ![Color.upL, Color.upL] (fun | 0 => 0 | 1 => 1)
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+ basisVector ![Color.upL, Color.upL] (fun | 0 => 1 | 1 => 0) := by
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rw [tensorNode_leftMetric]
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
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Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue]
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erw [Fermion.leftMetric_apply_one, Fermion.leftMetricVal_expand_tmul]
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simp only [Fin.isValue, map_add, map_neg]
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congr 1
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congr 1
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all_goals
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erw [pairIsoSep_tmul, basisVector]
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apply congrArg
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funext i
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fin_cases i
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· rfl
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· rfl
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/-- The expansion of the Fermionic left metric in terms of basis vectors as a structured
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tensor tree. -/
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lemma leftMetric_expand_tree : {εL | α β}ᵀ.tensor =
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(TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.upL, Color.upL]
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(fun | 0 => 0 | 1 => 1)))) <|
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(tensorNode (basisVector ![Color.upL, Color.upL] (fun | 0 => 1 | 1 => 0)))).tensor :=
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leftMetric_expand
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/-- The expansion of the Fermionic alt-left metric in terms of basis vectors. -/
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lemma altLeftMetric_expand : {εL' | α β}ᵀ.tensor =
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basisVector ![Color.downL, Color.downL] (fun | 0 => 0 | 1 => 1)
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- basisVector ![Color.downL, Color.downL] (fun | 0 => 1 | 1 => 0) := by
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rw [tensorNode_altLeftMetric]
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
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Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue]
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erw [Fermion.altLeftMetric_apply_one, Fermion.altLeftMetricVal_expand_tmul]
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simp only [Fin.isValue, map_sub]
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congr 1
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all_goals
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erw [pairIsoSep_tmul, basisVector]
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apply congrArg
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funext i
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fin_cases i
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· rfl
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· rfl
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/-- The expansion of the Fermionic alt-left metric in terms of basis vectors as a
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structured tensor tree. -/
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lemma altLeftMetric_expand_tree : {εL' | α β}ᵀ.tensor =
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(TensorTree.add (tensorNode (basisVector ![Color.downL, Color.downL]
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(fun | 0 => 0 | 1 => 1))) <|
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(smul (-1) (tensorNode (basisVector ![Color.downL, Color.downL]
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(fun | 0 => 1 | 1 => 0))))).tensor :=
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altLeftMetric_expand
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/-- The expansion of the Fermionic right metric in terms of basis vectors. -/
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lemma rightMetric_expand : {εR | α β}ᵀ.tensor =
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- basisVector ![Color.upR, Color.upR] (fun | 0 => 0 | 1 => 1)
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+ basisVector ![Color.upR, Color.upR] (fun | 0 => 1 | 1 => 0) := by
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rw [tensorNode_rightMetric]
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
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Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue]
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erw [Fermion.rightMetric_apply_one, Fermion.rightMetricVal_expand_tmul]
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simp only [Fin.isValue, map_add, map_neg]
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congr 1
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congr 1
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all_goals
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erw [pairIsoSep_tmul, basisVector]
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apply congrArg
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funext i
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fin_cases i
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· rfl
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· rfl
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/-- The expansion of the Fermionic right metric in terms of basis vectors as a
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structured tensor tree. -/
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lemma rightMetric_expand_tree : {εR | α β}ᵀ.tensor =
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(TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.upR, Color.upR]
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(fun | 0 => 0 | 1 => 1)))) <|
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(tensorNode (basisVector ![Color.upR, Color.upR] (fun | 0 => 1 | 1 => 0)))).tensor :=
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rightMetric_expand
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/-- The expansion of the Fermionic alt-right metric in terms of basis vectors. -/
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lemma altRightMetric_expand : {εR' | α β}ᵀ.tensor =
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basisVector ![Color.downR, Color.downR] (fun | 0 => 0 | 1 => 1)
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- basisVector ![Color.downR, Color.downR] (fun | 0 => 1 | 1 => 0) := by
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rw [tensorNode_altRightMetric]
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
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Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue]
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erw [Fermion.altRightMetric_apply_one, Fermion.altRightMetricVal_expand_tmul]
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simp only [Fin.isValue, map_sub]
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congr 1
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all_goals
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erw [pairIsoSep_tmul, basisVector]
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apply congrArg
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funext i
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fin_cases i
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· rfl
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· rfl
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/-- The expansion of the Fermionic alt-right metric in terms of basis vectors as a
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structured tensor tree. -/
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lemma altRightMetric_expand_tree : {εR' | α β}ᵀ.tensor =
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(TensorTree.add (tensorNode (basisVector
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![Color.downR, Color.downR] (fun | 0 => 0 | 1 => 1))) <|
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(smul (-1) (tensorNode (basisVector ![Color.downR, Color.downR]
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(fun | 0 => 1 | 1 => 0))))).tensor :=
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altRightMetric_expand
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end complexLorentzTensor
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