f8f94979ab
* make informal_definition and informal_lemma commands * drop the fields "math", "physics", and "proof" from InformalDefinition/InformalLemma and use docstrings instead * render informal docstring in dependency graph
157 lines
5.9 KiB
Text
157 lines
5.9 KiB
Text
/-
|
||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Authors: Joseph Tooby-Smith
|
||
-/
|
||
import HepLean.Lorentz.ComplexTensor.Metrics.Basis
|
||
import HepLean.Lorentz.ComplexTensor.Units.Basic
|
||
/-!
|
||
|
||
## Basic lemmas regarding metrics
|
||
|
||
-/
|
||
open IndexNotation
|
||
open CategoryTheory
|
||
open MonoidalCategory
|
||
open Matrix
|
||
open MatrixGroups
|
||
open Complex
|
||
open TensorProduct
|
||
open IndexNotation
|
||
open CategoryTheory
|
||
open TensorTree
|
||
open OverColor.Discrete
|
||
noncomputable section
|
||
|
||
namespace complexLorentzTensor
|
||
|
||
/-!
|
||
|
||
## Symmetry properties
|
||
|
||
-/
|
||
|
||
/-- The covariant metric is symmetric `{η' | μ ν = η' | ν μ}ᵀ`. -/
|
||
informal_lemma coMetric_symm where
|
||
deps := [``coMetric]
|
||
|
||
/-- The contravariant metric is symmetric `{η | μ ν = η | ν μ}ᵀ`. -/
|
||
informal_lemma contrMetric_symm where
|
||
deps := [``contrMetric]
|
||
|
||
/-- The left metric is antisymmetric `{εL | α α' = - εL | α' α}ᵀ`. -/
|
||
informal_lemma leftMetric_antisymm where
|
||
deps := [``leftMetric]
|
||
|
||
/-- The right metric is antisymmetric `{εR | β β' = - εR | β' β}ᵀ`. -/
|
||
informal_lemma rightMetric_antisymm where
|
||
deps := [``rightMetric]
|
||
|
||
/-- The alt-left metric is antisymmetric `{εL' | α α' = - εL' | α' α}ᵀ`. -/
|
||
informal_lemma altLeftMetric_antisymm where
|
||
deps := [``altLeftMetric]
|
||
|
||
/-- The alt-right metric is antisymmetric `{εR' | β β' = - εR' | β' β}ᵀ`. -/
|
||
informal_lemma altRightMetric_antisymm where
|
||
deps := [``altRightMetric]
|
||
|
||
/-!
|
||
|
||
## Contractions with each other
|
||
|
||
-/
|
||
|
||
/-- The contraction of the covariant metric with the contravariant metric is the unit
|
||
`{η' | μ ρ ⊗ η | ρ ν = δ' | μ ν}ᵀ`.
|
||
-/
|
||
informal_lemma coMetric_contr_contrMetric where
|
||
deps := [``coMetric, ``contrMetric, ``coContrUnit]
|
||
|
||
/-- The contraction of the contravariant metric with the covariant metric is the unit
|
||
`{η | μ ρ ⊗ η' | ρ ν = δ | μ ν}ᵀ`.
|
||
-/
|
||
informal_lemma contrMetric_contr_coMetric where
|
||
deps := [``contrMetric, ``coMetric, ``contrCoUnit]
|
||
|
||
/-- The contraction of the left metric with the alt-left metric is the unit
|
||
`{εL | α β ⊗ εL' | β γ = δL | α γ}ᵀ`.
|
||
-/
|
||
informal_lemma leftMetric_contr_altLeftMetric where
|
||
deps := [``leftMetric, ``altLeftMetric, ``leftAltLeftUnit]
|
||
|
||
/-- The contraction of the right metric with the alt-right metric is the unit
|
||
`{εR | α β ⊗ εR' | β γ = δR | α γ}ᵀ`.
|
||
-/
|
||
informal_lemma rightMetric_contr_altRightMetric where
|
||
deps := [``rightMetric, ``altRightMetric, ``rightAltRightUnit]
|
||
|
||
/-- The contraction of the alt-left metric with the left metric is the unit
|
||
`{εL' | α β ⊗ εL | β γ = δL' | α γ}ᵀ`.
|
||
-/
|
||
informal_lemma altLeftMetric_contr_leftMetric where
|
||
deps := [``altLeftMetric, ``leftMetric, ``altLeftLeftUnit]
|
||
|
||
/-- The contraction of the alt-right metric with the right metric is the unit
|
||
`{εR' | α β ⊗ εR | β γ = δR' | α γ}ᵀ`.
|
||
-/
|
||
informal_lemma altRightMetric_contr_rightMetric where
|
||
deps := [``altRightMetric, ``rightMetric, ``altRightRightUnit]
|
||
|
||
/-!
|
||
|
||
## Other relations
|
||
|
||
-/
|
||
/-- The map to color one gets when multiplying left and right metrics. -/
|
||
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. -/
|
||
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)
|
||
- basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
|
||
+ basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
|
||
rw [prod_tensor_eq_fst (leftMetric_expand_tree)]
|
||
rw [prod_tensor_eq_snd (rightMetric_expand_tree)]
|
||
rw [prod_add_both]
|
||
rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_prod _ _ _]
|
||
rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_tensor_eq <| prod_smul _ _ _]
|
||
rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_smul _ _ _]
|
||
rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_eq_one _ _ (by simp)]
|
||
rw [add_tensor_eq_fst <| add_tensor_eq_snd <| smul_prod _ _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
|
||
rw [add_tensor_eq_fst <| add_tensor_eq_fst <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_fst <| add_tensor_eq_snd <| smul_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| prod_basisVector_tree _ _]
|
||
rw [← add_assoc]
|
||
simp only [add_tensor, smul_tensor, tensorNode_tensor]
|
||
change _ = basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
|
||
+- basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
|
||
+- basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
|
||
+ basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0)
|
||
congr 1
|
||
congr 1
|
||
congr 1
|
||
all_goals
|
||
congr
|
||
funext x
|
||
fin_cases x <;> rfl
|
||
|
||
/-- 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))) <|
|
||
TensorTree.add (TensorTree.smul (-1 : ℂ) (tensorNode
|
||
(basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)))) <|
|
||
TensorTree.add (TensorTree.smul (-1 : ℂ) (tensorNode
|
||
(basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)))) <|
|
||
(tensorNode
|
||
(basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0)))).tensor := by
|
||
rw [leftMetric_prod_rightMetric]
|
||
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, add_tensor, tensorNode_tensor,
|
||
smul_tensor, neg_smul, one_smul]
|
||
rfl
|
||
|
||
end complexLorentzTensor
|