90 lines
3.8 KiB
Text
90 lines
3.8 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 Mathlib.LinearAlgebra.StdBasis
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import HepLean.SpaceTime.LorentzTensor.Basic
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import HepLean.SpaceTime.LorentzTensor.IndexNotation.Basic
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/-!
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# Einstein notation for real tensors
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Einstein notation is a specific example of index notation, with only one color.
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In this file we define Einstein notation for generic ring `R`.
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-/
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open TensorProduct
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/-- Einstein tensors have only one color, corresponding to a `down` index. . -/
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def einsteinTensorColor : TensorColor where
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Color := Unit
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τ a := a
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τ_involutive μ := by rfl
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instance : Fintype einsteinTensorColor.Color := Unit.fintype
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instance : DecidableEq einsteinTensorColor.Color := instDecidableEqPUnit
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variable {R : Type} [CommSemiring R]
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/-- The `TensorStructure` associated with `n`-dimensional tensors. -/
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noncomputable def einsteinTensor (R : Type) [CommSemiring R] (n : ℕ) : TensorStructure R where
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toTensorColor := einsteinTensorColor
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ColorModule _ := Fin n → R
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colorModule_addCommMonoid _ := Pi.addCommMonoid
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colorModule_module _ := Pi.Function.module (Fin n) R R
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contrDual _ := TensorProduct.lift (Fintype.total R R)
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contrDual_symm a x y := by
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simp only [lift.tmul, Fintype.total_apply, smul_eq_mul, mul_comm, Equiv.cast_refl,
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Equiv.refl_apply]
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unit a := ∑ i, Pi.basisFun R (Fin n) i ⊗ₜ[R] Pi.basisFun R (Fin n) i
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unit_rid a x:= by
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simp only [Pi.basisFun_apply]
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rw [tmul_sum, map_sum]
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trans ∑ i, x i • Pi.basisFun R (Fin n) i
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· refine Finset.sum_congr rfl (fun i _ => ?_)
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simp only [TensorStructure.contrLeftAux, LinearEquiv.refl_toLinearMap, LinearMap.coe_comp,
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LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul, lift.tmul,
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Fintype.total_apply, LinearMap.stdBasis_apply', smul_eq_mul, ite_mul, one_mul, zero_mul,
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Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte, LinearMap.id_coe, id_eq, lid_tmul,
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Pi.basisFun_apply]
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· funext a
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simp only [Pi.basisFun_apply, Finset.sum_apply, Pi.smul_apply, LinearMap.stdBasis_apply',
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smul_eq_mul, mul_ite, mul_one, mul_zero, Finset.sum_ite_eq', Finset.mem_univ, ↓reduceIte]
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metric a := ∑ i, Pi.basisFun R (Fin n) i ⊗ₜ[R] Pi.basisFun R (Fin n) i
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metric_dual a := by
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simp only [Pi.basisFun_apply, map_sum, comm_tmul]
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rw [tmul_sum, map_sum]
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refine Finset.sum_congr rfl (fun i _ => ?_)
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rw [sum_tmul, map_sum, Fintype.sum_eq_single i]
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· simp only [TensorStructure.contrMidAux, LinearEquiv.refl_toLinearMap,
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TensorStructure.contrLeftAux, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
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assoc_tmul, map_tmul, LinearMap.id_coe, id_eq, assoc_symm_tmul, lift.tmul,
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Fintype.total_apply, LinearMap.stdBasis_apply', smul_eq_mul, mul_ite, mul_one, mul_zero,
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Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte, lid_tmul, one_smul]
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· intro x hi
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simp only [TensorStructure.contrMidAux, LinearEquiv.refl_toLinearMap,
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TensorStructure.contrLeftAux, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
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assoc_tmul, map_tmul, LinearMap.id_coe, id_eq, assoc_symm_tmul, lift.tmul,
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Fintype.total_apply, LinearMap.stdBasis_apply', smul_eq_mul, mul_ite, mul_one, mul_zero,
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Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte, lid_tmul, ite_smul, one_smul, zero_smul]
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rw [if_neg]
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simp only [tmul_zero]
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exact id (Ne.symm hi)
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instance : IndexNotation einsteinTensorColor.Color where
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charList := {'ᵢ'}
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notaEquiv :=
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⟨fun _ => ⟨'ᵢ', Finset.mem_singleton.mpr rfl⟩,
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fun _ => Unit.unit,
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fun _ => rfl,
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by
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intro c
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have hc2 := c.2
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simp only [↓Char.isValue, Finset.mem_singleton] at hc2
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exact SetCoe.ext (id (Eq.symm hc2))⟩
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