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feat: Add contracting equivalence of index sets

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jstoobysmith 2024-08-02 09:51:13 -04:00
parent 238233b02c
commit 17139a6cf1
3 changed files with 330 additions and 101 deletions
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50
HepLean/SpaceTime/LorentzTensor/Real/Basic.lean
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@ -14,7 +14,8 @@ import HepLean.SpaceTime.LorentzTensor.MulActionTensor
open TensorProduct
open minkowskiMatrix
namespace realTensor
namespace realTensorColor
variable {d : ℕ}
/-!
@ -51,31 +52,25 @@ def colorTypEquivFin1Fin1 : ColorType ≃ Fin 1 ⊕ Fin 1 where
rename_i f
exact (Fin.fin_one_eq_zero f).symm
instance : DecidableEq realTensor.ColorType :=
instance : DecidableEq ColorType :=
Equiv.decidableEq colorTypEquivFin1Fin1
instance : Fintype realTensor.ColorType where
elems := {realTensor.ColorType.up, realTensor.ColorType.down}
instance : Fintype ColorType where
elems := {ColorType.up, ColorType.down}
complete := by
intro x
cases x
simp only [Finset.mem_insert, Finset.mem_singleton, or_false]
simp only [Finset.mem_insert, Finset.mem_singleton, or_true]
end realTensor
end realTensorColor
noncomputable section
open realTensor
open realTensorColor
/-! TODO: Set up the notation `𝓛𝓣ℝ` or similar. -/
/-- The `LorentzTensorStructure` associated with real Lorentz tensors. -/
def realLorentzTensor (d : ℕ) : TensorStructure ℝ where
def realTensorColor : TensorColor where
Color := ColorType
ColorModule μ :=
match μ with
| .up => LorentzVector d
| .down => CovariantLorentzVector d
τ μ :=
match μ with
| .up => .down
@ -84,6 +79,19 @@ def realLorentzTensor (d : ℕ) : TensorStructure ℝ where
match μ with
| .up => rfl
| .down => rfl
instance : Fintype realTensorColor.Color := realTensorColor.instFintypeColorType
instance : DecidableEq realTensorColor.Color := realTensorColor.instDecidableEqColorType
/-! TODO: Set up the notation `𝓛𝓣ℝ` or similar. -/
/-- The `LorentzTensorStructure` associated with real Lorentz tensors. -/
def realLorentzTensor (d : ℕ) : TensorStructure ℝ where
toTensorColor := realTensorColor
ColorModule μ :=
match μ with
| .up => LorentzVector d
| .down => CovariantLorentzVector d
colorModule_addCommMonoid μ :=
match μ with
| .up => instAddCommMonoidLorentzVector d
@ -100,11 +108,11 @@ def realLorentzTensor (d : ℕ) : TensorStructure ℝ where
match μ with
| .up => by
intro x y
simp only [LorentzVector.contrDownUp, Equiv.cast_refl, Equiv.refl_apply, LinearMap.coe_comp,
simp only [realTensorColor, LorentzVector.contrDownUp, Equiv.cast_refl, Equiv.refl_apply, LinearMap.coe_comp,
LinearEquiv.coe_coe, Function.comp_apply, comm_tmul]
| .down => by
intro x y
simp only [LorentzVector.contrDownUp, LinearMap.coe_comp, LinearEquiv.coe_coe,
simp only [realTensorColor, LorentzVector.contrDownUp, LinearMap.coe_comp, LinearEquiv.coe_coe,
Function.comp_apply, comm_tmul, Equiv.cast_refl, Equiv.refl_apply]
unit μ :=
match μ with
@ -116,16 +124,12 @@ def realLorentzTensor (d : ℕ) : TensorStructure ℝ where
| .down => LorentzVector.unitDown_rid
metric μ :=
match μ with
| realTensor.ColorType.up => asTenProd
| realTensor.ColorType.down => asCoTenProd
| realTensorColor.ColorType.up => asTenProd
| realTensorColor.ColorType.down => asCoTenProd
metric_dual μ :=
match μ with
| realTensor.ColorType.up => asTenProd_contr_asCoTenProd
| realTensor.ColorType.down => asCoTenProd_contr_asTenProd
instance : Fintype (realLorentzTensor d).Color := realTensor.instFintypeColorType
instance : DecidableEq (realLorentzTensor d).Color := realTensor.instDecidableEqColorType
| realTensorColor.ColorType.up => asTenProd_contr_asCoTenProd
| realTensorColor.ColorType.down => asCoTenProd_contr_asTenProd
/-- The action of the Lorentz group on real Lorentz tensors. -/
instance : MulActionTensor (LorentzGroup d) (realLorentzTensor d) where