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feat: Add real lorentz tensors

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jstoobysmith 2024-07-29 16:54:59 -04:00
parent 44b26efdaf
commit 99f4e85839
13 changed files with 602 additions and 48 deletions
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28
HepLean/SpaceTime/LorentzTensor/Basic.lean
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@ -34,6 +34,18 @@ open TensorProduct
variable {R : Type} [CommSemiring R]
def contrDualLeftAux {V1 V2 V3 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCommMonoid V3]
[Module R V1] [Module R V2] [Module R V3] (f : V1 ⊗[R] V2 →ₗ[R] R) :
V1 ⊗[R] V2 ⊗[R] V3 →ₗ[R] V3 :=
(TensorProduct.lid R _).toLinearMap ∘ₗ
TensorProduct.map (f) (LinearEquiv.refl R V3).toLinearMap
∘ₗ (TensorProduct.assoc R _ _ _).symm.toLinearMap
def contrDualMidAux {V1 V2 V3 V4 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCommMonoid V3]
[AddCommMonoid V4] [Module R V1] [Module R V2] [Module R V3] [Module R V4] (f : V1 ⊗[R] V2 →ₗ[R] R) :
(V4 ⊗[R] V1) ⊗[R] (V2 ⊗[R] V3) →ₗ[R] V4 ⊗[R] V3 :=
(TensorProduct.map (LinearEquiv.refl R V4).toLinearMap (contrDualLeftAux f)) ∘ₗ
(TensorProduct.assoc R _ _ _).toLinearMap
/-- An initial structure specifying a tensor system (e.g. a system in which you can
define real Lorentz tensors or Einstein notation convention). -/
structure TensorStructure (R : Type) [CommSemiring R] where
@ -56,12 +68,18 @@ structure TensorStructure (R : Type) [CommSemiring R] where
contrDual_symm : ∀ μ x y, (contrDual μ) (x ⊗ₜ[R] y) =
(contrDual (τ μ)) (y ⊗ₜ[R] (Equiv.cast (congrArg ColorModule (τ_involutive μ).symm) x))
/-- The unit of the contraction. -/
unit : (μ : Color) → ColorModule (τ μ) ⊗[R] ColorModule μ
unit : (μ : Color) → ColorModule μ ⊗[R] ColorModule (τ μ)
/-- The unit is a right identity. -/
unit_rid : ∀ μ (x : ColorModule μ),
TensorProduct.lid R _
(TensorProduct.map (contrDual μ) (LinearEquiv.refl R (ColorModule μ)).toLinearMap
((TensorProduct.assoc R _ _ _).symm (x ⊗ₜ[R] unit μ))) = x
unit_lid : ∀ μ (x : ColorModule μ),
TensorProduct.rid R _
(TensorProduct.map (LinearEquiv.refl R (ColorModule μ)).toLinearMap
(contrDual μ ∘ₗ (TensorProduct.comm R _ _).toLinearMap )
((TensorProduct.assoc R _ _ _) (unit μ ⊗ₜ[R] x ))) = x
/-- The metric for a given color. -/
metric : (μ : Color) → ColorModule μ ⊗[R] ColorModule μ
/-- The metric contracted with its dual is the unit. -/
metric_dual : ∀ (μ : Color), (contrDualMidAux (contrDual μ)
(metric μ ⊗ₜ[R] metric (τ μ))) = unit μ
namespace TensorStructure