143 lines
3.9 KiB
Text
143 lines
3.9 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.SpaceTime.LorentzVector.Contraction
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import HepLean.SpaceTime.LorentzVector.LorentzAction
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import HepLean.SpaceTime.LorentzTensor.MulActionTensor
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/-!
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# Real Lorentz tensors
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-/
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open TensorProduct
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open minkowskiMatrix
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namespace realTensor
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variable {d : ℕ}
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/-!
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## Definitions and lemmas needed to define a `LorentzTensorStructure`
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-/
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/-- The type colors for real Lorentz tensors. -/
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inductive ColorType
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| up
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| down
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def colorTypEquivFin1Fin1 : ColorType ≃ Fin 1 ⊕ Fin 1 where
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toFun
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| ColorType.up => Sum.inl ⟨0, Nat.zero_lt_one⟩
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| ColorType.down => Sum.inr ⟨0, Nat.zero_lt_one⟩
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invFun
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| Sum.inl _ => ColorType.up
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| Sum.inr _ => ColorType.down
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left_inv := by
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intro x
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cases x
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simp
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simp
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right_inv := by
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intro x
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cases x
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simp
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rename_i f
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exact (Fin.fin_one_eq_zero f).symm
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simp
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rename_i f
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exact (Fin.fin_one_eq_zero f).symm
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instance : DecidableEq realTensor.ColorType :=
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Equiv.decidableEq colorTypEquivFin1Fin1
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instance : Fintype realTensor.ColorType where
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elems := {realTensor.ColorType.up, realTensor.ColorType.down}
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complete := by
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intro x
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cases x
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simp
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simp
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end realTensor
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noncomputable section
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open realTensor
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/-! TODO: Set up the notation `𝓛𝓣ℝ` or similar. -/
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/-- The `LorentzTensorStructure` associated with real Lorentz tensors. -/
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def realLorentzTensor (d : ℕ) : TensorStructure ℝ where
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Color := ColorType
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ColorModule μ :=
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match μ with
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| .up => LorentzVector d
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| .down => CovariantLorentzVector d
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τ μ :=
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match μ with
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| .up => .down
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| .down => .up
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τ_involutive μ :=
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match μ with
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| .up => rfl
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| .down => rfl
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colorModule_addCommMonoid μ :=
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match μ with
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| .up => instAddCommMonoidLorentzVector d
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| .down => instAddCommMonoidCovariantLorentzVector d
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colorModule_module μ :=
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match μ with
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| .up => instModuleRealLorentzVector d
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| .down => instModuleRealCovariantLorentzVector d
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contrDual μ :=
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match μ with
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| .up => LorentzVector.contrUpDown
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| .down => LorentzVector.contrDownUp
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contrDual_symm μ :=
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match μ with
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| .up => by
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intro x y
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simp only [LorentzVector.contrDownUp, Equiv.cast_refl, Equiv.refl_apply, LinearMap.coe_comp,
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LinearEquiv.coe_coe, Function.comp_apply, comm_tmul]
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| .down => by
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intro x y
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simp only [LorentzVector.contrDownUp, LinearMap.coe_comp, LinearEquiv.coe_coe,
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Function.comp_apply, comm_tmul, Equiv.cast_refl, Equiv.refl_apply]
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unit μ :=
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match μ with
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| .up => LorentzVector.unitUp
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| .down => LorentzVector.unitDown
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unit_rid μ :=
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match μ with
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| .up => LorentzVector.unitUp_rid
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| .down => LorentzVector.unitDown_rid
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metric μ :=
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match μ with
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| realTensor.ColorType.up => asTenProd
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| realTensor.ColorType.down => asCoTenProd
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metric_dual μ :=
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match μ with
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| realTensor.ColorType.up => asTenProd_contr_asCoTenProd
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| realTensor.ColorType.down => asCoTenProd_contr_asTenProd
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instance : Fintype (realLorentzTensor d).Color := realTensor.instFintypeColorType
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instance : DecidableEq (realLorentzTensor d).Color := realTensor.instDecidableEqColorType
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/-- The action of the Lorentz group on real Lorentz tensors. -/
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instance : MulActionTensor (LorentzGroup d) (realLorentzTensor d) where
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repColorModule μ :=
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match μ with
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| .up => LorentzVector.rep
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| .down => CovariantLorentzVector.rep
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contrDual_inv μ _ :=
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match μ with
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| .up => TensorProduct.ext' (fun _ _ => LorentzVector.contrUpDown_invariant_lorentzAction)
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| .down => TensorProduct.ext' (fun _ _ => LorentzVector.contrDownUp_invariant_lorentzAction)
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metric_inv μ g :=
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match μ with
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| .up => asTenProd_invariant g
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| .down => asCoTenProd_invariant g
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end
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