86 lines
2.9 KiB
Text
86 lines
2.9 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.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.SpaceTime.LorentzTensor.GraphicalSpecies
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import HepLean.SpaceTime.LorentzVector.Basic
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/-!
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# Lorentz Tensors
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This file is currently a work-in-progress.
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The aim is to define Lorentz tensors, and devlop a systematic way to manipulate them.
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To manipulate them we will use the theory of modular operads
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(see e.g. [Raynor][raynor2021graphical]).
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-/
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/-- A Lorentz Tensor defined by its coordinate map. -/
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def LorentzTensor (d : ℕ) (X : FintypeCat) : Type :=
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(X → Fin 1 ⊕ Fin d) → ℝ
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/-- An instance of a additive commutative monoid on `LorentzTensor`. -/
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instance (d : ℕ) (X : FintypeCat) : AddCommMonoid (LorentzTensor d X) := Pi.addCommMonoid
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/-- An instance of a module on `LorentzVector`. -/
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noncomputable instance (d : ℕ) (X : FintypeCat) : Module ℝ (LorentzTensor d X) := Pi.module _ _ _
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namespace LorentzTensor
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open BigOperators
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open elGr
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open CategoryTheory
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variable {d : ℕ} {X Y : FintypeCat}
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/-- The map taking a list of `LorentzVector d` indexed by `X` to a ` LorentzTensor d X`. -/
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def tmul (t : X → LorentzVector d) : LorentzTensor d X :=
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fun f => ∏ x, (t x) (f x)
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/- An equivalence between `X → Fin 1 ⊕ Fin d` and `Y → Fin 1 ⊕ Fin d` given an isomorphism
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between `X` and `Y`. -/
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def indexEquivOfIndexHom (f : X ≅ Y) : (X → Fin 1 ⊕ Fin d) ≃ (Y → Fin 1 ⊕ Fin d) :=
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Equiv.piCongrLeft' _ (FintypeCat.equivEquivIso.symm f)
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/-- Given an isomorphism of indexing sets, a linear equivalence on Lorentz tensors. -/
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noncomputable def mapOfIndexHom (f : X ≅ Y) : LorentzTensor d Y ≃ₗ[ℝ] LorentzTensor d X :=
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LinearEquiv.piCongrLeft' ℝ _ (indexEquivOfIndexHom f).symm
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/-!
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## Graphical species and Lorentz tensors
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-/
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/-- The graphical species defined by Lorentz tensors.
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For this simple case, 𝓣 gets mapped to `PUnit`, if one wishes to include fermions etc,
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then `PUnit` will change to account for the colouring of edges. -/
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noncomputable def graphicalSpecies (d : ℕ) : GraphicalSpecies where
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obj x :=
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match x with
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| ⟨𝓣⟩ => PUnit
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| ⟨as f⟩ => LorentzTensor d f
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map {x y} f :=
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match x, y, f with
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| ⟨𝓣⟩, ⟨𝓣⟩, _ => 𝟙 PUnit
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| ⟨𝓣⟩, ⟨as x⟩, ⟨f⟩ => Empty.elim f
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| ⟨as f⟩, ⟨𝓣⟩, _ => fun _ => PUnit.unit
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| ⟨as f⟩, ⟨as g⟩, ⟨h⟩ => (mapOfIndexHom h).toEquiv.toFun
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map_id X := by
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match X with
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| ⟨𝓣⟩ => rfl
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| ⟨as f⟩ => rfl
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map_comp {x y z} f g := by
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match x, y, z, f, g with
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| ⟨𝓣⟩, ⟨𝓣⟩, ⟨𝓣⟩, _, _ => rfl
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| _, ⟨𝓣⟩, ⟨as _⟩, _, ⟨g⟩ => exact Empty.elim g
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| ⟨𝓣⟩, ⟨as _⟩, _, ⟨f⟩, _ => exact Empty.elim f
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| ⟨as x⟩, ⟨as y⟩, ⟨as z⟩, ⟨f⟩, ⟨g⟩ => rfl
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| ⟨as x⟩, ⟨𝓣⟩, ⟨𝓣⟩, _, _ => rfl
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| ⟨as x⟩, ⟨as y⟩, ⟨𝓣⟩, _, _ => rfl
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end LorentzTensor
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