113 lines
2.9 KiB
Text
113 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 Mathlib.Data.Complex.Exponential
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import Mathlib.Analysis.InnerProductSpace.PiL2
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/-!
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# Lorentz vectors
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(aka 4-vectors)
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In this file we define a Lorentz vector (in 4d, this is more often called a 4-vector).
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One of the most important example of a Lorentz vector is SpaceTime.
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-/
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/- The number of space dimensions . -/
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variable (d : ℕ)
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/-- The type of Lorentz Vectors in `d`-space dimensions. -/
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def LorentzVector : Type := (Fin 1 ⊕ Fin d) → ℝ
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/-- An instance of a additive commutative monoid on `LorentzVector`. -/
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instance : AddCommMonoid (LorentzVector d) := Pi.addCommMonoid
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/-- An instance of a module on `LorentzVector`. -/
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noncomputable instance : Module ℝ (LorentzVector d) := Pi.module _ _ _
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instance : AddCommGroup (LorentzVector d) := Pi.addCommGroup
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/-- The structure of a topological space `LorentzVector d`. -/
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instance : TopologicalSpace (LorentzVector d) :=
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haveI : NormedAddCommGroup (LorentzVector d) := Pi.normedAddCommGroup
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UniformSpace.toTopologicalSpace
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namespace LorentzVector
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variable {d : ℕ}
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variable (v : LorentzVector d)
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/-- The space components. -/
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@[simp]
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def space : EuclideanSpace ℝ (Fin d) := v ∘ Sum.inr
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/-- The time component. -/
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@[simp]
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def time : ℝ := v (Sum.inl 0)
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/-!
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# The standard basis
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-/
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/-- The standard basis of `LorentzVector` indexed by `Fin 1 ⊕ Fin (d)`. -/
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@[simps!]
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noncomputable def stdBasis : Basis (Fin 1 ⊕ Fin (d)) ℝ (LorentzVector d) := Pi.basisFun ℝ _
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/-- Notation for `stdBasis`. -/
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scoped[LorentzVector] notation "e" => stdBasis
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lemma stdBasis_apply (μ ν : Fin 1 ⊕ Fin d) : e μ ν = if μ = ν then 1 else 0 := by
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rw [stdBasis]
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erw [Pi.basisFun_apply]
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exact LinearMap.stdBasis_apply' ℝ μ ν
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/-- The standard unit time vector. -/
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noncomputable abbrev timeVec : (LorentzVector d) := e (Sum.inl 0)
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lemma timeVec_space : (@timeVec d).space = 0 := by
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funext i
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simp only [space, Function.comp_apply, stdBasis_apply, Fin.isValue, ↓reduceIte, PiLp.zero_apply]
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lemma timeVec_time: (@timeVec d).time = 1 := by
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simp only [time, Fin.isValue, stdBasis_apply, ↓reduceIte]
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/-!
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# Reflection of space
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-/
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/-- The reflection of space as a linear map. -/
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@[simps!]
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def spaceReflectionLin : LorentzVector d →ₗ[ℝ] LorentzVector d where
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toFun x := Sum.elim (x ∘ Sum.inl) (- x ∘ Sum.inr)
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map_add' x y := by
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funext i
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rcases i with i | i
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· rfl
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· simp only [Sum.elim_inr, Pi.neg_apply]
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apply neg_add
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map_smul' c x := by
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funext i
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rcases i with i | i
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· rfl
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· simp [HSMul.hSMul, SMul.smul]
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/-- The reflection of space. -/
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@[simp]
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def spaceReflection : LorentzVector d := spaceReflectionLin v
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lemma spaceReflection_space : v.spaceReflection.space = - v.space := by
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rfl
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lemma spaceReflection_time : v.spaceReflection.time = v.time := by
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rfl
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end LorentzVector
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