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refactor: Large refactor of Lorentz vecs

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jstoobysmith 2024-11-08 16:24:58 +00:00
parent a69cf91919
commit 3eb5da875f
8 changed files with 402 additions and 88 deletions
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29
HepLean/SpaceTime/LorentzVector/Real/Modules.lean
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@ -93,6 +93,10 @@ lemma stdBasis_toFin1dℝEquiv_apply_same (μ : Fin 1 ⊕ Fin d) :
rw [@LinearEquiv.apply_symm_apply]
exact Pi.single_eq_same μ 1
@[simp]
lemma stdBasis_apply_same (μ : Fin 1 ⊕ Fin d) : (stdBasis μ).val μ = 1 :=
stdBasis_toFin1dℝEquiv_apply_same μ
lemma stdBasis_toFin1dℝEquiv_apply_ne {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) :
toFin1dℝEquiv (stdBasis μ) ν = 0 := by
simp only [stdBasis, Basis.ofEquivFun, Basis.coe_ofRepr, LinearEquiv.trans_symm,
@ -100,6 +104,11 @@ lemma stdBasis_toFin1dℝEquiv_apply_ne {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν
rw [@LinearEquiv.apply_symm_apply]
exact Pi.single_eq_of_ne' h 1
@[simp]
lemma stdBasis_inl_apply_inr (i : Fin d) : (stdBasis (Sum.inl 0)).val (Sum.inr i) = 0 := by
refine stdBasis_toFin1dℝEquiv_apply_ne ?_
simp
/-- Decomposition of a contrvariant Lorentz vector into the standard basis. -/
lemma stdBasis_decomp (v : ContrMod d) : v = ∑ i, v.toFin1dℝ i • stdBasis i := by
apply toFin1dℝEquiv.injective
@ -151,6 +160,22 @@ lemma mulVec_mulVec (M N : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) (v :
/-!
## The norm
(Not the Minkowski norm, but the norm of a vector in `ContrℝModule d`.)
-/
def norm : NormedAddCommGroup (ContrMod d) where
norm v := ‖v.val‖₊
dist_self x := Pi.normedAddCommGroup.dist_self x.val
dist_triangle x y z := Pi.normedAddCommGroup.dist_triangle x.val y.val z.val
dist_comm x y := Pi.normedAddCommGroup.dist_comm x.val y.val
eq_of_dist_eq_zero {x y} := fun h => ext (MetricSpace.eq_of_dist_eq_zero h)
def toSpace (v : ContrMod d) : EuclideanSpace ℝ (Fin d) := v.val ∘ Sum.inr
/-!
## The representation.
-/
@ -230,6 +255,10 @@ lemma stdBasis_toFin1dℝEquiv_apply_same (μ : Fin 1 ⊕ Fin d) :
rw [@LinearEquiv.apply_symm_apply]
exact Pi.single_eq_same μ 1
@[simp]
lemma stdBasis_apply_same (μ : Fin 1 ⊕ Fin d) : (stdBasis μ).val μ = 1 :=
stdBasis_toFin1dℝEquiv_apply_same μ
lemma stdBasis_toFin1dℝEquiv_apply_ne {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) :
toFin1dℝEquiv (stdBasis μ) ν = 0 := by
simp only [stdBasis, Basis.ofEquivFun, Basis.coe_ofRepr, LinearEquiv.trans_symm,