feat: Add properties of Lorentz group
This commit is contained in:
jstoobysmith
parent
b9c20861d1
commit
db0f6de59c
2 changed files with 173 additions and 21 deletions
|
|
@ -50,16 +50,94 @@ namespace spaceTime
|
|||
|
||||
def asSmoothManifold : ModelWithCorners ℝ spaceTime spaceTime := 𝓘(ℝ, spaceTime)
|
||||
|
||||
@[simp]
|
||||
def stdBasis : Basis (Fin 4) ℝ spaceTime := Pi.basisFun ℝ (Fin 4)
|
||||
|
||||
lemma stdBasis_apply (μ ν : Fin 4) : stdBasis μ ν = if μ = ν then 1 else 0 := by
|
||||
erw [stdBasis, Pi.basisFun_apply, LinearMap.stdBasis_apply']
|
||||
|
||||
lemma stdBasis_not_eq {μ ν : Fin 4} (h : μ ≠ ν) : stdBasis μ ν = 0 := by
|
||||
rw [stdBasis_apply]
|
||||
exact if_neg h
|
||||
|
||||
lemma stdBasis_0 : stdBasis 0 = ![1, 0, 0, 0] := by
|
||||
funext i
|
||||
fin_cases i
|
||||
<;> simp [stdBasis_apply]
|
||||
|
||||
lemma stdBasis_1 : stdBasis 1 = ![0, 1, 0, 0] := by
|
||||
funext i
|
||||
fin_cases i
|
||||
<;> simp [stdBasis_apply]
|
||||
|
||||
lemma stdBasis_2 : stdBasis 2 = ![0, 0, 1, 0] := by
|
||||
funext i
|
||||
fin_cases i
|
||||
<;> simp [stdBasis_apply]
|
||||
|
||||
lemma stdBasis_3 : stdBasis 3 = ![0, 0, 0, 1] := by
|
||||
funext i
|
||||
fin_cases i
|
||||
<;> simp [stdBasis_apply]
|
||||
|
||||
lemma stdBasis_mulVec (μ ν : Fin 4) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
|
||||
(Λ *ᵥ stdBasis μ) ν = Λ ν μ := by
|
||||
rw [mulVec, dotProduct, Fintype.sum_eq_single μ, stdBasis_apply]
|
||||
simp
|
||||
intro x h
|
||||
rw [stdBasis_apply, if_neg (Ne.symm h)]
|
||||
simp
|
||||
|
||||
|
||||
|
||||
lemma explicit (x : spaceTime) : x = ![x 0, x 1, x 2, x 3] := by
|
||||
funext i
|
||||
fin_cases i <;> rfl
|
||||
|
||||
|
||||
|
||||
def η : Matrix (Fin 4) (Fin 4) ℝ :=
|
||||
![![1, 0, 0, 0], ![0, -1, 0, 0], ![0, 0, -1, 0], ![0, 0, 0, -1]]
|
||||
|
||||
lemma η_symmetric (μ ν : Fin 4) : η μ ν = η ν μ := by
|
||||
lemma η_off_diagonal {μ ν : Fin 4} (h : μ ≠ ν) : η μ ν = 0 := by
|
||||
fin_cases μ <;>
|
||||
fin_cases ν <;>
|
||||
simp only [η, Fin.zero_eta, Matrix.cons_val', Matrix.cons_val_fin_one, Matrix.cons_val_one,
|
||||
simp_all [η, Fin.zero_eta, Matrix.cons_val', Matrix.cons_val_fin_one, Matrix.cons_val_one,
|
||||
Matrix.cons_val_succ', Matrix.cons_val_zero, Matrix.empty_val', Matrix.head_cons,
|
||||
Matrix.head_fin_const, Matrix.head_cons, Matrix.vecCons_const, Fin.mk_one, Fin.mk_one]
|
||||
Matrix.head_fin_const, Matrix.head_cons, Matrix.vecCons_const, Fin.mk_one, Fin.mk_one,
|
||||
vecHead, vecTail, Function.comp_apply]
|
||||
|
||||
lemma η_symmetric (μ ν : Fin 4) : η μ ν = η ν μ := by
|
||||
by_cases h : μ = ν
|
||||
rw [h]
|
||||
rw [η_off_diagonal h]
|
||||
refine (η_off_diagonal ?_).symm
|
||||
exact fun a => h (id a.symm)
|
||||
|
||||
lemma η_transpose : η.transpose = η := by
|
||||
funext μ ν
|
||||
rw [transpose_apply, η_symmetric]
|
||||
|
||||
lemma η_sq : η * η = 1 := by
|
||||
funext μ ν
|
||||
rw [mul_apply, Fin.sum_univ_four]
|
||||
fin_cases μ <;> fin_cases ν <;>
|
||||
simp [η, Fin.zero_eta, Matrix.cons_val', Matrix.cons_val_fin_one, Matrix.cons_val_one,
|
||||
Matrix.cons_val_succ', Matrix.cons_val_zero, Matrix.empty_val', Matrix.head_cons,
|
||||
Matrix.head_fin_const, Matrix.head_cons, Matrix.vecCons_const, Fin.mk_one, Fin.mk_one,
|
||||
vecHead, vecTail, Function.comp_apply]
|
||||
|
||||
|
||||
|
||||
lemma η_mulVec (x : spaceTime) : η *ᵥ x = ![x 0, -x 1, -x 2, -x 3] := by
|
||||
rw [explicit x]
|
||||
rw [η]
|
||||
funext i
|
||||
rw [mulVec, dotProduct, Fin.sum_univ_four]
|
||||
fin_cases i <;>
|
||||
simp [η, Fin.zero_eta, Matrix.cons_val', Matrix.cons_val_fin_one, Matrix.cons_val_one,
|
||||
Matrix.cons_val_succ', Matrix.cons_val_zero, Matrix.empty_val', Matrix.head_cons,
|
||||
Matrix.head_fin_const, Matrix.head_cons, Matrix.vecCons_const, Fin.mk_one, Fin.mk_one,
|
||||
vecHead, vecTail, Function.comp_apply]
|
||||
|
||||
@[simps!]
|
||||
def linearMapForSpaceTime (x : spaceTime) : spaceTime →ₗ[ℝ] ℝ where
|
||||
|
|
@ -72,6 +150,7 @@ def linearMapForSpaceTime (x : spaceTime) : spaceTime →ₗ[ℝ] ℝ where
|
|||
rw [mulVec_smul, dotProduct_smul]
|
||||
rfl
|
||||
|
||||
@[simps!]
|
||||
def ηLin : LinearMap.BilinForm ℝ spaceTime where
|
||||
toFun x := linearMapForSpaceTime x
|
||||
map_add' x y := by
|
||||
|
|
@ -86,13 +165,74 @@ def ηLin : LinearMap.BilinForm ℝ spaceTime where
|
|||
rw [smul_dotProduct]
|
||||
rfl
|
||||
|
||||
lemma ηLin_expand (x y : spaceTime) : ηLin x y = x 0 * y 0 - x 1 * y 1 - x 2 * y 2 - x 3 * y 3 := by
|
||||
rw [ηLin]
|
||||
simp only [LinearMap.coe_mk, AddHom.coe_mk, linearMapForSpaceTime_apply, Fin.isValue]
|
||||
erw [η_mulVec]
|
||||
nth_rewrite 1 [explicit x]
|
||||
simp only [dotProduct, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, Fin.sum_univ_four,
|
||||
cons_val_zero, cons_val_one, head_cons, mul_neg, cons_val_two, tail_cons, cons_val_three]
|
||||
ring
|
||||
|
||||
lemma ηLin_symm (x y : spaceTime) : ηLin x y = ηLin y x := by
|
||||
rw [ηLin_expand, ηLin_expand]
|
||||
ring
|
||||
|
||||
lemma ηLin_stdBasis_apply (μ : Fin 4) (x : spaceTime) : ηLin (stdBasis μ) x = η μ μ * x μ := by
|
||||
rw [ηLin_expand]
|
||||
fin_cases μ
|
||||
<;> simp [stdBasis_0, stdBasis_1, stdBasis_2, stdBasis_3, η]
|
||||
|
||||
|
||||
lemma ηLin_η_stdBasis (μ ν : Fin 4) : ηLin (stdBasis μ) (stdBasis ν) = η μ ν := by
|
||||
rw [ηLin_stdBasis_apply]
|
||||
by_cases h : μ = ν
|
||||
rw [stdBasis_apply]
|
||||
subst h
|
||||
simp
|
||||
rw [stdBasis_not_eq, η_off_diagonal h]
|
||||
simp
|
||||
exact fun a => h (id a.symm)
|
||||
|
||||
lemma ηLin_mulVec_left (x y : spaceTime) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
|
||||
ηLin (Λ *ᵥ x) y = ηLin x ((η * Λᵀ * η) *ᵥ y) := by
|
||||
simp only [ηLin_apply_apply, mulVec_mulVec]
|
||||
rw [(vecMul_transpose Λ x).symm, ← dotProduct_mulVec, mulVec_mulVec]
|
||||
rw [← mul_assoc, ← mul_assoc, η_sq, one_mul]
|
||||
|
||||
lemma ηLin_mulVec_right (x y : spaceTime) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
|
||||
ηLin x (Λ *ᵥ y) = ηLin ((η * Λᵀ * η) *ᵥ x) y := by
|
||||
rw [ηLin_symm, ηLin_symm ((η * Λᵀ * η) *ᵥ x) _ ]
|
||||
exact ηLin_mulVec_left y x Λ
|
||||
|
||||
lemma ηLin_matrix_stdBasis (μ ν : Fin 4) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
|
||||
ηLin (stdBasis ν) (Λ *ᵥ stdBasis μ) = η ν ν * Λ ν μ := by
|
||||
rw [ηLin_stdBasis_apply, stdBasis_mulVec]
|
||||
|
||||
lemma ηLin_matrix_eq_identity_iff (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
|
||||
Λ = 1 ↔ ∀ (x y : spaceTime), ηLin x y = ηLin x (Λ *ᵥ y) := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
subst h
|
||||
simp
|
||||
intro h
|
||||
funext μ ν
|
||||
have h1 := h (stdBasis μ) (stdBasis ν)
|
||||
rw [ηLin_matrix_stdBasis, ηLin_η_stdBasis] at h1
|
||||
fin_cases μ <;> fin_cases ν <;>
|
||||
simp_all [η, Fin.zero_eta, Matrix.cons_val', Matrix.cons_val_fin_one, Matrix.cons_val_one,
|
||||
Matrix.cons_val_succ', Matrix.cons_val_zero, Matrix.empty_val', Matrix.head_cons,
|
||||
Matrix.head_fin_const, Matrix.head_cons, Matrix.vecCons_const, Fin.mk_one, Fin.mk_one,
|
||||
vecHead, vecTail, Function.comp_apply]
|
||||
|
||||
|
||||
def quadraticForm : QuadraticForm ℝ spaceTime := ηLin.toQuadraticForm
|
||||
|
||||
def lorentzGroup : Subgroup (GeneralLinearGroup (Fin 4) ℝ) where
|
||||
carrier := {Λ | ∀ (x y : spaceTime), ηLin (Λ *ᵥ x) (Λ *ᵥ y) = ηLin x y}
|
||||
mul_mem' {a b} := by
|
||||
intros ha hb x y
|
||||
simp
|
||||
simp only [Units.val_mul, mulVec_mulVec]
|
||||
rw [← mulVec_mulVec, ← mulVec_mulVec, ha, hb]
|
||||
one_mem' := by
|
||||
intros x y
|
||||
|
|
@ -105,6 +245,31 @@ def lorentzGroup : Subgroup (GeneralLinearGroup (Fin 4) ℝ) where
|
|||
simp [hx]
|
||||
|
||||
|
||||
lemma mem_lorentzGroup_iff (Λ : GeneralLinearGroup (Fin 4) ℝ) :
|
||||
Λ ∈ lorentzGroup ↔ ∀ (x y : spaceTime), ηLin (Λ *ᵥ x) (Λ *ᵥ y) = ηLin x y := by
|
||||
rfl
|
||||
|
||||
lemma mem_lorentzGroup_iff' (Λ : GeneralLinearGroup (Fin 4) ℝ) :
|
||||
Λ ∈ lorentzGroup ↔ ∀ (x y : spaceTime), ηLin (x) ((η * Λ.1ᵀ * η * Λ.1) *ᵥ y) = ηLin x y := by
|
||||
rw [mem_lorentzGroup_iff]
|
||||
apply Iff.intro
|
||||
intro h
|
||||
intro x y
|
||||
have h1 := h x y
|
||||
rw [ηLin_mulVec_left, mulVec_mulVec] at h1
|
||||
exact h1
|
||||
intro h
|
||||
intro x y
|
||||
rw [ηLin_mulVec_left, mulVec_mulVec]
|
||||
exact h x y
|
||||
|
||||
lemma mem_lorentzGroup_iff'' (Λ : GeneralLinearGroup (Fin 4) ℝ) :
|
||||
Λ ∈ lorentzGroup ↔ η * Λ.1ᵀ * η * Λ.1 = 1 := by
|
||||
rw [mem_lorentzGroup_iff', ηLin_matrix_eq_identity_iff (η * Λ.1ᵀ * η * Λ.1)]
|
||||
apply Iff.intro
|
||||
· simp_all only [ηLin_apply_apply, implies_true, iff_true, one_mulVec]
|
||||
· simp_all only [ηLin_apply_apply, mulVec_mulVec, implies_true]
|
||||
|
||||
abbrev cliffordAlgebra := CliffordAlgebra quadraticForm
|
||||
|
||||
end spaceTime
|
||||
|
|
|
|||
|
|
@ -13,7 +13,7 @@ This file defines the Gamma matrices.
|
|||
|
||||
- Prove that the algebra generated by the gamma matrices is ismorphic to the
|
||||
Clifford algebra assocaited with spacetime.
|
||||
-
|
||||
- Include relations for gamma matrices.
|
||||
-/
|
||||
|
||||
namespace StandardModel
|
||||
|
|
@ -41,22 +41,9 @@ def γ : Fin 4 → Matrix (Fin 4) (Fin 4) ℂ := ![γ0, γ1, γ2, γ3]
|
|||
|
||||
namespace γ
|
||||
|
||||
variable (μ : Fin 4)
|
||||
|
||||
|
||||
|
||||
|
||||
/-- The trace of the gamma matrices is zero. -/
|
||||
lemma trace_eq_zero (μ : Fin 4) : Matrix.trace (γ μ) = 0 := by
|
||||
fin_cases μ
|
||||
<;> simp [γ, γ0, γ1, γ2, γ3]
|
||||
<;> rw [Matrix.trace, Fin.sum_univ_four]
|
||||
<;> simp
|
||||
any_goals rfl
|
||||
change 0 + 0 = 0
|
||||
simp [add_zero]
|
||||
|
||||
open spaceTime
|
||||
|
||||
variable (μ ν : Fin 4)
|
||||
|
||||
@[simp]
|
||||
def γSet : Set (Matrix (Fin 4) (Fin 4) ℂ) := {γ i | i : Fin 4}
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue