Merge pull request #52 from HEPLean/LorentzAlgebra
Feat: Relation between spacetime and self-adjoint matrices
This commit is contained in:
Joseph Tooby-Smith
GitHub
commit
bfb9e01a2f
5 changed files with 104 additions and 3 deletions
|
|
@ -56,6 +56,7 @@ import HepLean.FlavorPhysics.CKMMatrix.Rows
|
|||
import HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.Basic
|
||||
import HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.StandardParameters
|
||||
import HepLean.GroupTheory.SO3.Basic
|
||||
import HepLean.SpaceTime.AsSelfAdjointMatrix
|
||||
import HepLean.SpaceTime.Basic
|
||||
import HepLean.SpaceTime.CliffordAlgebra
|
||||
import HepLean.SpaceTime.FourVelocity
|
||||
|
|
|
|||
93
HepLean/SpaceTime/AsSelfAdjointMatrix.lean
Normal file
93
HepLean/SpaceTime/AsSelfAdjointMatrix.lean
Normal file
|
|
@ -0,0 +1,93 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.SpaceTime.Metric
|
||||
import HepLean.SpaceTime.FourVelocity
|
||||
import Mathlib.GroupTheory.SpecificGroups.KleinFour
|
||||
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
|
||||
/-!
|
||||
# Spacetime as a self-adjoint matrix
|
||||
|
||||
The main result of this file is a linear equivalence `spaceTimeToHerm` between the vector space
|
||||
of space-time points and the vector space of 2×2-complex self-adjoint matrices.
|
||||
|
||||
-/
|
||||
namespace spaceTime
|
||||
|
||||
open Matrix
|
||||
open MatrixGroups
|
||||
open Complex
|
||||
|
||||
/-- A 2×2-complex matrix formed from a space-time point. -/
|
||||
@[simp]
|
||||
def toMatrix (x : spaceTime) : Matrix (Fin 2) (Fin 2) ℂ :=
|
||||
!![x 0 + x 3, x 1 - x 2 * I; x 1 + x 2 * I, x 0 - x 3]
|
||||
|
||||
/-- The matrix `x.toMatrix` for `x ∈ spaceTime` is self adjoint. -/
|
||||
lemma toMatrix_isSelfAdjoint (x : spaceTime) : IsSelfAdjoint x.toMatrix := by
|
||||
rw [isSelfAdjoint_iff, star_eq_conjTranspose, ← Matrix.ext_iff]
|
||||
intro i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
simp [toMatrix, conj_ofReal]
|
||||
ring
|
||||
|
||||
/-- A self-adjoint matrix formed from a space-time point. -/
|
||||
@[simps!]
|
||||
def toSelfAdjointMatrix' (x : spaceTime) : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) :=
|
||||
⟨x.toMatrix, toMatrix_isSelfAdjoint x⟩
|
||||
|
||||
/-- A self-adjoint matrix formed from a space-time point. -/
|
||||
@[simp]
|
||||
noncomputable def fromSelfAdjointMatrix' (x : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) : spaceTime :=
|
||||
![1/2 * (x.1 0 0 + x.1 1 1).re, (x.1 1 0).re, (x.1 1 0).im , (x.1 0 0 - x.1 1 1).re/2]
|
||||
|
||||
/-- The linear equivalence between the vector-space `spaceTime` and self-adjoint
|
||||
2×2-complex matrices. -/
|
||||
noncomputable def spaceTimeToHerm : spaceTime ≃ₗ[ℝ] selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) where
|
||||
toFun := toSelfAdjointMatrix'
|
||||
invFun := fromSelfAdjointMatrix'
|
||||
left_inv x := by
|
||||
simp only [fromSelfAdjointMatrix', one_div, Fin.isValue, toSelfAdjointMatrix'_coe, of_apply,
|
||||
cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_one, head_cons,
|
||||
head_fin_const, add_add_sub_cancel, add_re, ofReal_re, mul_re, I_re, mul_zero, ofReal_im,
|
||||
I_im, mul_one, sub_self, add_zero, add_im, mul_im, zero_add, add_sub_sub_cancel,
|
||||
half_add_self]
|
||||
funext i
|
||||
fin_cases i <;> field_simp
|
||||
rfl
|
||||
rfl
|
||||
right_inv x := by
|
||||
simp only [toSelfAdjointMatrix', toMatrix, fromSelfAdjointMatrix', one_div, Fin.isValue, add_re,
|
||||
sub_re, cons_val_zero, ofReal_mul, ofReal_inv, ofReal_ofNat, ofReal_add, cons_val_three,
|
||||
Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons, head_cons, ofReal_div, ofReal_sub,
|
||||
cons_val_one, cons_val_two, re_add_im]
|
||||
ext i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
field_simp [fromSelfAdjointMatrix', toMatrix, conj_ofReal]
|
||||
exact conj_eq_iff_re.mp (congrArg (fun M => M 0 0) $ selfAdjoint.mem_iff.mp x.2 )
|
||||
have h01 := congrArg (fun M => M 0 1) $ selfAdjoint.mem_iff.mp x.2
|
||||
simp only [Fin.isValue, star_apply, RCLike.star_def] at h01
|
||||
rw [← h01]
|
||||
rw [RCLike.conj_eq_re_sub_im]
|
||||
simp only [Fin.isValue, RCLike.re_to_complex, RCLike.im_to_complex, RCLike.I_to_complex]
|
||||
rfl
|
||||
exact conj_eq_iff_re.mp (congrArg (fun M => M 1 1) $ selfAdjoint.mem_iff.mp x.2 )
|
||||
map_add' x y := by
|
||||
simp only [toSelfAdjointMatrix', toMatrix, Fin.isValue, add_apply, ofReal_add,
|
||||
AddSubmonoid.mk_add_mk, of_add_of, add_cons, head_cons, tail_cons, empty_add_empty,
|
||||
Subtype.mk.injEq, EmbeddingLike.apply_eq_iff_eq]
|
||||
ext i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
field_simp [fromSelfAdjointMatrix', toMatrix, conj_ofReal, add_apply]
|
||||
<;> ring
|
||||
map_smul' r x := by
|
||||
simp only [toSelfAdjointMatrix', toMatrix, Fin.isValue, smul_apply, ofReal_mul,
|
||||
RingHom.id_apply]
|
||||
ext i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
field_simp [fromSelfAdjointMatrix', toMatrix, conj_ofReal, smul_apply]
|
||||
<;> ring
|
||||
|
||||
end spaceTime
|
||||
|
|
@ -89,6 +89,11 @@ lemma explicit (x : spaceTime) : x = ![x 0, x 1, x 2, x 3] := by
|
|||
funext i
|
||||
fin_cases i <;> rfl
|
||||
|
||||
@[simp]
|
||||
lemma add_apply (x y : spaceTime) (i : Fin 4) : (x + y) i = x i + y i := rfl
|
||||
|
||||
@[simp]
|
||||
lemma smul_apply (x : spaceTime) (a : ℝ) (i : Fin 4) : (a • x) i = a * x i := rfl
|
||||
|
||||
end spaceTime
|
||||
|
||||
|
|
|
|||
|
|
@ -65,7 +65,7 @@ lemma σ_comm (α β γ δ : Fin 4) :
|
|||
change σMat α β * σ γ δ - σ γ δ * σ α β = _
|
||||
funext a b
|
||||
simp only [σ_coe, sub_apply, AddSubmonoid.coe_add, Submodule.coe_toAddSubmonoid,
|
||||
Submodule.coe_smul_of_tower, add_apply, smul_apply, σMat, smul_eq_mul]
|
||||
Submodule.coe_smul_of_tower, Matrix.add_apply, Matrix.smul_apply, σMat, smul_eq_mul]
|
||||
rw [σMat_mul, σMat_mul, η_symmetric α γ, η_symmetric α δ, η_symmetric β γ, η_symmetric β δ]
|
||||
ring
|
||||
|
||||
|
|
@ -77,7 +77,7 @@ lemma eq_span_σ (Λ : lorentzAlgebra) :
|
|||
fin_cases a <;> fin_cases b <;>
|
||||
simp only [Fin.zero_eta, Fin.isValue, Fin.mk_one, Fin.reduceFinMk, AddSubmonoid.coe_add,
|
||||
Submodule.coe_smul_of_tower, σ_coe,
|
||||
add_apply, smul_apply, σMat, ηUpDown, ne_eq, zero_ne_one, not_false_eq_true,
|
||||
Matrix.add_apply, Matrix.smul_apply, σMat, ηUpDown, ne_eq, zero_ne_one, not_false_eq_true,
|
||||
one_apply_ne, η_explicit, of_apply, cons_val_zero,
|
||||
mul_zero, one_apply_eq, mul_one, sub_neg_eq_add,
|
||||
zero_add, smul_eq_mul, Fin.reduceEq, cons_val_one, vecHead, vecTail,
|
||||
|
|
@ -121,7 +121,7 @@ noncomputable def σCoordinateMap : lorentzAlgebra ≃ₗ[ℝ] Fin 6 →₀ ℝ
|
|||
fin_cases i <;> simp only [Fin.isValue, Set.Finite.toFinset_setOf, ne_eq, Finsupp.coe_mk,
|
||||
Fin.zero_eta, Fin.isValue, Fin.mk_one, Fin.reduceFinMk, AddSubmonoid.coe_add,
|
||||
Submodule.coe_toAddSubmonoid, Submodule.coe_smul_of_tower, σ_coe,
|
||||
add_apply, smul_apply, σMat, ηUpDown, ne_eq, zero_ne_one, not_false_eq_true,
|
||||
Matrix.add_apply, Matrix.smul_apply, σMat, ηUpDown, ne_eq, zero_ne_one, not_false_eq_true,
|
||||
one_apply_ne, η_explicit, of_apply, cons_val', cons_val_zero, empty_val',
|
||||
cons_val_fin_one, vecCons_const, mul_zero, one_apply_eq, mul_one, sub_neg_eq_add,
|
||||
zero_add, smul_eq_mul, Fin.reduceEq, cons_val_one, vecHead, vecTail, Nat.succ_eq_add_one,
|
||||
|
|
|
|||
|
|
@ -6,6 +6,8 @@ Authors: Joseph Tooby-Smith
|
|||
import HepLean.SpaceTime.Metric
|
||||
import HepLean.SpaceTime.FourVelocity
|
||||
import Mathlib.GroupTheory.SpecificGroups.KleinFour
|
||||
import Mathlib.Geometry.Manifold.Algebra.LieGroup
|
||||
import Mathlib.Analysis.Matrix
|
||||
/-!
|
||||
# The Lorentz Group
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue