141 lines
6.2 KiB
Text
141 lines
6.2 KiB
Text
/-
|
||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||
Released under Apache 2.0 license.
|
||
Authors: Joseph Tooby-Smith
|
||
-/
|
||
import HepLean.SpaceTime.MinkowskiMetric
|
||
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
|
||
/-!
|
||
# Spacetime as a self-adjoint matrix
|
||
|
||
There is a linear equivalence between the vector space of space-time points
|
||
and the vector space of 2×2-complex self-adjoint matrices.
|
||
|
||
In this file we define this linear equivalence in `toSelfAdjointMatrix`.
|
||
|
||
-/
|
||
/-! TODO: Generalize rep of Lorentz vector as a self-adjoint matrix to arbitrary dimension. -/
|
||
namespace SpaceTime
|
||
|
||
open Matrix
|
||
open MatrixGroups
|
||
open Complex
|
||
|
||
/-- A 2×2-complex matrix formed from a space-time point. -/
|
||
@[simp]
|
||
def toMatrix (x : LorentzVector 3) : Matrix (Fin 2) (Fin 2) ℂ :=
|
||
!![x.time + x.space 2, x.space 0 - x.space 1 * I; x.space 0 + x.space 1 * I, x.time - x.space 2]
|
||
|
||
/-- The matrix `x.toMatrix` for `x ∈ spaceTime` is self adjoint. -/
|
||
lemma toMatrix_isSelfAdjoint (x : LorentzVector 3) : IsSelfAdjoint (toMatrix x) := by
|
||
rw [isSelfAdjoint_iff, star_eq_conjTranspose, ← Matrix.ext_iff]
|
||
intro i j
|
||
fin_cases i <;> fin_cases j <;>
|
||
simp [toMatrix, conj_ofReal]
|
||
rfl
|
||
|
||
/-- A self-adjoint matrix formed from a space-time point. -/
|
||
@[simps!]
|
||
def toSelfAdjointMatrix' (x : LorentzVector 3) : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) :=
|
||
⟨toMatrix x, 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)) :
|
||
LorentzVector 3 := fun i =>
|
||
match i with
|
||
| Sum.inl 0 => 1/2 * (x.1 0 0 + x.1 1 1).re
|
||
| Sum.inr 0 => (x.1 1 0).re
|
||
| Sum.inr 1 => (x.1 1 0).im
|
||
| Sum.inr 2 => 1/2 * (x.1 0 0 - x.1 1 1).re
|
||
|
||
/-- The linear equivalence between the vector-space `spaceTime` and self-adjoint
|
||
2×2-complex matrices. -/
|
||
noncomputable def toSelfAdjointMatrix :
|
||
LorentzVector 3 ≃ₗ[ℝ] selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) where
|
||
toFun := toSelfAdjointMatrix'
|
||
invFun := fromSelfAdjointMatrix'
|
||
left_inv x := by
|
||
funext i
|
||
match i with
|
||
| Sum.inl 0 =>
|
||
simp [fromSelfAdjointMatrix', toSelfAdjointMatrix', toMatrix, toMatrix_isSelfAdjoint]
|
||
ring_nf
|
||
| Sum.inr 0 =>
|
||
simp [fromSelfAdjointMatrix', toSelfAdjointMatrix', toMatrix, toMatrix_isSelfAdjoint]
|
||
| Sum.inr 1 =>
|
||
simp [fromSelfAdjointMatrix', toSelfAdjointMatrix', toMatrix, toMatrix_isSelfAdjoint]
|
||
| Sum.inr 2 =>
|
||
simp [fromSelfAdjointMatrix', toSelfAdjointMatrix', toMatrix, toMatrix_isSelfAdjoint]
|
||
ring
|
||
right_inv x := by
|
||
simp only [toSelfAdjointMatrix', toMatrix, LorentzVector.time, fromSelfAdjointMatrix', one_div,
|
||
Fin.isValue, add_re, ofReal_mul, ofReal_inv, ofReal_ofNat, ofReal_add, LorentzVector.space,
|
||
Function.comp_apply, sub_re, ofReal_sub, 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, RCLike.conj_eq_re_sub_im]
|
||
rfl
|
||
exact conj_eq_iff_re.mp (congrArg (fun M => M 1 1) $ selfAdjoint.mem_iff.mp x.2)
|
||
map_add' x y := by
|
||
ext i j : 2
|
||
simp only [toSelfAdjointMatrix'_coe, add_apply, ofReal_add, of_apply, cons_val', empty_val',
|
||
cons_val_fin_one, AddSubmonoid.coe_add, AddSubgroup.coe_toAddSubmonoid, Matrix.add_apply]
|
||
fin_cases i <;> fin_cases j
|
||
· rw [show (x + y) (Sum.inl 0) = x (Sum.inl 0) + y (Sum.inl 0) from rfl]
|
||
rw [show (x + y) (Sum.inr 2) = x (Sum.inr 2) + y (Sum.inr 2) from rfl]
|
||
simp only [Fin.isValue, ofReal_add, Fin.zero_eta, cons_val_zero]
|
||
ring
|
||
· rw [show (x + y) (Sum.inr 0) = x (Sum.inr 0) + y (Sum.inr 0) from rfl]
|
||
rw [show (x + y) (Sum.inr 1) = x (Sum.inr 1) + y (Sum.inr 1) from rfl]
|
||
simp only [Fin.isValue, ofReal_add, Fin.mk_one, cons_val_one, head_cons, Fin.zero_eta,
|
||
cons_val_zero]
|
||
ring
|
||
· rw [show (x + y) (Sum.inr 0) = x (Sum.inr 0) + y (Sum.inr 0) from rfl]
|
||
rw [show (x + y) (Sum.inr 1) = x (Sum.inr 1) + y (Sum.inr 1) from rfl]
|
||
simp only [Fin.isValue, ofReal_add, Fin.zero_eta, cons_val_zero, Fin.mk_one, cons_val_one,
|
||
head_fin_const]
|
||
ring
|
||
· rw [show (x + y) (Sum.inl 0) = x (Sum.inl 0) + y (Sum.inl 0) from rfl]
|
||
rw [show (x + y) (Sum.inr 2) = x (Sum.inr 2) + y (Sum.inr 2) from rfl]
|
||
simp only [Fin.isValue, ofReal_add, Fin.mk_one, cons_val_one, head_cons, head_fin_const]
|
||
ring
|
||
map_smul' r x := by
|
||
ext i j : 2
|
||
simp only [toSelfAdjointMatrix'_coe, Fin.isValue, of_apply, cons_val', empty_val',
|
||
cons_val_fin_one, RingHom.id_apply, selfAdjoint.val_smul, smul_apply, real_smul]
|
||
fin_cases i <;> fin_cases j
|
||
· rw [show (r • x) (Sum.inl 0) = r * x (Sum.inl 0) from rfl]
|
||
rw [show (r • x) (Sum.inr 2) = r * x (Sum.inr 2) from rfl]
|
||
simp only [Fin.isValue, ofReal_mul, Fin.zero_eta, cons_val_zero]
|
||
ring
|
||
· rw [show (r • x) (Sum.inr 0) = r * x (Sum.inr 0) from rfl]
|
||
rw [show (r • x) (Sum.inr 1) = r * x (Sum.inr 1) from rfl]
|
||
simp only [Fin.isValue, ofReal_mul, Fin.mk_one, cons_val_one, head_cons, Fin.zero_eta,
|
||
cons_val_zero]
|
||
ring
|
||
· rw [show (r • x) (Sum.inr 0) = r * x (Sum.inr 0) from rfl]
|
||
rw [show (r • x) (Sum.inr 1) = r * x (Sum.inr 1) from rfl]
|
||
simp only [Fin.isValue, ofReal_mul, Fin.zero_eta, cons_val_zero, Fin.mk_one, cons_val_one,
|
||
head_fin_const]
|
||
ring
|
||
· rw [show (r • x) (Sum.inl 0) = r * x (Sum.inl 0) from rfl]
|
||
rw [show (r • x) (Sum.inr 2) = r * x (Sum.inr 2) from rfl]
|
||
simp only [Fin.isValue, ofReal_mul, Fin.mk_one, cons_val_one, head_cons, head_fin_const]
|
||
ring
|
||
|
||
open minkowskiMetric in
|
||
lemma det_eq_ηLin (x : LorentzVector 3) : det (toSelfAdjointMatrix x).1 = ⟪x, x⟫ₘ := by
|
||
simp only [toSelfAdjointMatrix, LinearEquiv.coe_mk, toSelfAdjointMatrix'_coe, Fin.isValue,
|
||
det_fin_two_of, eq_time_minus_inner_prod, LorentzVector.time, LorentzVector.space,
|
||
PiLp.inner_apply, Function.comp_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_three,
|
||
ofReal_sub, ofReal_mul, ofReal_add]
|
||
ring_nf
|
||
simp only [Fin.isValue, I_sq, mul_neg, mul_one]
|
||
ring
|
||
|
||
end SpaceTime
|