feat: Add Pauli-matrices as tensor.
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jstoobysmith
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13 changed files with 989 additions and 154 deletions
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@ -4,7 +4,10 @@ Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.SpaceTime.MinkowskiMetric
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import HepLean.SpaceTime.PauliMatrices.SelfAdjoint
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import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
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import Mathlib.Tactic.Polyrith
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import LLMLean
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/-!
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# Lorentz vector as a self-adjoint matrix
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@ -20,150 +23,79 @@ namespace SpaceTime
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open Matrix
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open MatrixGroups
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open Complex
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/-- A 2×2-complex matrix formed from a Lorentz vector point. -/
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@[simp]
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def toMatrix (x : LorentzVector 3) : Matrix (Fin 2) (Fin 2) ℂ :=
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!![x.time + x.space 2, x.space 0 - x.space 1 * I; x.space 0 + x.space 1 * I, x.time - x.space 2]
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/-- The matrix `x.toMatrix` for `x ∈ spaceTime` is self adjoint. -/
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lemma toMatrix_isSelfAdjoint (x : LorentzVector 3) : IsSelfAdjoint (toMatrix x) := by
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rw [isSelfAdjoint_iff, star_eq_conjTranspose, ← Matrix.ext_iff]
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intro i j
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fin_cases i <;> fin_cases j <;>
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simp only [toMatrix, LorentzVector.time, Fin.isValue, LorentzVector.space, Function.comp_apply,
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Fin.zero_eta, conjTranspose_apply, of_apply, cons_val', cons_val_zero, empty_val',
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cons_val_fin_one, star_add, RCLike.star_def, conj_ofReal, Fin.mk_one, cons_val_one,
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head_fin_const, star_mul', conj_I, mul_neg, head_cons, star_sub, sub_neg_eq_add]
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rfl
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/-- A self-adjoint matrix formed from a Lorentz vector point. -/
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@[simps!]
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def toSelfAdjointMatrix' (x : LorentzVector 3) : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) :=
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⟨toMatrix x, toMatrix_isSelfAdjoint x⟩
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/-- A self-adjoint matrix formed from a Lorentz vector point. -/
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@[simp]
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noncomputable def fromSelfAdjointMatrix' (x : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
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LorentzVector 3 := fun i =>
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match i with
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| Sum.inl 0 => 1/2 * (x.1 0 0 + x.1 1 1).re
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| Sum.inr 0 => (x.1 1 0).re
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| Sum.inr 1 => (x.1 1 0).im
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| Sum.inr 2 => 1/2 * (x.1 0 0 - x.1 1 1).re
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noncomputable section
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/-- The linear equivalence between the vector-space `spaceTime` and self-adjoint
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2×2-complex matrices. -/
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noncomputable def toSelfAdjointMatrix :
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LorentzVector 3 ≃ₗ[ℝ] selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) where
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toFun := toSelfAdjointMatrix'
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invFun := fromSelfAdjointMatrix'
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left_inv x := by
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funext i
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match i with
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| Sum.inl 0 =>
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simp only [fromSelfAdjointMatrix', one_div, toSelfAdjointMatrix', toMatrix,
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LorentzVector.time, Fin.isValue, LorentzVector.space, Function.comp_apply, of_apply,
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cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_one, head_cons,
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head_fin_const, add_add_sub_cancel, add_re, ofReal_re]
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ring_nf
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| Sum.inr 0 =>
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simp [fromSelfAdjointMatrix', toSelfAdjointMatrix', toMatrix, toMatrix_isSelfAdjoint]
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| Sum.inr 1 =>
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simp only [fromSelfAdjointMatrix', toSelfAdjointMatrix', toMatrix, LorentzVector.time,
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Fin.isValue, LorentzVector.space, Function.comp_apply, of_apply, cons_val', cons_val_zero,
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empty_val', cons_val_fin_one, cons_val_one, head_fin_const, add_im, ofReal_im, mul_im,
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ofReal_re, I_im, mul_one, I_re, mul_zero, add_zero, zero_add]
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| Sum.inr 2 =>
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simp only [fromSelfAdjointMatrix', one_div, toSelfAdjointMatrix', toMatrix,
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LorentzVector.time, Fin.isValue, LorentzVector.space, Function.comp_apply, of_apply,
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cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_one, head_cons,
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head_fin_const, add_sub_sub_cancel, add_re, ofReal_re]
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ring
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right_inv x := by
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simp only [toSelfAdjointMatrix', toMatrix, LorentzVector.time, fromSelfAdjointMatrix', one_div,
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Fin.isValue, add_re, ofReal_mul, ofReal_inv, ofReal_ofNat, ofReal_add, LorentzVector.space,
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Function.comp_apply, sub_re, ofReal_sub, re_add_im]
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ext i j
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fin_cases i <;> fin_cases j <;>
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field_simp [fromSelfAdjointMatrix', toMatrix, conj_ofReal]
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· exact conj_eq_iff_re.mp (congrArg (fun M => M 0 0) $ selfAdjoint.mem_iff.mp x.2)
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· have h01 := congrArg (fun M => M 0 1) $ selfAdjoint.mem_iff.mp x.2
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simp only [Fin.isValue, star_apply, RCLike.star_def] at h01
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rw [← h01, RCLike.conj_eq_re_sub_im]
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rfl
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· exact conj_eq_iff_re.mp (congrArg (fun M => M 1 1) $ selfAdjoint.mem_iff.mp x.2)
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map_add' x y := by
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ext i j : 2
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simp only [toSelfAdjointMatrix'_coe, add_apply, ofReal_add, of_apply, cons_val', empty_val',
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cons_val_fin_one, AddSubmonoid.coe_add, AddSubgroup.coe_toAddSubmonoid, Matrix.add_apply]
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fin_cases i <;> fin_cases j
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· rw [show (x + y) (Sum.inl 0) = x (Sum.inl 0) + y (Sum.inl 0) from rfl]
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rw [show (x + y) (Sum.inr 2) = x (Sum.inr 2) + y (Sum.inr 2) from rfl]
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simp only [Fin.isValue, ofReal_add, Fin.zero_eta, cons_val_zero]
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simp only [Fin.isValue, toSelfAdjointMatrix', toMatrix, LorentzVector.time,
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LorentzVector.space, Function.comp_apply, AddMemClass.mk_add_mk, of_add_of, add_cons,
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head_cons, tail_cons, empty_add_empty, of_apply, cons_val', cons_val_zero, empty_val',
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cons_val_fin_one]
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ring
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· rw [show (x + y) (Sum.inr 0) = x (Sum.inr 0) + y (Sum.inr 0) from rfl]
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rw [show (x + y) (Sum.inr 1) = x (Sum.inr 1) + y (Sum.inr 1) from rfl]
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simp only [Fin.isValue, ofReal_add, Fin.mk_one, cons_val_one, head_cons, Fin.zero_eta,
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cons_val_zero]
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simp only [Fin.isValue, toSelfAdjointMatrix', toMatrix, LorentzVector.time,
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LorentzVector.space, Function.comp_apply, AddMemClass.mk_add_mk, of_add_of, add_cons,
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head_cons, tail_cons, empty_add_empty, of_apply, cons_val', cons_val_one, empty_val',
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cons_val_fin_one, cons_val_zero]
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ring
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· rw [show (x + y) (Sum.inr 0) = x (Sum.inr 0) + y (Sum.inr 0) from rfl]
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rw [show (x + y) (Sum.inr 1) = x (Sum.inr 1) + y (Sum.inr 1) from rfl]
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simp only [Fin.isValue, ofReal_add, Fin.zero_eta, cons_val_zero, Fin.mk_one, cons_val_one,
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head_fin_const]
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simp only [Fin.isValue, toSelfAdjointMatrix', toMatrix, LorentzVector.time,
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LorentzVector.space, Function.comp_apply, AddMemClass.mk_add_mk, of_add_of, add_cons,
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head_cons, tail_cons, empty_add_empty, of_apply, cons_val', cons_val_zero, empty_val',
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cons_val_fin_one, cons_val_one, head_fin_const]
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ring
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· rw [show (x + y) (Sum.inl 0) = x (Sum.inl 0) + y (Sum.inl 0) from rfl]
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rw [show (x + y) (Sum.inr 2) = x (Sum.inr 2) + y (Sum.inr 2) from rfl]
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simp only [Fin.isValue, ofReal_add, Fin.mk_one, cons_val_one, head_cons, head_fin_const]
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simp only [Fin.isValue, toSelfAdjointMatrix', toMatrix, LorentzVector.time,
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LorentzVector.space, Function.comp_apply, AddMemClass.mk_add_mk, of_add_of, add_cons,
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head_cons, tail_cons, empty_add_empty, of_apply, cons_val', cons_val_one, empty_val',
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cons_val_fin_one, head_fin_const]
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ring
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map_smul' r x := by
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ext i j : 2
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simp only [toSelfAdjointMatrix'_coe, Fin.isValue, of_apply, cons_val', empty_val',
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cons_val_fin_one, RingHom.id_apply, selfAdjoint.val_smul, smul_apply, real_smul]
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fin_cases i <;> fin_cases j
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· rw [show (r • x) (Sum.inl 0) = r * x (Sum.inl 0) from rfl]
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rw [show (r • x) (Sum.inr 2) = r * x (Sum.inr 2) from rfl]
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simp only [Fin.isValue, ofReal_mul, Fin.zero_eta, cons_val_zero]
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ring
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· rw [show (r • x) (Sum.inr 0) = r * x (Sum.inr 0) from rfl]
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rw [show (r • x) (Sum.inr 1) = r * x (Sum.inr 1) from rfl]
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simp only [Fin.isValue, ofReal_mul, Fin.mk_one, cons_val_one, head_cons, Fin.zero_eta,
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cons_val_zero]
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ring
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· rw [show (r • x) (Sum.inr 0) = r * x (Sum.inr 0) from rfl]
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rw [show (r • x) (Sum.inr 1) = r * x (Sum.inr 1) from rfl]
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simp only [Fin.isValue, ofReal_mul, Fin.zero_eta, cons_val_zero, Fin.mk_one, cons_val_one,
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head_fin_const]
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ring
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· rw [show (r • x) (Sum.inl 0) = r * x (Sum.inl 0) from rfl]
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rw [show (r • x) (Sum.inr 2) = r * x (Sum.inr 2) from rfl]
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simp only [Fin.isValue, ofReal_mul, Fin.mk_one, cons_val_one, head_cons, head_fin_const]
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ring
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`2×2`-complex matrices. -/
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def toSelfAdjointMatrix : LorentzVector 3 ≃ₗ[ℝ] selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) :=
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(Finsupp.linearEquivFunOnFinite ℝ ℝ (Fin 1 ⊕ Fin 3)).symm ≪≫ₗ PauliMatrix.σSAL.repr.symm
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lemma toSelfAdjointMatrix_apply (x : LorentzVector 3) : toSelfAdjointMatrix x =
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x (Sum.inl 0) • ⟨PauliMatrix.σ0, PauliMatrix.σ0_selfAdjoint⟩
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- x (Sum.inr 0) • ⟨PauliMatrix.σ1, PauliMatrix.σ1_selfAdjoint⟩
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- x (Sum.inr 1) • ⟨PauliMatrix.σ2, PauliMatrix.σ2_selfAdjoint⟩
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- x (Sum.inr 2) • ⟨PauliMatrix.σ3, PauliMatrix.σ3_selfAdjoint⟩ := by
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simp only [toSelfAdjointMatrix, PauliMatrix.σSAL, LinearEquiv.trans_apply, Basis.repr_symm_apply,
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Basis.coe_mk, Fin.isValue]
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rw [Finsupp.linearCombination_apply_of_mem_supported ℝ (s := Finset.univ)]
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· change (∑ i : Fin 1 ⊕ Fin 3, x i • PauliMatrix.σSAL' i) = _
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simp [Fin.sum_univ_three, PauliMatrix.σSAL']
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apply Subtype.ext
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simp only [Fin.isValue, AddSubgroup.coe_add, selfAdjoint.val_smul, smul_neg,
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AddSubgroupClass.coe_sub]
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simp only [neg_add, add_assoc, sub_eq_add_neg]
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· simp_all only [Finset.coe_univ, Finsupp.supported_univ, Submodule.mem_top]
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lemma toSelfAdjointMatrix_apply_coe (x : LorentzVector 3) : (toSelfAdjointMatrix x).1 =
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x (Sum.inl 0) • PauliMatrix.σ0
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- x (Sum.inr 0) • PauliMatrix.σ1
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- x (Sum.inr 1) • PauliMatrix.σ2
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- x (Sum.inr 2) • PauliMatrix.σ3 := by
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rw [toSelfAdjointMatrix_apply]
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simp only [Fin.isValue, AddSubgroupClass.coe_sub, selfAdjoint.val_smul]
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lemma toSelfAdjointMatrix_stdBasis (i : Fin 1 ⊕ Fin 3) :
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toSelfAdjointMatrix (LorentzVector.stdBasis i) = PauliMatrix.σSAL i := by
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rw [toSelfAdjointMatrix_apply]
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match i with
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| Sum.inl 0 =>
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simp [LorentzVector.stdBasis]
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erw [Pi.basisFun_apply]
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simp [PauliMatrix.σSAL, PauliMatrix.σSAL' ]
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| Sum.inr 0 =>
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simp [LorentzVector.stdBasis]
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erw [Pi.basisFun_apply]
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simp [PauliMatrix.σSAL, PauliMatrix.σSAL' ]
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refine Eq.symm (PauliMatrix.selfAdjoint_ext rfl rfl rfl rfl)
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| Sum.inr 1 =>
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simp [LorentzVector.stdBasis]
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erw [Pi.basisFun_apply]
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simp [PauliMatrix.σSAL, PauliMatrix.σSAL' ]
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refine Eq.symm (PauliMatrix.selfAdjoint_ext rfl rfl rfl rfl)
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| Sum.inr 2 =>
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simp [LorentzVector.stdBasis]
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erw [Pi.basisFun_apply]
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simp [PauliMatrix.σSAL, PauliMatrix.σSAL' ]
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refine Eq.symm (PauliMatrix.selfAdjoint_ext rfl rfl rfl rfl)
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@[simp]
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lemma toSelfAdjointMatrix_symm_basis (i : Fin 1 ⊕ Fin 3) :
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toSelfAdjointMatrix.symm (PauliMatrix.σSAL i) = (LorentzVector.stdBasis i) := by
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refine (LinearEquiv.symm_apply_eq toSelfAdjointMatrix).mpr ?_
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rw [toSelfAdjointMatrix_stdBasis]
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open minkowskiMetric in
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lemma det_eq_ηLin (x : LorentzVector 3) : det (toSelfAdjointMatrix x).1 = ⟪x, x⟫ₘ := by
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simp only [toSelfAdjointMatrix, LinearEquiv.coe_mk, toSelfAdjointMatrix'_coe, Fin.isValue,
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det_fin_two_of, eq_time_minus_inner_prod, LorentzVector.time, LorentzVector.space,
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rw [toSelfAdjointMatrix_apply_coe]
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simp only [Fin.isValue, eq_time_minus_inner_prod, LorentzVector.time, LorentzVector.space,
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PiLp.inner_apply, Function.comp_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_three,
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ofReal_sub, ofReal_mul, ofReal_add]
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simp only [Fin.isValue, PauliMatrix.σ0, smul_of, smul_cons, real_smul, mul_one, smul_zero,
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smul_empty, PauliMatrix.σ1, of_sub_of, sub_cons, head_cons, sub_zero, tail_cons, zero_sub,
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sub_self, zero_empty, PauliMatrix.σ2, smul_neg, sub_neg_eq_add, PauliMatrix.σ3, det_fin_two_of]
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ring_nf
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simp only [Fin.isValue, I_sq, mul_neg, mul_one]
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ring
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end
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end SpaceTime
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