92 lines
3.7 KiB
Text
92 lines
3.7 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.SpaceTime.Metric
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import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
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/-!
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# Spacetime as a self-adjoint matrix
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There is a linear equivalence between the vector space of space-time points
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and the vector space of 2×2-complex self-adjoint matrices.
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In this file we define this linear equivalence in `toSelfAdjointMatrix`.
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-/
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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 space-time point. -/
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@[simp]
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def toMatrix (x : spaceTime) : Matrix (Fin 2) (Fin 2) ℂ :=
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!![x 0 + x 3, x 1 - x 2 * I; x 1 + x 2 * I, x 0 - x 3]
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/-- The matrix `x.toMatrix` for `x ∈ spaceTime` is self adjoint. -/
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lemma toMatrix_isSelfAdjoint (x : spaceTime) : IsSelfAdjoint x.toMatrix := 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 [toMatrix, conj_ofReal]
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rfl
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/-- A self-adjoint matrix formed from a space-time point. -/
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@[simps!]
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def toSelfAdjointMatrix' (x : spaceTime) : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) :=
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⟨x.toMatrix, toMatrix_isSelfAdjoint x⟩
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/-- A self-adjoint matrix formed from a space-time point. -/
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@[simp]
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noncomputable def fromSelfAdjointMatrix' (x : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) : spaceTime :=
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![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]
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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 : spaceTime ≃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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simp only [fromSelfAdjointMatrix', one_div, toSelfAdjointMatrix'_coe, of_apply, cons_val',
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cons_val_zero, empty_val', cons_val_fin_one, cons_val_one, head_cons, head_fin_const,
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add_add_sub_cancel, add_re, ofReal_re, mul_re, I_re, mul_zero, ofReal_im, I_im, mul_one,
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sub_self, add_zero, add_im, mul_im, zero_add, add_sub_sub_cancel, half_add_self]
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field_simp [spaceTime]
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ext1 x
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fin_cases x <;> rfl
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right_inv x := by
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simp only [toSelfAdjointMatrix', toMatrix, fromSelfAdjointMatrix', one_div, Fin.isValue, add_re,
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sub_re, cons_val_zero, ofReal_mul, ofReal_inv, ofReal_ofNat, ofReal_add, cons_val_three,
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Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons, head_cons, ofReal_div, ofReal_sub,
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cons_val_one, cons_val_two, 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 <;> simp <;> 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', toMatrix, Fin.isValue, smul_apply, ofReal_mul,
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RingHom.id_apply]
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fin_cases i <;> fin_cases j <;>
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field_simp [fromSelfAdjointMatrix', toMatrix, conj_ofReal, smul_apply]
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<;> ring
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lemma det_eq_ηLin (x : spaceTime) : det (toSelfAdjointMatrix x).1 = ηLin x x := by
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simp [toSelfAdjointMatrix, ηLin_expand]
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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 spaceTime
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