feat(LaplaceRungeLenzVector): prove angularMomentum_commutation_lrl

[pitmonticone]

This PR adds proofs autoformalised by @Aristotle-Harmonic.

Co-authored-by: Aristotle (Harmonic) aristotle-harmonic@harmonic.fun

[pitmonticone]

Submitted your review comments to @Aristotle-Harmonic and it should have resolved them all.

[jstoobysmith]

The build here emits a bunch of warnings of the form

Hint: Omit it from the simp argument list

It is also possible to combine some of the simps without breaking things

[adomasbaliuka]

how about this (I couldn't further combine the simps without breaking it)

Suggested change

	/-- `⁅𝐋ᵢⱼ, 𝐀(ε)ₖ⁆ = iℏ(δᵢₖ𝐀(ε)ⱼ - δⱼₖ𝐀(ε)ᵢ)` -/
	@[sorryful]
	lemma angularMomentum_commutation_lrl (ε : ℝˣ) (i j k : Fin H.d) :
	⁅𝐋[i,j], H.lrlOperator ε k⁆ = (Complex.I * ℏ * δ[i,k]) • H.lrlOperator ε j
	- (Complex.I * ℏ * δ[j,k]) • H.lrlOperator ε i := by
	sorry
	simp only [lrlOperator_eq' H ε]
	simp only [lie_sub, lie_add, lie_sum, lie_smul, lie_leibniz]
	simp only [angularMomentum_commutation_angularMomentum,
	angularMomentum_commutation_momentum,
	angularMomentum_commutation_radiusRegPow,
	angularMomentum_commutation_position]
	dsimp only [kroneckerDelta]
	simp only [comp_sub, comp_smul, zero_comp, add_zero,
	smul_sub, smul_add, smul_smul]
	simp only [Finset.sum_add_distrib, Finset.sum_sub_distrib]
	simp only [Nat.cast_ite, Nat.cast_one, CharP.cast_eq_zero, mul_ite, mul_one, mul_zero,
	ite_smul, zero_smul, smul_ite, smul_zero,
	Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte]
	rw [angularMomentumOperator_antisymm k i, angularMomentumOperator_antisymm k j]
	simp only [neg_comp, smul_neg]
	have ite_comp_right : ∀ (p : Prop) [Decidable p]
	(A B : 𝓢(Space H.d, ℂ) →L[ℂ] 𝓢(Space H.d, ℂ)),
	(if p then A else 0).comp B = if p then A.comp B else 0 :=
	fun p _ A B ↦ by split_ifs <;> simp
	simp only [ite_comp_right,
	sub_comp, add_comp, smul_comp,
	Finset.sum_add_distrib, Finset.sum_sub_distrib,
	Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte]
	rw [angularMomentumOperator_antisymm i k, angularMomentumOperator_antisymm j k]
	simp only [neg_comp, smul_neg, neg_neg]
	split_ifs with hik hjk <;>
	simp_all only [smul_zero, sub_zero, zero_sub, zero_smul, zero_mul,
	Finset.sum_const_zero, add_zero, sub_self, neg_zero, neg_neg,
	← Finset.smul_sum, angularMomentumOperator_eq_zero, zero_comp] <;>
	(try simp only [smul_comm (H.m * H.k) (Complex.I * ↑↑ℏ)]) <;>
	(try conv_lhs => rw [show
	2⁻¹ * Complex.I * ↑↑ℏ * (↑H.d - 1) * (Complex.I * ↑↑ℏ) =
	Complex.I * ↑↑ℏ * (2⁻¹ * Complex.I * ↑↑ℏ * (↑H.d - 1)) from by ring]) <;>
	abel
	/-- `⁅𝐋ᵢⱼ, 𝐀(ε)ₖ⁆ = iℏ(δᵢₖ𝐀(ε)ⱼ - δⱼₖ𝐀(ε)ᵢ)` -/
	lemma angularMomentum_commutation_lrl (ε : ℝˣ) (i j k : Fin H.d) :
	⁅𝐋[i,j], H.lrlOperator ε k⁆ = (Complex.I * ℏ * δ[i,k]) • H.lrlOperator ε j
	- (Complex.I * ℏ * δ[j,k]) • H.lrlOperator ε i := by
	simp only [lrlOperator_eq, lie_sub, lie_add, lie_leibniz, angularMomentum_commutation_momentumSqr,
	angularMomentum_commutation_position, angularMomentum_commutation_momentum, lie_sum, lie_smul,
	angularMomentum_commutation_radiusRegPow]
	simp only [comp_zero, smul_sub, sub_comp, smul_comp, zero_add, comp_sub, comp_smul,
	Finset.sum_add_distrib, Finset.sum_sub_distrib, ← Finset.smul_sum, sum_smul, zero_comp,
	add_zero]
	simp only [sub_add_sub_cancel, sub_self, zero_comp, add_zero, ← Nat.cast_smul_eq_nsmul ℂ, smul_smul]
	ext f x
	simp only [ContinuousLinearMap.sub_apply, ContinuousLinearMap.add_apply,
	ContinuousLinearMap.smul_apply, SchwartzMap.sub_apply, SchwartzMap.add_apply,
	SchwartzMap.smul_apply, smul_eq_mul, real_smul, ofReal_mul]
	ring

[jstoobysmith]

This looks good to me! Will give @pitmonticone chance to comment here. This PR currently has a merge conflict (since ./PhysLean got renamed to ./PhysLib), which will also need addressing (maybe easier to open a new PR).

[adomasbaliuka]

Is anyone working on getting the main results for hydrogen? E.g. Spectrum?

Apparently, there are some derivations of the hydrogen spectrum using the Laplace-Runge-Lenz vector (I've never seen such a derivation). Is that the planned approach and the reason why LRL is being developed here?

[jstoobysmith]

@adomasbaliuka this work is been done by @gloges who might be able to provide you more details. Maybe worth taking the conversation to the Zulip rather than here though