feat: PhyslibAlpha/QuantumMechanics: operators on one coordinate of a product of finite targets

PhyslibAlpha.lean

@@ -285,6 +285,7 @@ public import PhyslibAlpha.ProbabilisticTheory.Weight.Extension
 public import PhyslibAlpha.QuantumMechanics.HarmonicOscillator.LadderSystem
 public import PhyslibAlpha.QuantumMechanics.HarmonicOscillator.Vacuum
 public import PhyslibAlpha.QuantumMechanics.HilbertSpaces.FiniteTarget.Operators
+public import PhyslibAlpha.QuantumMechanics.HilbertSpaces.FiniteTarget.Product
 public import PhyslibAlpha.QuantumMechanics.QuantumHarmonicOscillator
 public import PhyslibAlpha.QuantumMechanics.StinespringDilation
 public import PhyslibAlpha.Relativity.General.Schwarzschild.IncompressibleSphere

PhyslibAlpha/QuantumMechanics/HilbertSpaces/FiniteTarget/Product.lean

@@ -0,0 +1,176 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernandez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernandez
+-/
+module
+
+public import Physlib.QuantumMechanics.HilbertSpaces.FiniteTarget.Basic
+/-!
+
+# Operators on one coordinate of a product of finite targets
+
+## i. Overview
+
+A system whose configurations are pairs `(a, b)` has the Hilbert space `𝓗[α × β]`. An operator
+`A` on `𝓗[α]` acts on the first coordinate alone, `(A Ψ) (a, b) = ∑ a', ⟨a|A|a'⟩ Ψ (a', b)`,
+and leaves the second in place; an operator on `𝓗[β]` acts on the second coordinate in the same
+way. Operators on different coordinates commute, acting twice on one coordinate composes the
+operators, and hermitian operators stay hermitian.
+
+## ii. Key results
+
+- `onFst`, `onSnd` : an operator acting on one coordinate of `𝓗[α × β]`.
+- `onFst_comp`, `onSnd_comp` : acting twice on the same coordinate.
+- `onFst_comp_onSnd` : operators on different coordinates commute.
+- `onFst_isSymmetric`, `onSnd_isSymmetric` : hermitian operators stay hermitian.
+
+## iii. Table of contents
+
+- A. Amplitudes
+- B. Operators on one coordinate
+- C. Composition
+- D. Hermiticity
+
+-/
+
+@[expose] public section
+
+namespace QuantumMechanics
+
+namespace FiniteHilbertSpace
+
+open InnerProductSpace
+
+variable {α β : Type*} [Fintype α] [DecidableEq α] [Fintype β] [DecidableEq β]
+
+/-!
+
+## A. Amplitudes
+
+-/
+
+/-- The amplitude `⟨a|A|a'⟩` of an operator is the `a` component of `A |a'⟩`. -/
+lemma val_apply_basisFun (A : 𝓗[α] →ₗ[ℂ] 𝓗[α]) (a a' : α) :
+    (A (basisFun α a')).val a = ⟪basisFun α a, A (basisFun α a')⟫_ℂ := by
+  rw [inner_eq_val, basisFun_apply a, EuclideanSpace.inner_single_left, map_one, one_mul]
+
+/-- An operator in components: `(A ψ) a = ∑ a', ⟨a|A|a'⟩ ψ a'`. -/
+lemma val_apply (A : 𝓗[α] →ₗ[ℂ] 𝓗[α]) (ψ : 𝓗[α]) (a : α) :
+    (A ψ).val a = ∑ a', (A (basisFun α a')).val a * ψ.val a' := by
+  conv_lhs => rw [← (basisFun α).sum_repr ψ]
+  simp only [map_sum, map_smul]
+  change (linearEquivEuclidean (∑ a', _)) a = _
+  rw [map_sum]
+  simp only [WithLp.ofLp_sum, Finset.sum_apply, map_smul, OrthonormalBasis.repr_apply_apply,
+    inner_eq_val, basisFun_apply, EuclideanSpace.inner_single_left, map_one, one_mul]
+  exact Finset.sum_congr rfl fun a' _ => by rw [mul_comm]; rfl
+
+/-!
+
+## B. Operators on one coordinate
+
+-/
+
+/-- The operator `A` acting on the first coordinate: `(A Ψ) (a, b) = ∑ a', ⟨a|A|a'⟩ Ψ (a', b)`. -/
+noncomputable def onFst (A : 𝓗[α] →ₗ[ℂ] 𝓗[α]) : 𝓗[α × β] →ₗ[ℂ] 𝓗[α × β] where
+  toFun Ψ := ⟨WithLp.toLp 2 fun p => ∑ a, (A (basisFun α a)).val p.1 * Ψ.val (a, p.2)⟩
+  map_add' Ψ Φ := by
+    ext p
+    simp [mul_add, Finset.sum_add_distrib]
+  map_smul' c Ψ := by
+    ext p
+    simp [Finset.mul_sum, mul_left_comm]
+
+/-- The operator `B` acting on the second coordinate: `(B Ψ) (a, b) = ∑ b', ⟨b|B|b'⟩ Ψ (a, b')`. -/
+noncomputable def onSnd (B : 𝓗[β] →ₗ[ℂ] 𝓗[β]) : 𝓗[α × β] →ₗ[ℂ] 𝓗[α × β] where
+  toFun Ψ := ⟨WithLp.toLp 2 fun p => ∑ b, (B (basisFun β b)).val p.2 * Ψ.val (p.1, b)⟩
+  map_add' Ψ Φ := by
+    ext p
+    simp [mul_add, Finset.sum_add_distrib]
+  map_smul' c Ψ := by
+    ext p
+    simp [Finset.mul_sum, mul_left_comm]
+
+lemma onFst_val (A : 𝓗[α] →ₗ[ℂ] 𝓗[α]) (Ψ : 𝓗[α × β]) (p : α × β) :
+    (onFst A Ψ).val p = ∑ a, (A (basisFun α a)).val p.1 * Ψ.val (a, p.2) := rfl
+
+lemma onSnd_val (B : 𝓗[β] →ₗ[ℂ] 𝓗[β]) (Ψ : 𝓗[α × β]) (p : α × β) :
+    (onSnd B Ψ).val p = ∑ b, (B (basisFun β b)).val p.2 * Ψ.val (p.1, b) := rfl
+
+/-!
+
+## C. Composition
+
+-/
+
+/-- Acting twice on the first coordinate composes the operators. -/
+lemma onFst_comp (A A' : 𝓗[α] →ₗ[ℂ] 𝓗[α]) :
+    onFst (β := β) (A ∘ₗ A') = onFst A ∘ₗ onFst A' := by
+  ext Ψ p
+  simp only [LinearMap.comp_apply, onFst_val]
+  simp only [val_apply A (A' _), Finset.sum_mul, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  exact Finset.sum_congr rfl fun _ _ => Finset.sum_congr rfl fun _ _ => by ring
+
+/-- Acting twice on the second coordinate composes the operators. -/
+lemma onSnd_comp (B B' : 𝓗[β] →ₗ[ℂ] 𝓗[β]) :
+    onSnd (α := α) (B ∘ₗ B') = onSnd B ∘ₗ onSnd B' := by
+  ext Ψ p
+  simp only [LinearMap.comp_apply, onSnd_val]
+  simp only [val_apply B (B' _), Finset.sum_mul, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  exact Finset.sum_congr rfl fun _ _ => Finset.sum_congr rfl fun _ _ => by ring
+
+/-- Operators on different coordinates commute. -/
+lemma onFst_comp_onSnd (A : 𝓗[α] →ₗ[ℂ] 𝓗[α]) (B : 𝓗[β] →ₗ[ℂ] 𝓗[β]) :
+    onFst A ∘ₗ onSnd B = onSnd B ∘ₗ onFst A := by
+  ext Ψ p
+  simp only [LinearMap.comp_apply, onFst_val, onSnd_val, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  exact Finset.sum_congr rfl fun _ _ => Finset.sum_congr rfl fun _ _ => by ring
+
+/-!
+
+## D. Hermiticity
+
+-/
+
+/-- The inner product of `𝓗[α × β]` in components. -/
+lemma inner_eq_sum (Ψ Φ : 𝓗[α × β]) :
+    ⟪Ψ, Φ⟫_ℂ = ∑ p, (starRingEnd ℂ) (Ψ.val p) * Φ.val p := by
+  rw [inner_eq_val, PiLp.inner_apply]
+  exact Finset.sum_congr rfl fun p _ => by rw [RCLike.inner_apply, mul_comm]
+
+/-- Hermitian amplitudes: `conj ⟨a'|A|a⟩ = ⟨a|A|a'⟩`. -/
+lemma conj_val_apply_basisFun {A : 𝓗[α] →ₗ[ℂ] 𝓗[α]} (hA : A.IsSymmetric) (a a' : α) :
+    (starRingEnd ℂ) ((A (basisFun α a)).val a') = (A (basisFun α a')).val a := by
+  rw [val_apply_basisFun, val_apply_basisFun, inner_conj_symm, hA]
+
+/-- A hermitian operator on the first coordinate is hermitian. -/
+lemma onFst_isSymmetric {A : 𝓗[α] →ₗ[ℂ] 𝓗[α]} (hA : A.IsSymmetric) :
+    (onFst (β := β) A).IsSymmetric := fun Ψ Φ => by
+  simp only [inner_eq_sum, onFst_val, map_sum, map_mul, Finset.sum_mul, Finset.mul_sum,
+    Fintype.sum_prod_type]
+  conv_lhs => rw [Finset.sum_comm]
+  conv_rhs => rw [Finset.sum_comm]
+  refine Finset.sum_congr rfl fun b _ => ?_
+  conv_lhs => rw [Finset.sum_comm]
+  exact Finset.sum_congr rfl fun a _ => Finset.sum_congr rfl fun a' _ => by
+    rw [conj_val_apply_basisFun hA]
+    ring
+
+/-- A hermitian operator on the second coordinate is hermitian. -/
+lemma onSnd_isSymmetric {B : 𝓗[β] →ₗ[ℂ] 𝓗[β]} (hB : B.IsSymmetric) :
+    (onSnd (α := α) B).IsSymmetric := fun Ψ Φ => by
+  simp only [inner_eq_sum, onSnd_val, map_sum, map_mul, Finset.sum_mul, Finset.mul_sum,
+    Fintype.sum_prod_type]
+  refine Finset.sum_congr rfl fun a _ => ?_
+  conv_lhs => rw [Finset.sum_comm]
+  exact Finset.sum_congr rfl fun b _ => Finset.sum_congr rfl fun b' _ => by
+    rw [conj_val_apply_basisFun hB]
+    ring
+
+end FiniteHilbertSpace
+
+end QuantumMechanics