feat: unit extremal vector of the path observables on ℂ^d

PhyslibAlpha.lean

@@ -102,6 +102,7 @@ public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.Uncertainty
 public import PhyslibAlpha.AlgebraicFramework.WStarAlgebra.Basic
 public import PhyslibAlpha.AlgebraicFramework.WStarAlgebra.ConjSpace
 public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.Dynamics.Automorphism
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathExtremalState
 public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.Dynamics.Hamiltonian
 public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.Trace
 public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.TraceClass.Basic

PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathExtremalState.lean

@@ -0,0 +1,95 @@
+/-
+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 Mathlib
+
+/-!
+
+# The extremal state of the path observables
+
+The unit vector `ψ_d ∈ ℂ^d` with coordinates `(-i)^j √(2 / (d + 1)) sin ((j + 1) π / (d + 1))`.
+
+It is called extremal because it is the eigenvector of `i [T_d, P_d]` (the path observables of
+`PathObservables`) for the largest eigenvalue `2 / (d - 1)`. This file only defines `ψ_d` and
+computes its norm: that it is extremal is neither proved nor used here.
+
+-/
+
+@[expose] public section
+
+open Finset
+
+namespace PathObservables
+
+/-- The angle `π / (d + 1)`. -/
+noncomputable def angle (d : ℕ) : ℝ := Real.pi / ((d : ℝ) + 1)
+
+/-- The unnormalised extremal vector, with coordinates `(-i)^j sin ((j + 1) π / (d + 1))`. -/
+noncomputable def sineMode (d : ℕ) : EuclideanSpace ℂ (Fin d) :=
+  WithLp.toLp 2 fun j => (-Complex.I) ^ (j : ℕ) * (Real.sin (((j : ℕ) + 1) * angle d) : ℂ)
+
+/-- The cosines `cos (2π k / n)` sum to zero over a full period. -/
+lemma sum_cos_two_pi_div {n : ℕ} (hn : 1 < n) :
+    ∑ k ∈ range n, Real.cos (2 * Real.pi * k / n) = 0 := by
+  have hn0 : n ≠ 0 := by omega
+  have h := (Complex.isPrimitiveRoot_exp n hn0).geom_sum_eq_zero hn
+  have h' := congrArg Complex.re h
+  rw [Complex.re_sum] at h'
+  simp only [← Complex.exp_nat_mul, Complex.zero_re] at h'
+  rw [← h']
+  refine sum_congr rfl fun k _ => ?_
+  rw [← Complex.exp_ofReal_mul_I_re]
+  congr 2
+  push_cast
+  ring
+
+/-- The squared sines of the multiples of `π / (d + 1)` sum to `(d + 1) / 2`. -/
+lemma sum_sin_sq_angle (d : ℕ) (hd : 1 ≤ d) :
+    ∑ j : Fin d, Real.sin (((j : ℕ) + 1) * angle d) ^ 2 = ((d : ℝ) + 1) / 2 := by
+  have h := sum_cos_two_pi_div (n := d + 1) (by omega)
+  push_cast at h
+  have hsq : ∀ k : ℕ, Real.sin (k * angle d) ^ 2 =
+      (1 - Real.cos (2 * Real.pi * k / ((d : ℝ) + 1))) / 2 := by
+    intro k
+    rw [Real.sin_sq_eq_half_sub, show 2 * (k * angle d) = 2 * Real.pi * k / ((d : ℝ) + 1) by
+      unfold angle; ring]
+    ring
+  have hL : ∑ k ∈ range (d + 1), (1 - Real.cos (2 * Real.pi * k / ((d : ℝ) + 1))) / 2 =
+      ((d : ℝ) + 1) / 2 := by
+    rw [← sum_div, sum_sub_distrib, h]
+    simp
+  have hS : ∑ k ∈ range (d + 1), Real.sin ((k : ℝ) * angle d) ^ 2 = ((d : ℝ) + 1) / 2 := by
+    rw [sum_congr rfl fun k _ => hsq k]
+    exact hL
+  have h0 := sum_range_succ' (fun k => Real.sin ((k : ℝ) * angle d) ^ 2) d
+  rw [hS] at h0
+  rw [Fin.sum_univ_eq_sum_range (fun j => Real.sin (((j : ℕ) + 1) * angle d) ^ 2) d]
+  simpa using h0.symm
+
+/-- The squared norm of the unnormalised extremal vector is `(d + 1) / 2`. -/
+lemma norm_sq_sineMode (d : ℕ) (hd : 1 ≤ d) : ‖sineMode d‖ ^ 2 = ((d : ℝ) + 1) / 2 := by
+  rw [EuclideanSpace.norm_sq_eq, ← sum_sin_sq_angle d hd]
+  refine sum_congr rfl fun j _ => ?_
+  rw [sineMode, WithLp.ofLp_toLp, norm_mul, norm_pow, norm_neg, Complex.norm_I, one_pow, one_mul,
+    Complex.norm_real, Real.norm_eq_abs, sq_abs]
+
+/-- The extremal state: the unit vector `√(2 / (d + 1)) • sineMode d`. By definition it is this
+explicit vector; extremality is not part of the definition. -/
+noncomputable def extremalState (d : ℕ) : EuclideanSpace ℂ (Fin d) :=
+  ((Real.sqrt (2 / ((d : ℝ) + 1)) : ℝ) : ℂ) • sineMode d
+
+/-- The extremal state is a unit vector. -/
+lemma norm_extremalState (d : ℕ) (hd : 1 ≤ d) : ‖extremalState d‖ = 1 := by
+  have hpos : (0 : ℝ) < (d : ℝ) + 1 := by positivity
+  have hs : ‖sineMode d‖ = Real.sqrt (((d : ℝ) + 1) / 2) := by
+    rw [← norm_sq_sineMode d hd, Real.sqrt_sq (norm_nonneg _)]
+  rw [extremalState, norm_smul, hs, Complex.norm_real, Real.norm_of_nonneg (Real.sqrt_nonneg _),
+    ← Real.sqrt_mul (by positivity)]
+  rw [show 2 / ((d : ℝ) + 1) * (((d : ℝ) + 1) / 2) = 1 by field_simp]
+  simp
+
+end PathObservables