PhysLean/HepLean/StandardModel/HiggsBoson/PointwiseInnerProd.lean

/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license.
Authors: Joseph Tooby-Smith
-/
import HepLean.StandardModel.HiggsBoson.Basic
/-!
# The pointwise inner product

We define the pointwise inner product of two Higgs fields.

The notation for the inner product is `⟪φ, ψ⟫_H`.

We also define the pointwise norm squared of a Higgs field `∥φ∥_H ^ 2`.

-/

noncomputable section

namespace StandardModel

namespace HiggsField

open Manifold
open Matrix
open Complex
open ComplexConjugate
open SpaceTime

/-!

## The pointwise inner product

-/

/-- Given two `HiggsField`, the map `SpaceTime → ℂ` obtained by taking their pointwise
  inner product. -/
def innerProd (φ1 φ2 : HiggsField) : SpaceTime → ℂ := fun x => ⟪φ1 x, φ2 x⟫_ℂ

/-- Notation for the inner product of two Higgs fields. -/
scoped[StandardModel.HiggsField] notation "⟪" φ1 "," φ2 "⟫_H" => innerProd φ1 φ2

/-!

## Properties of the inner product

-/

@[simp]
lemma innerProd_left_zero (φ : HiggsField) : ⟪0, φ⟫_H = 0 := by
  funext x
  simp [innerProd]

@[simp]
lemma innerProd_right_zero (φ : HiggsField) : ⟪φ, 0⟫_H = 0 := by
  funext x
  simp [innerProd]
example (x : ℝ): RCLike.ofReal x = ofReal' x := by
  rfl
lemma innerProd_expand (φ1 φ2 : HiggsField) :
    ⟪φ1, φ2⟫_H = fun x => equivRealProdCLM.symm (((φ1 x 0).re * (φ2 x 0).re
    + (φ1 x 1).re * (φ2 x 1).re+ (φ1 x 0).im * (φ2 x 0).im + (φ1 x 1).im * (φ2 x 1).im),
    ((φ1 x 0).re * (φ2 x 0).im + (φ1 x 1).re * (φ2 x 1).im
    - (φ1 x 0).im * (φ2 x 0).re - (φ1 x 1).im * (φ2 x 1).re)) := by
  funext x
  simp only [innerProd, PiLp.inner_apply, RCLike.inner_apply, Fin.sum_univ_two,
    equivRealProdCLM_symm_apply, ofReal_add, ofReal_mul, ofReal_sub]
  rw [RCLike.conj_eq_re_sub_im, RCLike.conj_eq_re_sub_im]
  nth_rewrite 1 [← RCLike.re_add_im (φ2 x 0)]
  nth_rewrite 1 [← RCLike.re_add_im (φ2 x 1)]
  ring_nf
  repeat rw [show RCLike.ofReal _ = ofReal' _ by rfl]
  simp only [Algebra.id.map_eq_id, RCLike.re_to_complex, RingHom.id_apply, RCLike.I_to_complex,
    RCLike.im_to_complex, I_sq, mul_neg, mul_one, neg_mul, sub_neg_eq_add, one_mul]
  ring

lemma smooth_innerProd (φ1 φ2 : HiggsField) : Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℂ) ⟪φ1, φ2⟫_H := by
  rw [innerProd_expand]
  exact (ContinuousLinearMap.smooth (equivRealProdCLM.symm : ℝ × ℝ →L[ℝ] ℂ)).comp $
    (((((φ1.apply_re_smooth 0).smul (φ2.apply_re_smooth 0)).add
    ((φ1.apply_re_smooth 1).smul (φ2.apply_re_smooth 1))).add
    ((φ1.apply_im_smooth 0).smul (φ2.apply_im_smooth 0))).add
    ((φ1.apply_im_smooth 1).smul (φ2.apply_im_smooth 1))).prod_mk_space $
    ((((φ1.apply_re_smooth 0).smul (φ2.apply_im_smooth 0)).add
    ((φ1.apply_re_smooth 1).smul (φ2.apply_im_smooth 1))).sub
    ((φ1.apply_im_smooth 0).smul (φ2.apply_re_smooth 0))).sub
    ((φ1.apply_im_smooth 1).smul (φ2.apply_re_smooth 1))

/-!

## The pointwise norm squared

-/

/-- Given a `HiggsField`, the map `SpaceTime → ℝ` obtained by taking the square norm of the
  pointwise Higgs vector. -/
@[simp]
def normSq (φ : HiggsField) : SpaceTime → ℝ := fun x => ‖φ x‖ ^ 2

/-- Notation for the norm squared of a Higgs field. -/
scoped[StandardModel.HiggsField] notation "‖" φ1 "‖_H ^ 2" => normSq φ1

/-!

## Relation between inner prod and norm squared

-/

lemma innerProd_self_eq_normSq (φ : HiggsField) (x : SpaceTime) :
    ⟪φ, φ⟫_H x = ‖φ‖_H ^ 2 x := by
  erw [normSq, @PiLp.norm_sq_eq_of_L2, Fin.sum_univ_two]
  simp only [innerProd, PiLp.inner_apply, RCLike.inner_apply, conj_mul', norm_eq_abs,
    Fin.sum_univ_two, ofReal_add, ofReal_pow]

lemma normSq_eq_innerProd_self (φ : HiggsField) (x : SpaceTime) :
    ‖φ x‖ ^ 2 = (⟪φ, φ⟫_H x).re := by
  rw [innerProd_self_eq_normSq]
  rfl

/-!

# Properties of the norm squared

-/

lemma toHiggsVec_norm (φ : HiggsField) (x : SpaceTime) :
    ‖φ x‖ = ‖φ.toHiggsVec x‖ := rfl

lemma normSq_expand (φ : HiggsField) :
    φ.normSq = fun x => (conj (φ x 0) * (φ x 0) + conj (φ x 1) * (φ x 1)).re := by
  funext x
  simp [normSq, add_re, mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add, @norm_sq_eq_inner ℂ]

lemma normSq_nonneg (φ : HiggsField) (x : SpaceTime) : 0 ≤ φ.normSq x := by
  simp [normSq, ge_iff_le, norm_nonneg, pow_nonneg]

lemma normSq_zero (φ : HiggsField) (x : SpaceTime) : φ.normSq x = 0 ↔ φ x = 0 := by
  simp [normSq, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff, norm_eq_zero]

lemma normSq_smooth (φ : HiggsField) : Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℝ) φ.normSq := by
  rw [normSq_expand]
  refine Smooth.add ?_ ?_
  simp only [mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
  exact ((φ.apply_re_smooth 0).smul (φ.apply_re_smooth 0)).add $
    (φ.apply_im_smooth 0).smul (φ.apply_im_smooth 0)
  simp only [mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
  exact ((φ.apply_re_smooth 1).smul (φ.apply_re_smooth 1)).add $
    (φ.apply_im_smooth 1).smul (φ.apply_im_smooth 1)

end HiggsField

end StandardModel
end