152 lines
5 KiB
Text
152 lines
5 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 as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.StandardModel.HiggsBoson.Basic
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/-!
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# The pointwise inner product
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We define the pointwise inner product of two Higgs fields.
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The notation for the inner product is `⟪φ, ψ⟫_H`.
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We also define the pointwise norm squared of a Higgs field `∥φ∥_H ^ 2`.
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-/
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noncomputable section
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namespace StandardModel
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namespace HiggsField
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open Manifold
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open Matrix
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open Complex
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open ComplexConjugate
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open SpaceTime
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/-!
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## The pointwise inner product
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-/
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/-- Given two `HiggsField`, the map `SpaceTime → ℂ` obtained by taking their pointwise
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inner product. -/
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def innerProd (φ1 φ2 : HiggsField) : SpaceTime → ℂ := fun x => ⟪φ1 x, φ2 x⟫_ℂ
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/-- Notation for the inner product of two Higgs fields. -/
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scoped[StandardModel.HiggsField] notation "⟪" φ1 "," φ2 "⟫_H" => innerProd φ1 φ2
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/-!
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## Properties of the inner product
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-/
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@[simp]
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lemma innerProd_left_zero (φ : HiggsField) : ⟪0, φ⟫_H = 0 := by
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funext x
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simp [innerProd]
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@[simp]
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lemma innerProd_right_zero (φ : HiggsField) : ⟪φ, 0⟫_H = 0 := by
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funext x
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simp [innerProd]
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lemma innerProd_expand (φ1 φ2 : HiggsField) :
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⟪φ1, φ2⟫_H = fun x => equivRealProdCLM.symm (((φ1 x 0).re * (φ2 x 0).re
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+ (φ1 x 1).re * (φ2 x 1).re+ (φ1 x 0).im * (φ2 x 0).im + (φ1 x 1).im * (φ2 x 1).im),
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((φ1 x 0).re * (φ2 x 0).im + (φ1 x 1).re * (φ2 x 1).im
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- (φ1 x 0).im * (φ2 x 0).re - (φ1 x 1).im * (φ2 x 1).re)) := by
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funext x
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simp only [innerProd, PiLp.inner_apply, RCLike.inner_apply, Fin.sum_univ_two,
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equivRealProdCLM_symm_apply, ofReal_add, ofReal_mul, ofReal_sub]
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rw [RCLike.conj_eq_re_sub_im, RCLike.conj_eq_re_sub_im]
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nth_rewrite 1 [← RCLike.re_add_im (φ2 x 0)]
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nth_rewrite 1 [← RCLike.re_add_im (φ2 x 1)]
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ring_nf
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repeat rw [show RCLike.ofReal _ = ofReal' _ by rfl]
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simp only [Algebra.id.map_eq_id, RCLike.re_to_complex, RingHom.id_apply, RCLike.I_to_complex,
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RCLike.im_to_complex, I_sq, mul_neg, mul_one, neg_mul, sub_neg_eq_add, one_mul]
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ring
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lemma smooth_innerProd (φ1 φ2 : HiggsField) : Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℂ) ⟪φ1, φ2⟫_H := by
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rw [innerProd_expand]
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exact (ContinuousLinearMap.smooth (equivRealProdCLM.symm : ℝ × ℝ →L[ℝ] ℂ)).comp $
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(((((φ1.apply_re_smooth 0).smul (φ2.apply_re_smooth 0)).add
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((φ1.apply_re_smooth 1).smul (φ2.apply_re_smooth 1))).add
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((φ1.apply_im_smooth 0).smul (φ2.apply_im_smooth 0))).add
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((φ1.apply_im_smooth 1).smul (φ2.apply_im_smooth 1))).prod_mk_space $
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((((φ1.apply_re_smooth 0).smul (φ2.apply_im_smooth 0)).add
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((φ1.apply_re_smooth 1).smul (φ2.apply_im_smooth 1))).sub
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((φ1.apply_im_smooth 0).smul (φ2.apply_re_smooth 0))).sub
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((φ1.apply_im_smooth 1).smul (φ2.apply_re_smooth 1))
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/-!
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## The pointwise norm squared
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-/
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/-- Given a `HiggsField`, the map `SpaceTime → ℝ` obtained by taking the square norm of the
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pointwise Higgs vector. -/
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@[simp]
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def normSq (φ : HiggsField) : SpaceTime → ℝ := fun x => ‖φ x‖ ^ 2
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/-- Notation for the norm squared of a Higgs field. -/
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scoped[StandardModel.HiggsField] notation "‖" φ1 "‖_H ^ 2" => normSq φ1
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/-!
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## Relation between inner prod and norm squared
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-/
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lemma innerProd_self_eq_normSq (φ : HiggsField) (x : SpaceTime) :
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⟪φ, φ⟫_H x = ‖φ‖_H ^ 2 x := by
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erw [normSq, @PiLp.norm_sq_eq_of_L2, Fin.sum_univ_two]
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simp only [innerProd, PiLp.inner_apply, RCLike.inner_apply, conj_mul', norm_eq_abs,
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Fin.sum_univ_two, ofReal_add, ofReal_pow]
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lemma normSq_eq_innerProd_self (φ : HiggsField) (x : SpaceTime) :
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‖φ x‖ ^ 2 = (⟪φ, φ⟫_H x).re := by
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rw [innerProd_self_eq_normSq]
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rfl
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/-!
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# Properties of the norm squared
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-/
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lemma toHiggsVec_norm (φ : HiggsField) (x : SpaceTime) :
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‖φ x‖ = ‖φ.toHiggsVec x‖ := rfl
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lemma normSq_expand (φ : HiggsField) :
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φ.normSq = fun x => (conj (φ x 0) * (φ x 0) + conj (φ x 1) * (φ x 1)).re := by
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funext x
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simp [normSq, add_re, mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add, @norm_sq_eq_inner ℂ]
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lemma normSq_nonneg (φ : HiggsField) (x : SpaceTime) : 0 ≤ φ.normSq x := by
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simp [normSq, ge_iff_le, norm_nonneg, pow_nonneg]
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lemma normSq_zero (φ : HiggsField) (x : SpaceTime) : φ.normSq x = 0 ↔ φ x = 0 := by
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simp [normSq, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff, norm_eq_zero]
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lemma normSq_smooth (φ : HiggsField) : Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℝ) φ.normSq := by
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rw [normSq_expand]
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refine Smooth.add ?_ ?_
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simp only [mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
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exact ((φ.apply_re_smooth 0).smul (φ.apply_re_smooth 0)).add $
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(φ.apply_im_smooth 0).smul (φ.apply_im_smooth 0)
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simp only [mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
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exact ((φ.apply_re_smooth 1).smul (φ.apply_re_smooth 1)).add $
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(φ.apply_im_smooth 1).smul (φ.apply_im_smooth 1)
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end HiggsField
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end StandardModel
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end
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