59 lines
2.3 KiB
Text
59 lines
2.3 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.BeyondTheStandardModel.TwoHDM.Basic
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import Mathlib.LinearAlgebra.Matrix.PosDef
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/-!
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# Gauge orbits for the 2HDM
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The main reference for material in this section is https://arxiv.org/pdf/hep-ph/0605184.
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-/
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namespace TwoHDM
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open StandardModel
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open ComplexConjugate
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open HiggsField
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noncomputable section
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/-- For two Higgs fields `Φ₁` and `Φ₂`, the map from space time to 2 x 2 complex matrices
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defined by ((Φ₁^†Φ₁, Φ₂^†Φ₁), (Φ₁^†Φ₂, Φ₂^†Φ₂)). -/
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def prodMatrix (Φ1 Φ2 : HiggsField) (x : SpaceTime) : Matrix (Fin 2) (Fin 2) ℂ :=
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!![⟪Φ1, Φ1⟫_H x, ⟪Φ2, Φ1⟫_H x; ⟪Φ1, Φ2⟫_H x, ⟪Φ2, Φ2⟫_H x]
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/-- The matrix `prodMatrix` is hermitian. -/
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lemma prodMatrix_hermitian (Φ1 Φ2 : HiggsField) (x : SpaceTime) :
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(prodMatrix Φ1 Φ2 x).IsHermitian := by
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rw [Matrix.IsHermitian]
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ext i j
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fin_cases i <;> fin_cases j
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· simp [prodMatrix]
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· simp only [prodMatrix, innerProd, PiLp.inner_apply, RCLike.inner_apply, Fin.sum_univ_two,
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Fin.isValue, Fin.zero_eta, Fin.mk_one, Matrix.conjTranspose_apply, Matrix.of_apply,
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Matrix.cons_val', Matrix.cons_val_zero, Matrix.empty_val', Matrix.cons_val_fin_one,
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Matrix.cons_val_one, Matrix.head_fin_const, star_add, star_mul', RCLike.star_def,
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RingHomCompTriple.comp_apply, RingHom.id_apply, Matrix.head_cons]
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ring
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· simp only [prodMatrix, innerProd, PiLp.inner_apply, RCLike.inner_apply, Fin.sum_univ_two,
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Fin.isValue, Fin.mk_one, Fin.zero_eta, Matrix.conjTranspose_apply, Matrix.of_apply,
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Matrix.cons_val', Matrix.cons_val_one, Matrix.head_cons, Matrix.empty_val',
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Matrix.cons_val_fin_one, Matrix.cons_val_zero, star_add, star_mul', RCLike.star_def,
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RingHomCompTriple.comp_apply, RingHom.id_apply, Matrix.head_fin_const]
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ring
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· simp [prodMatrix]
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informal_lemma prodMatrix_positive_semidefinite where
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math :≈ "For all x in ``SpaceTime, ``prodMatrix at `x` is positive semidefinite."
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deps :≈ [``prodMatrix, ``SpaceTime]
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informal_lemma prodMatrix_smooth where
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math :≈ "The map ``prodMatrix is a smooth function on spacetime."
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deps :≈ [``prodMatrix]
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end
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end TwoHDM
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