66 lines
1.6 KiB
Text
66 lines
1.6 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.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.StandardModel.Basic
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/-!
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# The Clifford Algebra
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This file defines the Gamma matrices.
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## TODO
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- Prove that the algebra generated by the gamma matrices is ismorphic to the
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Clifford algebra assocaited with spacetime.
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- Include relations for gamma matrices.
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-/
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namespace StandardModel
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open Complex
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noncomputable section diracRepresentation
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def γ0 : Matrix (Fin 4) (Fin 4) ℂ :=
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![![1, 0, 0, 0], ![0, 1, 0, 0], ![0, 0, -1, 0], ![0, 0, 0, -1]]
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def γ1 : Matrix (Fin 4) (Fin 4) ℂ :=
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![![0, 0, 0, 1], ![0, 0, 1, 0], ![0, -1, 0, 0], ![-1, 0, 0, 0]]
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def γ2 : Matrix (Fin 4) (Fin 4) ℂ :=
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![![0, 0, 0, - I], ![0, 0, I, 0], ![0, I, 0, 0], ![-I, 0, 0, 0]]
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def γ3 : Matrix (Fin 4) (Fin 4) ℂ :=
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![![0, 0, 1, 0], ![0, 0, 0, -1], ![-1, 0, 0, 0], ![0, 1, 0, 0]]
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def γ5 : Matrix (Fin 4) (Fin 4) ℂ := I • (γ0 * γ1 * γ2 * γ3)
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@[simp]
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def γ : Fin 4 → Matrix (Fin 4) (Fin 4) ℂ := ![γ0, γ1, γ2, γ3]
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namespace γ
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open spaceTime
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variable (μ ν : Fin 4)
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@[simp]
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def γSet : Set (Matrix (Fin 4) (Fin 4) ℂ) := {γ i | i : Fin 4}
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lemma γ_in_γSet (μ : Fin 4) : γ μ ∈ γSet := by
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simp [γSet]
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def diracAlgebra : Subalgebra ℝ (Matrix (Fin 4) (Fin 4) ℂ) :=
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Algebra.adjoin ℝ γSet
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lemma γSet_subset_diracAlgebra : γSet ⊆ diracAlgebra :=
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Algebra.subset_adjoin
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lemma γ_in_diracAlgebra (μ : Fin 4) : γ μ ∈ diracAlgebra :=
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γSet_subset_diracAlgebra (γ_in_γSet μ)
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end γ
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end diracRepresentation
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end StandardModel
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