61 lines
1.3 KiB
Text
61 lines
1.3 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.Tensors.OverColor.Basic
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import HepLean.Mathematics.PiTensorProduct
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/-!
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## Complex Lorentz tensors
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-/
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namespace Fermion
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open Matrix
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open MatrixGroups
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open Complex
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open TensorProduct
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open IndexNotation
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open CategoryTheory
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open MonoidalCategory
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/-- The colors associated with complex representations of SL(2, ℂ) of intrest to physics. -/
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inductive Color
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| upL : Color
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| downL : Color
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| upR : Color
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| downR : Color
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| up : Color
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| down : Color
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def τ : Color → Color
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| Color.upL => Color.downL
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| Color.downL => Color.upL
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| Color.upR => Color.downR
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| Color.downR => Color.upR
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| Color.up => Color.down
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| Color.down => Color.up
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def evalNo : Color → ℕ
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| Color.upL => 2
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| Color.downL => 2
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| Color.upR => 2
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| Color.downR => 2
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| Color.up => 4
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| Color.down => 4
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noncomputable section
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/-- The corresponding representations associated with a color. -/
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def colorToRep (c : Color) : Rep ℂ SL(2, ℂ) :=
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match c with
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| Color.upL => Fermion.altLeftHanded
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| Color.downL => Fermion.leftHanded
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| Color.upR => Fermion.altRightHanded
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| Color.downR => Fermion.rightHanded
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| Color.up => Lorentz.complexContr
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| Color.down => Lorentz.complexCo
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end
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end Fermion
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