59 lines
1.9 KiB
Text
59 lines
1.9 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.ComplexLorentz.PauliLower
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/-!
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## Bispinors
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-/
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open IndexNotation
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open CategoryTheory
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open MonoidalCategory
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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 TensorTree
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open OverColor.Discrete
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open Fermion
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noncomputable section
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namespace complexLorentzTensor
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open Lorentz
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/-- A bispinor `pᵃᵃ` created from a lorentz vector `p^μ`. -/
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def contrBispinorUp (p : complexContr) :=
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{p | μ ⊗ pauliCo | μ α β}ᵀ.tensor
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lemma tensorNode_contrBispinorUp (p : complexContr) :
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(tensorNode (contrBispinorUp p)).tensor = {p | μ ⊗ pauliCo | μ α β}ᵀ.tensor := by
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rw [contrBispinorUp, tensorNode_tensor]
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/-- A bispinor `pₐₐ` created from a lorentz vector `p^μ`. -/
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def contrBispinorDown (p : complexContr) :=
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{Fermion.altLeftMetric | α α' ⊗ Fermion.altRightMetric | β β' ⊗
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(contrBispinorUp p) | α β}ᵀ.tensor
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/-- Expands the tensor node of `contrBispinorDown` into a tensor tree based on
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`contrBispinorUp`. -/
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lemma tensorNode_contrBispinorDown (p : complexContr) :
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{contrBispinorDown p | α β}ᵀ.tensor = {Fermion.altLeftMetric | α α' ⊗
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Fermion.altRightMetric | β β' ⊗ (contrBispinorUp p) | α β}ᵀ.tensor := by
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rw [contrBispinorDown, tensorNode_tensor]
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/-- Expansion of a `contrBispinorDown` into the original contravariant tensor nested
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between pauli matrices and metrics. -/
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lemma contrBispinorDown_full_nested (p : complexContr) :
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{contrBispinorDown p | α β}ᵀ.tensor = {Fermion.altLeftMetric | α α' ⊗
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Fermion.altRightMetric | β β' ⊗ (p | μ ⊗ pauliCo | μ α β)}ᵀ.tensor := by
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conv =>
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lhs
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rw [tensorNode_contrBispinorDown]
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rw [contr_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_snd <| tensorNode_contrBispinorUp p]
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end complexLorentzTensor
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end
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