233 lines
9.4 KiB
Text
233 lines
9.4 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.Tensors.Tree.Dot
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import HepLean.Lorentz.Weyl.Contraction
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import HepLean.Lorentz.Weyl.Metric
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import HepLean.Lorentz.Weyl.Unit
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import HepLean.Lorentz.ComplexVector.Contraction
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import HepLean.Lorentz.ComplexVector.Metric
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import HepLean.Lorentz.ComplexVector.Unit
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import HepLean.Mathematics.PiTensorProduct
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import HepLean.Lorentz.PauliMatrices.AsTensor
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/-!
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## Complex Lorentz tensors
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-/
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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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namespace complexLorentzTensor
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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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/-- Color for complex Lorentz tensors is decidable. -/
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instance : DecidableEq Color := fun x y =>
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match x, y with
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| Color.upL, Color.upL => isTrue rfl
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| Color.downL, Color.downL => isTrue rfl
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| Color.upR, Color.upR => isTrue rfl
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| Color.downR, Color.downR => isTrue rfl
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| Color.up, Color.up => isTrue rfl
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| Color.down, Color.down => isTrue rfl
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/- The false -/
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| Color.upL, Color.downL => isFalse fun h => Color.noConfusion h
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| Color.upL, Color.upR => isFalse fun h => Color.noConfusion h
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| Color.upL, Color.downR => isFalse fun h => Color.noConfusion h
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| Color.upL, Color.up => isFalse fun h => Color.noConfusion h
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| Color.upL, Color.down => isFalse fun h => Color.noConfusion h
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| Color.downL, Color.upL => isFalse fun h => Color.noConfusion h
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| Color.downL, Color.upR => isFalse fun h => Color.noConfusion h
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| Color.downL, Color.downR => isFalse fun h => Color.noConfusion h
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| Color.downL, Color.up => isFalse fun h => Color.noConfusion h
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| Color.downL, Color.down => isFalse fun h => Color.noConfusion h
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| Color.upR, Color.upL => isFalse fun h => Color.noConfusion h
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| Color.upR, Color.downL => isFalse fun h => Color.noConfusion h
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| Color.upR, Color.downR => isFalse fun h => Color.noConfusion h
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| Color.upR, Color.up => isFalse fun h => Color.noConfusion h
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| Color.upR, Color.down => isFalse fun h => Color.noConfusion h
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| Color.downR, Color.upL => isFalse fun h => Color.noConfusion h
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| Color.downR, Color.downL => isFalse fun h => Color.noConfusion h
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| Color.downR, Color.upR => isFalse fun h => Color.noConfusion h
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| Color.downR, Color.up => isFalse fun h => Color.noConfusion h
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| Color.downR, Color.down => isFalse fun h => Color.noConfusion h
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| Color.up, Color.upL => isFalse fun h => Color.noConfusion h
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| Color.up, Color.downL => isFalse fun h => Color.noConfusion h
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| Color.up, Color.upR => isFalse fun h => Color.noConfusion h
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| Color.up, Color.downR => isFalse fun h => Color.noConfusion h
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| Color.up, Color.down => isFalse fun h => Color.noConfusion h
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| Color.down, Color.upL => isFalse fun h => Color.noConfusion h
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| Color.down, Color.downL => isFalse fun h => Color.noConfusion h
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| Color.down, Color.upR => isFalse fun h => Color.noConfusion h
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| Color.down, Color.downR => isFalse fun h => Color.noConfusion h
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| Color.down, Color.up => isFalse fun h => Color.noConfusion h
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end complexLorentzTensor
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noncomputable section
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open complexLorentzTensor in
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/-- The tensor structure for complex Lorentz tensors. -/
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def complexLorentzTensor : TensorSpecies where
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C := complexLorentzTensor.Color
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G := SL(2, ℂ)
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G_group := inferInstance
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k := ℂ
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k_commRing := inferInstance
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FD := Discrete.functor fun c =>
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match c with
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| Color.upL => Fermion.leftHanded
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| Color.downL => Fermion.altLeftHanded
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| Color.upR => Fermion.rightHanded
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| Color.downR => Fermion.altRightHanded
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| Color.up => Lorentz.complexContr
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| Color.down => Lorentz.complexCo
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τ := fun c =>
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match c with
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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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τ_involution c := by
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match c with
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| Color.upL => rfl
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| Color.downL => rfl
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| Color.upR => rfl
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| Color.downR => rfl
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| Color.up => rfl
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| Color.down => rfl
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contr := Discrete.natTrans fun c =>
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match c with
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| Discrete.mk Color.upL => Fermion.leftAltContraction
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| Discrete.mk Color.downL => Fermion.altLeftContraction
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| Discrete.mk Color.upR => Fermion.rightAltContraction
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| Discrete.mk Color.downR => Fermion.altRightContraction
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| Discrete.mk Color.up => Lorentz.contrCoContraction
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| Discrete.mk Color.down => Lorentz.coContrContraction
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metric := Discrete.natTrans fun c =>
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match c with
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| Discrete.mk Color.upL => Fermion.leftMetric
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| Discrete.mk Color.downL => Fermion.altLeftMetric
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| Discrete.mk Color.upR => Fermion.rightMetric
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| Discrete.mk Color.downR => Fermion.altRightMetric
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| Discrete.mk Color.up => Lorentz.contrMetric
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| Discrete.mk Color.down => Lorentz.coMetric
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unit := Discrete.natTrans fun c =>
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match c with
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| Discrete.mk Color.upL => Fermion.altLeftLeftUnit
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| Discrete.mk Color.downL => Fermion.leftAltLeftUnit
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| Discrete.mk Color.upR => Fermion.altRightRightUnit
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| Discrete.mk Color.downR => Fermion.rightAltRightUnit
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| Discrete.mk Color.up => Lorentz.coContrUnit
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| Discrete.mk Color.down => Lorentz.contrCoUnit
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repDim := fun c =>
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match c with
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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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repDim_neZero := fun c =>
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match c with
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| Color.upL => inferInstance
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| Color.downL => inferInstance
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| Color.upR => inferInstance
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| Color.downR => inferInstance
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| Color.up => inferInstance
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| Color.down => inferInstance
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basis := fun c =>
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match c with
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| Color.upL => Fermion.leftBasis
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| Color.downL => Fermion.altLeftBasis
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| Color.upR => Fermion.rightBasis
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| Color.downR => Fermion.altRightBasis
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| Color.up => Lorentz.complexContrBasisFin4
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| Color.down => Lorentz.complexCoBasisFin4
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contr_tmul_symm := fun c =>
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match c with
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| Color.upL => Fermion.leftAltContraction_tmul_symm
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| Color.downL => Fermion.altLeftContraction_tmul_symm
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| Color.upR => Fermion.rightAltContraction_tmul_symm
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| Color.downR => Fermion.altRightContraction_tmul_symm
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| Color.up => Lorentz.contrCoContraction_tmul_symm
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| Color.down => Lorentz.coContrContraction_tmul_symm
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contr_unit := fun c =>
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match c with
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| Color.upL => Fermion.contr_altLeftLeftUnit
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| Color.downL => Fermion.contr_leftAltLeftUnit
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| Color.upR => Fermion.contr_altRightRightUnit
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| Color.downR => Fermion.contr_rightAltRightUnit
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| Color.up => Lorentz.contr_coContrUnit
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| Color.down => Lorentz.contr_contrCoUnit
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unit_symm := fun c =>
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match c with
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| Color.upL => Fermion.altLeftLeftUnit_symm
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| Color.downL => Fermion.leftAltLeftUnit_symm
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| Color.upR => Fermion.altRightRightUnit_symm
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| Color.downR => Fermion.rightAltRightUnit_symm
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| Color.up => Lorentz.coContrUnit_symm
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| Color.down => Lorentz.contrCoUnit_symm
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contr_metric := fun c =>
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match c with
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| Color.upL => by simpa using Fermion.leftAltContraction_apply_metric
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| Color.downL => by simpa using Fermion.altLeftContraction_apply_metric
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| Color.upR => by simpa using Fermion.rightAltContraction_apply_metric
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| Color.downR => by simpa using Fermion.altRightContraction_apply_metric
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| Color.up => by simpa using Lorentz.contrCoContraction_apply_metric
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| Color.down => by simpa using Lorentz.coContrContraction_apply_metric
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namespace complexLorentzTensor
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/-- Color for complex Lorentz tensors is decidable. -/
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instance : DecidableEq complexLorentzTensor.C := complexLorentzTensor.instDecidableEqColor
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/-- Contracting two basis elements gives `1` if the index for the basis elements is the same,
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and `0` otherwise. Holds for any color of index. -/
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lemma basis_contr (c : complexLorentzTensor.C) (i : Fin (complexLorentzTensor.repDim c))
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(j : Fin (complexLorentzTensor.repDim (complexLorentzTensor.τ c))) :
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complexLorentzTensor.castToField
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((complexLorentzTensor.contr.app {as := c}).hom
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(complexLorentzTensor.basis c i ⊗ₜ complexLorentzTensor.basis (complexLorentzTensor.τ c) j)) =
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if i.val = j.val then 1 else 0 :=
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match c with
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| Color.upL => Fermion.leftAltContraction_basis _ _
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| Color.downL => Fermion.altLeftContraction_basis _ _
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| Color.upR => Fermion.rightAltContraction_basis _ _
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| Color.downR => Fermion.altRightContraction_basis _ _
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| Color.up => Lorentz.contrCoContraction_basis _ _
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| Color.down => Lorentz.coContrContraction_basis _ _
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/-- For any object in the over color category, with source `Fin n`, has a decidable source. -/
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instance {n : ℕ} {c : Fin n → complexLorentzTensor.C} :
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DecidableEq (OverColor.mk c).left := instDecidableEqFin n
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/-- For any object in the over color category, with source `Fin n`, has a finite source. -/
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instance {n : ℕ} {c : Fin n → complexLorentzTensor.C} :
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Fintype (OverColor.mk c).left := Fin.fintype n
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/-- The equality of two maps in `OverColor C` from objects based on `Fin _` is decidable. -/
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instance {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
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{c1 : Fin m → complexLorentzTensor.C} (σ σ' : OverColor.mk c ⟶ OverColor.mk c1) :
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Decidable (σ = σ') :=
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decidable_of_iff _ (OverColor.Hom.ext_iff σ σ')
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end complexLorentzTensor
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end
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