refactor: Move constructors
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jstoobysmith
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6 changed files with 421 additions and 169 deletions
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@ -70,7 +70,9 @@ import HepLean.SpaceTime.LorentzGroup.Orthochronous
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import HepLean.SpaceTime.LorentzGroup.Proper
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import HepLean.SpaceTime.LorentzGroup.Restricted
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import HepLean.SpaceTime.LorentzGroup.Rotations
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import HepLean.SpaceTime.LorentzTensor.Basic
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import HepLean.SpaceTime.LorentzTensor.Real.Basic
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import HepLean.SpaceTime.LorentzTensor.Real.Constructors
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import HepLean.SpaceTime.LorentzTensor.Real.LorentzAction
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import HepLean.SpaceTime.LorentzVector.AsSelfAdjointMatrix
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import HepLean.SpaceTime.LorentzVector.Basic
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import HepLean.SpaceTime.LorentzVector.NormOne
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@ -96,7 +96,7 @@ lemma standParamAsMatrix_unitary (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
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rw [sin_sq, sin_sq]
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ring
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/-- A CKM Matrix from four reals `θ₁₂`, `θ₁₃`, `θ₂₃`, and `δ₁₃`. This is the standard
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/-- A CKM Matrix from four reals `θ₁₂`, `θ₁₃`, `θ₂₃`, and `δ₁₃`. This is the standard
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parameterization of CKM matrices. -/
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def standParam (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : CKMMatrix :=
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⟨standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃, by
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@ -6,7 +6,6 @@ Authors: Joseph Tooby-Smith
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import Mathlib.Logic.Function.CompTypeclasses
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import Mathlib.Data.Real.Basic
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import Mathlib.Analysis.Normed.Field.Basic
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import Mathlib.LinearAlgebra.Matrix.Trace
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/-!
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# Real Lorentz Tensors
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@ -423,141 +422,6 @@ def contr {X : Type} (T : Marked d X 2)
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/-!
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# Tensors from reals, vectors and matrices
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Note that that these definitions are not equivariant with respect to an
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action of the Lorentz group. They are provided for constructive purposes.
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-/
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/-- A 0-tensor from a real number. -/
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def ofReal (d : ℕ) (r : ℝ) : RealLorentzTensor d Empty where
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color := fun _ => Colors.up
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coord := fun _ => r
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/-- A marked 1-tensor with a single up index constructed from a vector.
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Note: This is not the same as rising indices on `ofVecDown`. -/
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def ofVecUp {d : ℕ} (v : Fin 1 ⊕ Fin d → ℝ) :
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Marked d Empty 1 where
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color := fun _ => Colors.up
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coord := fun i => v <| i <| Sum.inr <| ⟨0, PUnit.unit⟩
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/-- A marked 1-tensor with a single down index constructed from a vector.
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Note: This is not the same as lowering indices on `ofVecUp`. -/
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def ofVecDown {d : ℕ} (v : Fin 1 ⊕ Fin d → ℝ) :
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Marked d Empty 1 where
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color := fun _ => Colors.down
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coord := fun i => v <| i <| Sum.inr <| ⟨0, PUnit.unit⟩
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/-- A tensor with two up indices constructed from a matrix.
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Note: This is not the same as rising or lowering indices on other `ofMat...`. -/
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def ofMatUpUp {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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Marked d Empty 2 where
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color := fun _ => Colors.up
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coord := fun i => m (i (Sum.inr ⟨0, PUnit.unit⟩)) (i (Sum.inr ⟨1, PUnit.unit⟩))
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/-- A tensor with two down indices constructed from a matrix.
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Note: This is not the same as rising or lowering indices on other `ofMat...`. -/
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def ofMatDownDown {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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Marked d Empty 2 where
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color := fun _ => Colors.down
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coord := fun i => m (i (Sum.inr ⟨0, PUnit.unit⟩)) (i (Sum.inr ⟨1, PUnit.unit⟩))
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/-- A marked 2-tensor with the first index up and the second index down.
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Note: This is not the same as rising or lowering indices on other `ofMat...`. -/
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@[simps!]
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def ofMatUpDown {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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Marked d Empty 2 where
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color := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => Colors.up
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| Sum.inr ⟨1, PUnit.unit⟩ => Colors.down
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coord := fun i => m (i (Sum.inr ⟨0, PUnit.unit⟩)) (i (Sum.inr ⟨1, PUnit.unit⟩))
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/-- A marked 2-tensor with the first index down and the second index up.
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Note: This is not the same as rising or lowering indices on other `ofMat...`. -/
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def ofMatDownUp {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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Marked d Empty 2 where
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color := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => Colors.down
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| Sum.inr ⟨1, PUnit.unit⟩ => Colors.up
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coord := fun i => m (i (Sum.inr ⟨0, PUnit.unit⟩)) (i (Sum.inr ⟨1, PUnit.unit⟩))
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/-- Contracting the indices of `ofMatUpDown` returns the trace of the matrix. -/
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lemma contr_ofMatUpDown_eq_trace {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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contr (ofMatUpDown M) (by rfl) = ofReal d M.trace := by
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refine ext' ?_ ?_
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· funext i
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exact Empty.elim i
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· funext i
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simp only [Fin.isValue, contr, IndexValue, Equiv.cast_apply, trace, diag_apply, ofReal,
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Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton]
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apply Finset.sum_congr rfl
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intro x _
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rfl
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/-- Contracting the indices of `ofMatDownUp` returns the trace of the matrix. -/
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lemma contr_ofMatDownUp_eq_trace {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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contr (ofMatDownUp M) (by rfl) = ofReal d M.trace := by
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refine ext' ?_ ?_
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· funext i
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exact Empty.elim i
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· funext i
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rfl
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/-- Multiplying `ofVecUp` with `ofVecDown` gives the dot product. -/
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@[simp]
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lemma mul_ofVecUp_ofVecDown_eq_dot_prod {d : ℕ} (v₁ v₂ : Fin 1 ⊕ Fin d → ℝ) :
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congrSet (@Equiv.equivEmpty (Empty ⊕ Empty) instIsEmptySum)
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(mul (ofVecUp v₁) (ofVecDown v₂) (by rfl)) = ofReal d (v₁ ⬝ᵥ v₂) := by
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refine ext' ?_ ?_
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· funext i
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exact Empty.elim i
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· funext i
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rfl
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/-- Multiplying `ofVecDown` with `ofVecUp` gives the dot product. -/
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@[simp]
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lemma mul_ofVecDown_ofVecUp_eq_dot_prod {d : ℕ} (v₁ v₂ : Fin 1 ⊕ Fin d → ℝ) :
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congrSet (Equiv.equivEmpty (Empty ⊕ Empty))
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(mul (ofVecDown v₁) (ofVecUp v₂) rfl) = ofReal d (v₁ ⬝ᵥ v₂) := by
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refine ext' ?_ ?_
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· funext i
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exact Empty.elim i
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· funext i
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rfl
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lemma mul_ofMatUpDown_ofVecUp_eq_mulVec {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
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(v : Fin 1 ⊕ Fin d → ℝ) :
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congrSet ((Equiv.sumEmpty (Empty ⊕ PUnit.{1}) Empty).trans equivPUnitToSigma)
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(mul (unmarkFirst $ ofMatUpDown M) (ofVecUp v) rfl) = ofVecUp (M *ᵥ v) := by
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refine ext' ?_ ?_
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· funext i
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, congrSet_apply_color, mul_color, Equiv.symm_symm]
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fin_cases i
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rfl
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· funext i
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rfl
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lemma mul_ofMatDownUp_ofVecDown_eq_mulVec {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
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(v : Fin 1 ⊕ Fin d → ℝ) :
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congrSet ((Equiv.sumEmpty (Empty ⊕ PUnit.{1}) Empty).trans equivPUnitToSigma)
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(mul (unmarkFirst $ ofMatDownUp M) (ofVecDown v) rfl) = ofVecDown (M *ᵥ v) := by
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refine ext' ?_ ?_
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· funext i
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, congrSet_apply_color, mul_color, Equiv.symm_symm]
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fin_cases i
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rfl
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· funext i
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rfl
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/-!
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## Rising and lowering indices
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Rising or lowering an index corresponds to changing the color of that index.
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@ -568,14 +432,6 @@ Rising or lowering an index corresponds to changing the color of that index.
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/-!
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## Action of the Lorentz group
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-/
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/-! TODO: Define the action of the Lorentz group on the sets of Tensors. -/
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/-!
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## Graphical species and Lorentz tensors
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-/
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403
HepLean/SpaceTime/LorentzTensor/Real/Constructors.lean
Normal file
403
HepLean/SpaceTime/LorentzTensor/Real/Constructors.lean
Normal file
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@ -0,0 +1,403 @@
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/-
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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.SpaceTime.LorentzTensor.Real.Basic
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import HepLean.SpaceTime.LorentzTensor.Real.LorentzAction
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/-!
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# Constructors for real Lorentz tensors
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In this file we will constructors of real Lorentz tensors from real numbers,
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vectors and matrices.
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We will derive properties of these constructors.
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-/
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namespace RealLorentzTensor
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/-!
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# Tensors from reals, vectors and matrices
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Note that that these definitions are not equivariant with respect to an
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action of the Lorentz group. They are provided for constructive purposes.
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-/
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/-- A 0-tensor from a real number. -/
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def ofReal (d : ℕ) (r : ℝ) : RealLorentzTensor d Empty where
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color := fun _ => Colors.up
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coord := fun _ => r
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/-- A marked 1-tensor with a single up index constructed from a vector.
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Note: This is not the same as rising indices on `ofVecDown`. -/
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def ofVecUp {d : ℕ} (v : Fin 1 ⊕ Fin d → ℝ) :
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Marked d Empty 1 where
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color := fun _ => Colors.up
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coord := fun i => v <| i <| Sum.inr <| ⟨0, PUnit.unit⟩
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/-- A marked 1-tensor with a single down index constructed from a vector.
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Note: This is not the same as lowering indices on `ofVecUp`. -/
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def ofVecDown {d : ℕ} (v : Fin 1 ⊕ Fin d → ℝ) :
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Marked d Empty 1 where
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color := fun _ => Colors.down
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coord := fun i => v <| i <| Sum.inr <| ⟨0, PUnit.unit⟩
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/-- A tensor with two up indices constructed from a matrix.
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Note: This is not the same as rising or lowering indices on other `ofMat...`. -/
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def ofMatUpUp {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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Marked d Empty 2 where
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color := fun _ => Colors.up
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coord := fun i => m (i (Sum.inr ⟨0, PUnit.unit⟩)) (i (Sum.inr ⟨1, PUnit.unit⟩))
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/-- A tensor with two down indices constructed from a matrix.
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Note: This is not the same as rising or lowering indices on other `ofMat...`. -/
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def ofMatDownDown {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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Marked d Empty 2 where
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color := fun _ => Colors.down
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coord := fun i => m (i (Sum.inr ⟨0, PUnit.unit⟩)) (i (Sum.inr ⟨1, PUnit.unit⟩))
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/-- A marked 2-tensor with the first index up and the second index down.
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Note: This is not the same as rising or lowering indices on other `ofMat...`. -/
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@[simps!]
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def ofMatUpDown {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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Marked d Empty 2 where
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color := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => Colors.up
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| Sum.inr ⟨1, PUnit.unit⟩ => Colors.down
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coord := fun i => m (i (Sum.inr ⟨0, PUnit.unit⟩)) (i (Sum.inr ⟨1, PUnit.unit⟩))
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/-- A marked 2-tensor with the first index down and the second index up.
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Note: This is not the same as rising or lowering indices on other `ofMat...`. -/
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def ofMatDownUp {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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Marked d Empty 2 where
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color := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => Colors.down
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| Sum.inr ⟨1, PUnit.unit⟩ => Colors.up
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coord := fun i => m (i (Sum.inr ⟨0, PUnit.unit⟩)) (i (Sum.inr ⟨1, PUnit.unit⟩))
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/-!
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## Equivalence of `IndexValue` for constructors
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-/
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/-- Index values for `ofVecUp v` are equivalent to elements of `Fin 1 ⊕ Fin d`. -/
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def ofVecUpIndexValue (v : Fin 1 ⊕ Fin d → ℝ) :
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IndexValue d (ofVecUp v).color ≃ (Fin 1 ⊕ Fin d) where
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toFun i := i (Sum.inr ⟨0, PUnit.unit⟩)
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invFun x := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => x
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left_inv i := by
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funext y
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match y with
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| Sum.inr ⟨0, PUnit.unit⟩ => rfl
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right_inv x := rfl
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/-- Index values for `ofVecDown v` are equivalent to elements of `Fin 1 ⊕ Fin d`. -/
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def ofVecDownIndexValue (v : Fin 1 ⊕ Fin d → ℝ) :
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IndexValue d (ofVecDown v).color ≃ (Fin 1 ⊕ Fin d) where
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toFun i := i (Sum.inr ⟨0, PUnit.unit⟩)
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invFun x := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => x
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left_inv i := by
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funext y
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match y with
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| Sum.inr ⟨0, PUnit.unit⟩ => rfl
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right_inv x := rfl
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/-- Index values for `ofMatUpUp v` are equivalent to elements of
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`(Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d)`. -/
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def ofMatUpUpIndexValue (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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IndexValue d (ofMatUpUp M).color ≃ (Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d) where
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toFun i := (i (Sum.inr ⟨0, PUnit.unit⟩), i (Sum.inr ⟨1, PUnit.unit⟩))
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invFun x := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => x.1
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| Sum.inr ⟨1, PUnit.unit⟩ => x.2
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left_inv i := by
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funext y
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match y with
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| Sum.inr ⟨0, PUnit.unit⟩ => rfl
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| Sum.inr ⟨1, PUnit.unit⟩ => rfl
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right_inv x := rfl
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/-- Index values for `ofMatDownDown v` are equivalent to elements of
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`(Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d)`. -/
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def ofMatDownDownIndexValue (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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IndexValue d (ofMatDownDown M).color ≃ (Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d) where
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toFun i := (i (Sum.inr ⟨0, PUnit.unit⟩), i (Sum.inr ⟨1, PUnit.unit⟩))
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invFun x := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => x.1
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| Sum.inr ⟨1, PUnit.unit⟩ => x.2
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left_inv i := by
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funext y
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match y with
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| Sum.inr ⟨0, PUnit.unit⟩ => rfl
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| Sum.inr ⟨1, PUnit.unit⟩ => rfl
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right_inv x := rfl
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/-- Index values for `ofMatUpDown v` are equivalent to elements of
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`(Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d)`. -/
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def ofMatUpDownIndexValue (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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IndexValue d (ofMatUpDown M).color ≃ (Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d) where
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toFun i := (i (Sum.inr ⟨0, PUnit.unit⟩), i (Sum.inr ⟨1, PUnit.unit⟩))
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invFun x := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => x.1
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| Sum.inr ⟨1, PUnit.unit⟩ => x.2
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left_inv i := by
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funext y
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match y with
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| Sum.inr ⟨0, PUnit.unit⟩ => rfl
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| Sum.inr ⟨1, PUnit.unit⟩ => rfl
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right_inv x := rfl
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/-- Index values for `ofMatDownUp v` are equivalent to elements of
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`(Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d)`. -/
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def ofMatDownUpIndexValue (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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IndexValue d (ofMatDownUp M).color ≃ (Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d) where
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toFun i := (i (Sum.inr ⟨0, PUnit.unit⟩), i (Sum.inr ⟨1, PUnit.unit⟩))
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invFun x := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => x.1
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| Sum.inr ⟨1, PUnit.unit⟩ => x.2
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left_inv i := by
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funext y
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match y with
|
||||
| Sum.inr ⟨0, PUnit.unit⟩ => rfl
|
||||
| Sum.inr ⟨1, PUnit.unit⟩ => rfl
|
||||
right_inv x := rfl
|
||||
|
||||
/-!
|
||||
|
||||
## Contraction of indices of tensors from matrices
|
||||
|
||||
-/
|
||||
open Matrix
|
||||
open Marked
|
||||
|
||||
/-- Contracting the indices of `ofMatUpDown` returns the trace of the matrix. -/
|
||||
lemma contr_ofMatUpDown_eq_trace {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
|
||||
contr (ofMatUpDown M) (by rfl) = ofReal d M.trace := by
|
||||
refine ext' ?_ ?_
|
||||
· funext i
|
||||
exact Empty.elim i
|
||||
· funext i
|
||||
simp only [Fin.isValue, contr, IndexValue, Equiv.cast_apply, trace, diag_apply, ofReal,
|
||||
Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton]
|
||||
apply Finset.sum_congr rfl
|
||||
intro x _
|
||||
rfl
|
||||
|
||||
/-- Contracting the indices of `ofMatDownUp` returns the trace of the matrix. -/
|
||||
lemma contr_ofMatDownUp_eq_trace {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
|
||||
contr (ofMatDownUp M) (by rfl) = ofReal d M.trace := by
|
||||
refine ext' ?_ ?_
|
||||
· funext i
|
||||
exact Empty.elim i
|
||||
· funext i
|
||||
rfl
|
||||
|
||||
/-!
|
||||
|
||||
## Multiplication of tensors from vectors and matrices
|
||||
|
||||
-/
|
||||
|
||||
/-- Multiplying `ofVecUp` with `ofVecDown` gives the dot product. -/
|
||||
@[simp]
|
||||
lemma mul_ofVecUp_ofVecDown_eq_dot_prod {d : ℕ} (v₁ v₂ : Fin 1 ⊕ Fin d → ℝ) :
|
||||
congrSet (@Equiv.equivEmpty (Empty ⊕ Empty) instIsEmptySum)
|
||||
(mul (ofVecUp v₁) (ofVecDown v₂) (by rfl)) = ofReal d (v₁ ⬝ᵥ v₂) := by
|
||||
refine ext' ?_ ?_
|
||||
· funext i
|
||||
exact Empty.elim i
|
||||
· funext i
|
||||
rfl
|
||||
|
||||
/-- Multiplying `ofVecDown` with `ofVecUp` gives the dot product. -/
|
||||
@[simp]
|
||||
lemma mul_ofVecDown_ofVecUp_eq_dot_prod {d : ℕ} (v₁ v₂ : Fin 1 ⊕ Fin d → ℝ) :
|
||||
congrSet (Equiv.equivEmpty (Empty ⊕ Empty))
|
||||
(mul (ofVecDown v₁) (ofVecUp v₂) rfl) = ofReal d (v₁ ⬝ᵥ v₂) := by
|
||||
refine ext' ?_ ?_
|
||||
· funext i
|
||||
exact Empty.elim i
|
||||
· funext i
|
||||
rfl
|
||||
|
||||
lemma mul_ofMatUpDown_ofVecUp_eq_mulVec {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
|
||||
(v : Fin 1 ⊕ Fin d → ℝ) :
|
||||
congrSet ((Equiv.sumEmpty (Empty ⊕ PUnit.{1}) Empty).trans equivPUnitToSigma)
|
||||
(mul (unmarkFirst $ ofMatUpDown M) (ofVecUp v) rfl) = ofVecUp (M *ᵥ v) := by
|
||||
refine ext' ?_ ?_
|
||||
· funext i
|
||||
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, congrSet_apply_color, mul_color, Equiv.symm_symm]
|
||||
fin_cases i
|
||||
rfl
|
||||
· funext i
|
||||
rfl
|
||||
|
||||
lemma mul_ofMatDownUp_ofVecDown_eq_mulVec {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
|
||||
(v : Fin 1 ⊕ Fin d → ℝ) :
|
||||
congrSet ((Equiv.sumEmpty (Empty ⊕ PUnit.{1}) Empty).trans equivPUnitToSigma)
|
||||
(mul (unmarkFirst $ ofMatDownUp M) (ofVecDown v) rfl) = ofVecDown (M *ᵥ v) := by
|
||||
refine ext' ?_ ?_
|
||||
· funext i
|
||||
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, congrSet_apply_color, mul_color, Equiv.symm_symm]
|
||||
fin_cases i
|
||||
rfl
|
||||
· funext i
|
||||
rfl
|
||||
|
||||
/-!
|
||||
|
||||
## The Lorentz action on constructors
|
||||
|
||||
-/
|
||||
section lorentzAction
|
||||
variable {d : ℕ} {X : Type} [Fintype X] [DecidableEq X] (T : RealLorentzTensor d X) (c : X → Colors)
|
||||
variable (Λ Λ' : LorentzGroup d)
|
||||
|
||||
open Matrix
|
||||
|
||||
/-- The action of the Lorentz group on `ofReal v` is trivial. -/
|
||||
@[simp]
|
||||
lemma lorentzAction_ofReal (r : ℝ) : Λ • ofReal d r = ofReal d r := by
|
||||
refine ext' rfl ?_
|
||||
funext i
|
||||
erw [lorentzAction_smul_coord]
|
||||
simp only [Finset.univ_unique, Finset.univ_eq_empty, Finset.prod_empty, one_mul,
|
||||
Finset.sum_singleton, IndexValue]
|
||||
rfl
|
||||
|
||||
/-- The action of the Lorentz group on `ofVecUp v` is by vector multiplication. -/
|
||||
@[simp]
|
||||
lemma lorentzAction_ofVecUp (v : Fin 1 ⊕ Fin d → ℝ) :
|
||||
Λ • ofVecUp v = ofVecUp (Λ *ᵥ v) := by
|
||||
refine ext' rfl ?_
|
||||
funext i
|
||||
erw [lorentzAction_smul_coord]
|
||||
simp only [ofVecUp, IndexValue, Fin.isValue, Fintype.prod_sum_type, Finset.univ_eq_empty,
|
||||
Finset.prod_empty, one_mul]
|
||||
rw [mulVec]
|
||||
simp only [Fin.isValue, dotProduct, Finset.univ_unique, Fin.default_eq_zero,
|
||||
Finset.sum_singleton]
|
||||
erw [Finset.sum_equiv (ofVecUpIndexValue v)]
|
||||
intro i
|
||||
simp_all only [Finset.mem_univ, Fin.isValue, Equiv.coe_fn_mk]
|
||||
intro j _
|
||||
erw [Finset.prod_singleton]
|
||||
rfl
|
||||
|
||||
/-- The action of the Lorentz group on `ofVecDown v` is
|
||||
by vector multiplication of the transpose-inverse. -/
|
||||
@[simp]
|
||||
lemma lorentzAction_ofVecDown (v : Fin 1 ⊕ Fin d → ℝ) :
|
||||
Λ • ofVecDown v = ofVecDown ((LorentzGroup.transpose Λ⁻¹) *ᵥ v) := by
|
||||
refine ext' rfl ?_
|
||||
funext i
|
||||
erw [lorentzAction_smul_coord]
|
||||
simp only [ofVecDown, IndexValue, Fin.isValue, Fintype.prod_sum_type, Finset.univ_eq_empty,
|
||||
Finset.prod_empty, one_mul, lorentzGroupIsGroup_inv]
|
||||
rw [mulVec]
|
||||
simp only [Fin.isValue, dotProduct, Finset.univ_unique, Fin.default_eq_zero,
|
||||
Finset.sum_singleton]
|
||||
erw [Finset.sum_equiv (ofVecUpIndexValue v)]
|
||||
intro i
|
||||
simp_all only [Finset.mem_univ, Fin.isValue, Equiv.coe_fn_mk]
|
||||
intro j _
|
||||
erw [Finset.prod_singleton]
|
||||
rfl
|
||||
|
||||
@[simp]
|
||||
lemma lorentzAction_ofMatUpUp (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
|
||||
Λ • ofMatUpUp M = ofMatUpUp (Λ.1 * M * (LorentzGroup.transpose Λ).1) := by
|
||||
refine ext' rfl ?_
|
||||
funext i
|
||||
erw [lorentzAction_smul_coord]
|
||||
erw [← Equiv.sum_comp (ofMatUpUpIndexValue M).symm]
|
||||
simp only [ofMatUpUp, IndexValue, Fin.isValue, Fintype.prod_sum_type, Finset.univ_eq_empty,
|
||||
Finset.prod_empty, one_mul, mul_apply]
|
||||
erw [Finset.sum_product]
|
||||
rw [Finset.sum_comm]
|
||||
refine Finset.sum_congr rfl (fun x _ => ?_)
|
||||
rw [Finset.sum_mul]
|
||||
refine Finset.sum_congr rfl (fun y _ => ?_)
|
||||
erw [← Equiv.prod_comp (Equiv.sigmaPUnit (Fin 2)).symm]
|
||||
rw [Fin.prod_univ_two]
|
||||
simp only [colorMatrix, Fin.isValue, MonoidHom.coe_mk, OneHom.coe_mk]
|
||||
rw [mul_assoc, mul_comm _ (M _ _), ← mul_assoc]
|
||||
congr
|
||||
|
||||
@[simp]
|
||||
lemma lorentzAction_ofMatDownDown (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
|
||||
Λ • ofMatDownDown M = ofMatDownDown ((LorentzGroup.transpose Λ⁻¹).1 * M * (Λ⁻¹).1) := by
|
||||
refine ext' rfl ?_
|
||||
funext i
|
||||
erw [lorentzAction_smul_coord]
|
||||
erw [← Equiv.sum_comp (ofMatDownDownIndexValue M).symm]
|
||||
simp only [ofMatDownDown, IndexValue, Fin.isValue, Fintype.prod_sum_type, Finset.univ_eq_empty,
|
||||
Finset.prod_empty, one_mul, mul_apply]
|
||||
erw [Finset.sum_product]
|
||||
rw [Finset.sum_comm]
|
||||
refine Finset.sum_congr rfl (fun x _ => ?_)
|
||||
rw [Finset.sum_mul]
|
||||
refine Finset.sum_congr rfl (fun y _ => ?_)
|
||||
erw [← Equiv.prod_comp (Equiv.sigmaPUnit (Fin 2)).symm]
|
||||
rw [Fin.prod_univ_two]
|
||||
simp only [colorMatrix, Fin.isValue, MonoidHom.coe_mk, OneHom.coe_mk]
|
||||
rw [mul_assoc, mul_comm _ (M _ _), ← mul_assoc]
|
||||
congr
|
||||
|
||||
@[simp]
|
||||
lemma lorentzAction_ofMatUpDown (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
|
||||
Λ • ofMatUpDown M = ofMatUpDown (Λ.1 * M * (Λ⁻¹).1) := by
|
||||
refine ext' rfl ?_
|
||||
funext i
|
||||
erw [lorentzAction_smul_coord]
|
||||
erw [← Equiv.sum_comp (ofMatUpDownIndexValue M).symm]
|
||||
simp only [ofMatUpDown, IndexValue, Fin.isValue, Fintype.prod_sum_type, Finset.univ_eq_empty,
|
||||
Finset.prod_empty, one_mul, mul_apply]
|
||||
erw [Finset.sum_product]
|
||||
rw [Finset.sum_comm]
|
||||
refine Finset.sum_congr rfl (fun x _ => ?_)
|
||||
rw [Finset.sum_mul]
|
||||
refine Finset.sum_congr rfl (fun y _ => ?_)
|
||||
erw [← Equiv.prod_comp (Equiv.sigmaPUnit (Fin 2)).symm]
|
||||
rw [Fin.prod_univ_two]
|
||||
simp only [colorMatrix, Fin.isValue, MonoidHom.coe_mk, OneHom.coe_mk]
|
||||
rw [mul_assoc, mul_comm _ (M _ _), ← mul_assoc]
|
||||
congr
|
||||
|
||||
@[simp]
|
||||
lemma lorentzAction_ofMatDownUp (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
|
||||
Λ • ofMatDownUp M =
|
||||
ofMatDownUp ((LorentzGroup.transpose Λ⁻¹).1 * M * (LorentzGroup.transpose Λ).1) := by
|
||||
refine ext' rfl ?_
|
||||
funext i
|
||||
erw [lorentzAction_smul_coord]
|
||||
erw [← Equiv.sum_comp (ofMatDownUpIndexValue M).symm]
|
||||
simp only [ofMatDownUp, IndexValue, Fin.isValue, Fintype.prod_sum_type, Finset.univ_eq_empty,
|
||||
Finset.prod_empty, one_mul, mul_apply]
|
||||
erw [Finset.sum_product]
|
||||
rw [Finset.sum_comm]
|
||||
refine Finset.sum_congr rfl (fun x _ => ?_)
|
||||
rw [Finset.sum_mul]
|
||||
refine Finset.sum_congr rfl (fun y _ => ?_)
|
||||
erw [← Equiv.prod_comp (Equiv.sigmaPUnit (Fin 2)).symm]
|
||||
rw [Fin.prod_univ_two]
|
||||
simp only [colorMatrix, Fin.isValue, MonoidHom.coe_mk, OneHom.coe_mk]
|
||||
rw [mul_assoc, mul_comm _ (M _ _), ← mul_assoc]
|
||||
congr
|
||||
|
||||
end lorentzAction
|
||||
|
||||
end RealLorentzTensor
|
||||
|
|
@ -28,7 +28,7 @@ variable {μ : Colors}
|
|||
/-- Monoid homomorphism from the Lorentz group to matrices indexed by `ColorsIndex d μ` for a
|
||||
color `μ`.
|
||||
|
||||
Thought of as the representation of the Lorentz group for that color index. -/
|
||||
This can be thought of as the representation of the Lorentz group for that color index. -/
|
||||
def colorMatrix (μ : Colors) : LorentzGroup d →* Matrix (ColorsIndex d μ) (ColorsIndex d μ) ℝ where
|
||||
toFun Λ := match μ with
|
||||
| .up => fun i j => Λ.1 i j
|
||||
|
|
@ -56,21 +56,23 @@ def colorMatrix (μ : Colors) : LorentzGroup d →* Matrix (ColorsIndex d μ) (C
|
|||
Matrix.transpose_mul, Matrix.transpose_apply]
|
||||
rfl
|
||||
|
||||
/-- A real number which occurs in the definition of -/
|
||||
/-- A real number occuring in the action of the Lorentz group on Lorentz tensors. -/
|
||||
@[simp]
|
||||
def prodColorMatrixOnIndexValue (i j : IndexValue d c) : ℝ :=
|
||||
∏ x, colorMatrix (c x) Λ (i x) (j x)
|
||||
|
||||
/-- `prodColorMatrixOnIndexValue` evaluated at `1` on the diagonal returns `1`. -/
|
||||
lemma one_prodColorMatrixOnIndexValue_on_diag (i : IndexValue d c) :
|
||||
prodColorMatrixOnIndexValue c 1 i i = 1 := by
|
||||
prodColorMatrixOnIndexValue c 1 i i = 1 := by
|
||||
simp only [prodColorMatrixOnIndexValue]
|
||||
rw [Finset.prod_eq_one]
|
||||
intro x _
|
||||
simp only [colorMatrix, MonoidHom.map_one, Matrix.one_apply]
|
||||
rfl
|
||||
|
||||
/-- `prodColorMatrixOnIndexValue` evaluated at `1` off the diagonal returns `0`. -/
|
||||
lemma one_prodColorMatrixOnIndexValue_off_diag {i j : IndexValue d c} (hij : j ≠ i) :
|
||||
prodColorMatrixOnIndexValue c 1 i j = 0 := by
|
||||
prodColorMatrixOnIndexValue c 1 i j = 0 := by
|
||||
simp only [prodColorMatrixOnIndexValue]
|
||||
obtain ⟨x, hijx⟩ := Function.ne_iff.mp hij
|
||||
rw [@Finset.prod_eq_zero _ _ _ _ _ x]
|
||||
|
|
@ -82,8 +84,8 @@ lemma mul_prodColorMatrixOnIndexValue (i j : IndexValue d c) :
|
|||
prodColorMatrixOnIndexValue c (Λ * Λ') i j =
|
||||
∑ (k : IndexValue d c),
|
||||
∏ x, (colorMatrix (c x) Λ (i x) (k x)) * (colorMatrix (c x) Λ' (k x) (j x)) := by
|
||||
have h1 : ∑ (k : IndexValue d c), ∏ x,
|
||||
(colorMatrix (c x) Λ (i x) (k x)) * (colorMatrix (c x) Λ' (k x) (j x)) =
|
||||
have h1 : ∑ (k : IndexValue d c), ∏ x,
|
||||
(colorMatrix (c x) Λ (i x) (k x)) * (colorMatrix (c x) Λ' (k x) (j x)) =
|
||||
∏ x, ∑ y, (colorMatrix (c x) Λ (i x) y) * (colorMatrix (c x) Λ' y (j x)) := by
|
||||
rw [Finset.prod_sum]
|
||||
simp only [Finset.prod_attach_univ, Finset.sum_univ_pi]
|
||||
|
|
@ -95,8 +97,9 @@ lemma mul_prodColorMatrixOnIndexValue (i j : IndexValue d c) :
|
|||
simp only [prodColorMatrixOnIndexValue, map_mul]
|
||||
exact Finset.prod_congr rfl (fun x _ => rfl)
|
||||
|
||||
/-- Action of the Lorentz group on the set of Real Lorentz Tensors indexed by `X`. -/
|
||||
def lorentzAction : MulAction (LorentzGroup d) (RealLorentzTensor d X) where
|
||||
/-- Action of the Lorentz group on `X`-indexed Real Lorentz Tensors. -/
|
||||
@[simps!]
|
||||
instance lorentzAction : MulAction (LorentzGroup d) (RealLorentzTensor d X) where
|
||||
smul Λ T := {color := T.color,
|
||||
coord := fun i => ∑ j, prodColorMatrixOnIndexValue T.color Λ i j * T.coord j}
|
||||
one_smul T := by
|
||||
|
|
@ -133,20 +136,7 @@ def lorentzAction : MulAction (LorentzGroup d) (RealLorentzTensor d X) where
|
|||
rw [Finset.prod_mul_distrib]
|
||||
rfl
|
||||
|
||||
/-!
|
||||
|
||||
## The Lorentz action on constructors
|
||||
|
||||
-/
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
@[simps!]
|
||||
instance : MulAction (LorentzGroup d) (Marked d X n) := lorentzAction
|
||||
|
||||
end RealLorentzTensor
|
||||
|
|
|
|||
1
scripts/nolints.json
Normal file
1
scripts/nolints.json
Normal file
|
|
@ -0,0 +1 @@
|
|||
[]
|
||||
Loading…
Add table
Add a link
Reference in a new issue