444 lines
18 KiB
Text
444 lines
18 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.SpaceTime.LorentzTensor.Real.Basic
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import HepLean.SpaceTime.LorentzGroup.Basic
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/-!
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# Lorentz group action on Real Lorentz Tensors
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We define the action of the Lorentz group on Real Lorentz Tensors.
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The Lorentz action is currently only defined for finite and decidable types `X`.
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-/
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namespace RealLorentzTensor
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variable {d : ℕ} {X Y : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
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(T : RealLorentzTensor d X) (c : X → Colors) (Λ Λ' : LorentzGroup d) {μ : Colors}
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open LorentzGroup BigOperators Marked
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/-- Monoid homomorphism from the Lorentz group to matrices indexed by `ColorsIndex d μ` for a
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color `μ`.
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This can be thought of as the representation of the Lorentz group for that color index. -/
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def colorMatrix (μ : Colors) : LorentzGroup d →* Matrix (ColorsIndex d μ) (ColorsIndex d μ) ℝ where
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toFun Λ := match μ with
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| .up => fun i j => Λ.1 i j
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| .down => fun i j => (LorentzGroup.transpose Λ⁻¹).1 i j
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map_one' := by
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match μ with
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| .up =>
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simp only [lorentzGroupIsGroup_one_coe]
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ext i j
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simp only [OfNat.ofNat, One.one, ColorsIndex]
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congr
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| .down =>
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simp only [transpose, inv_one, lorentzGroupIsGroup_one_coe, Matrix.transpose_one]
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ext i j
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simp only [OfNat.ofNat, One.one, ColorsIndex]
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congr
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map_mul' Λ Λ' := by
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match μ with
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| .up =>
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ext i j
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simp only [lorentzGroupIsGroup_mul_coe]
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| .down =>
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ext i j
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simp only [transpose, mul_inv_rev, lorentzGroupIsGroup_inv, lorentzGroupIsGroup_mul_coe,
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Matrix.transpose_mul, Matrix.transpose_apply]
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rfl
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lemma colorMatrix_cast {μ ν : Colors} (h : μ = ν) (Λ : LorentzGroup d) :
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colorMatrix μ Λ =
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Matrix.reindex (colorsIndexCast h).symm (colorsIndexCast h).symm (colorMatrix ν Λ) := by
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subst h
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rfl
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lemma colorMatrix_dual_cast {μ : Colors} (Λ : LorentzGroup d) :
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colorMatrix (τ μ) Λ = Matrix.reindex (colorsIndexDualCastSelf) (colorsIndexDualCastSelf)
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(colorMatrix μ (LorentzGroup.transpose Λ⁻¹)) := by
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match μ with
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| .up => rfl
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| .down =>
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ext i j
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simp only [τ, colorMatrix, MonoidHom.coe_mk, OneHom.coe_mk, colorsIndexDualCastSelf, transpose,
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lorentzGroupIsGroup_inv, Matrix.transpose_apply, minkowskiMetric.dual_transpose,
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minkowskiMetric.dual_dual, Matrix.reindex_apply, Equiv.coe_fn_symm_mk, Matrix.submatrix_apply]
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lemma colorMatrix_transpose {μ : Colors} (Λ : LorentzGroup d) :
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colorMatrix μ (LorentzGroup.transpose Λ) = (colorMatrix μ Λ).transpose := by
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match μ with
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| .up => rfl
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| .down =>
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ext i j
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simp only [colorMatrix, transpose, lorentzGroupIsGroup_inv, Matrix.transpose_apply,
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MonoidHom.coe_mk, OneHom.coe_mk, minkowskiMetric.dual_transpose]
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/-!
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## Lorentz group to tensor representation matrices.
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-/
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/-- The matrix representation of the Lorentz group given a color of index. -/
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@[simps!]
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def toTensorRepMat {c : X → Colors} :
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LorentzGroup d →* Matrix (IndexValue d c) (IndexValue d c) ℝ where
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toFun Λ := fun i j => ∏ x, colorMatrix (c x) Λ (i x) (j x)
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map_one' := by
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ext i j
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by_cases hij : i = j
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· subst hij
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simp only [map_one, Matrix.one_apply_eq, Finset.prod_const_one]
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· obtain ⟨x, hijx⟩ := Function.ne_iff.mp hij
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simp only [map_one]
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rw [@Finset.prod_eq_zero _ _ _ _ _ x]
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exact Eq.symm (Matrix.one_apply_ne' fun a => hij (id (Eq.symm a)))
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exact Finset.mem_univ x
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exact Matrix.one_apply_ne' (id (Ne.symm hijx))
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map_mul' Λ Λ' := by
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ext i j
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rw [Matrix.mul_apply]
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trans ∑ (k : IndexValue d c), ∏ x,
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(colorMatrix (c x) Λ (i x) (k x)) * (colorMatrix (c x) Λ' (k x) (j x))
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have h1 : ∑ (k : IndexValue d c), ∏ x,
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(colorMatrix (c x) Λ (i x) (k x)) * (colorMatrix (c x) Λ' (k x) (j x)) =
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∏ x, ∑ y, (colorMatrix (c x) Λ (i x) y) * (colorMatrix (c x) Λ' y (j x)) := by
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rw [Finset.prod_sum]
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simp only [Finset.prod_attach_univ, Finset.sum_univ_pi]
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rfl
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rw [h1]
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simp only [map_mul]
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rfl
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refine Finset.sum_congr rfl (fun k _ => ?_)
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rw [Finset.prod_mul_distrib]
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lemma toTensorRepMat_mul' (i j : IndexValue d c) :
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toTensorRepMat (Λ * Λ') i j = ∑ (k : IndexValue d c),
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∏ x, colorMatrix (c x) Λ (i x) (k x) * colorMatrix (c x) Λ' (k x) (j x) := by
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simp [Matrix.mul_apply, IndexValue]
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refine Finset.sum_congr rfl (fun k _ => ?_)
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rw [Finset.prod_mul_distrib]
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rfl
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lemma toTensorRepMat_of_indexValueSumEquiv {cX : X → Colors} {cY : Y → Colors}
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(i j : IndexValue d (Sum.elim cX cY)) :
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toTensorRepMat Λ i j = toTensorRepMat Λ (indexValueSumEquiv i).1 (indexValueSumEquiv j).1 *
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toTensorRepMat Λ (indexValueSumEquiv i).2 (indexValueSumEquiv j).2 :=
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Fintype.prod_sum_type fun x => (colorMatrix (Sum.elim cX cY x)) Λ (i x) (j x)
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lemma toTensorRepMat_of_indexValueSumEquiv' {cX : X → Colors} {cY : Y → Colors}
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(i j : IndexValue d cX) (k l : IndexValue d cY) :
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toTensorRepMat Λ i j * toTensorRepMat Λ k l =
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toTensorRepMat Λ (indexValueSumEquiv.symm (i, k)) (indexValueSumEquiv.symm (j, l)) :=
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(Fintype.prod_sum_type fun x => (colorMatrix (Sum.elim cX cY x)) Λ
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(indexValueSumEquiv.symm (i, k) x) (indexValueSumEquiv.symm (j, l) x)).symm
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/-!
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## Tensor representation matrices and marked tensors.
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-/
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lemma toTensorRepMat_of_splitIndexValue' (T : Marked d X n)
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(i j : T.UnmarkedIndexValue) (k l : T.MarkedIndexValue) :
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toTensorRepMat Λ i j * toTensorRepMat Λ k l =
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toTensorRepMat Λ (splitIndexValue.symm (i, k)) (splitIndexValue.symm (j, l)) :=
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(Fintype.prod_sum_type fun x =>
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(colorMatrix (T.color x)) Λ (splitIndexValue.symm (i, k) x) (splitIndexValue.symm (j, l) x)).symm
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lemma toTensorRepMat_oneMarkedIndexValue_dual (T : Marked d X 1) (S : Marked d Y 1)
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(h : T.markedColor 0 = τ (S.markedColor 0)) (x : ColorsIndex d (T.markedColor 0))
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(k : S.MarkedIndexValue) :
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toTensorRepMat Λ (oneMarkedIndexValue $ colorsIndexDualCast h x) k =
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toTensorRepMat Λ⁻¹ (oneMarkedIndexValue
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$ (colorsIndexDualCast h).symm $ oneMarkedIndexValue.symm k)
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(oneMarkedIndexValue x) := by
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rw [toTensorRepMat_apply, toTensorRepMat_apply]
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erw [Finset.prod_singleton, Finset.prod_singleton]
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simp only [Fin.zero_eta, Fin.isValue, lorentzGroupIsGroup_inv]
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rw [colorMatrix_cast h, colorMatrix_dual_cast]
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rw [Matrix.reindex_apply, Matrix.reindex_apply]
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simp only [Fin.isValue, lorentzGroupIsGroup_inv, minkowskiMetric.dual_dual, Subtype.coe_eta,
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Equiv.symm_symm, Matrix.submatrix_apply]
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rw [colorMatrix_transpose]
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simp only [Fin.isValue, Matrix.transpose_apply]
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apply congrArg
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simp only [Fin.isValue, oneMarkedIndexValue, colorsIndexDualCast, Equiv.coe_fn_symm_mk,
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Equiv.symm_trans_apply, Equiv.symm_symm, Equiv.coe_fn_mk, Equiv.apply_symm_apply,
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Equiv.symm_apply_apply]
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lemma toTensorRepMap_sum_dual (T : Marked d X 1) (S : Marked d Y 1)
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(h : T.markedColor 0 = τ (S.markedColor 0)) (j : T.MarkedIndexValue) (k : S.MarkedIndexValue) :
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∑ x, toTensorRepMat Λ (oneMarkedIndexValue $ colorsIndexDualCast h x) k
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* toTensorRepMat Λ (oneMarkedIndexValue x) j =
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toTensorRepMat 1
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(oneMarkedIndexValue $ (colorsIndexDualCast h).symm $ oneMarkedIndexValue.symm k) j := by
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trans ∑ x, toTensorRepMat Λ⁻¹ (oneMarkedIndexValue$ (colorsIndexDualCast h).symm $
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oneMarkedIndexValue.symm k) (oneMarkedIndexValue x) * toTensorRepMat Λ (oneMarkedIndexValue x) j
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apply Finset.sum_congr rfl (fun x _ => ?_)
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rw [toTensorRepMat_oneMarkedIndexValue_dual]
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rw [← Equiv.sum_comp oneMarkedIndexValue.symm]
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change ∑ i, toTensorRepMat Λ⁻¹ (oneMarkedIndexValue $ (colorsIndexDualCast h).symm $
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oneMarkedIndexValue.symm k) i * toTensorRepMat Λ i j = _
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rw [← Matrix.mul_apply, ← toTensorRepMat.map_mul, inv_mul_self Λ]
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lemma toTensorRepMat_one_coord_sum (T : Marked d X n) (i : T.UnmarkedIndexValue)
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(k : T.MarkedIndexValue) : T.coord (splitIndexValue.symm (i, k)) = ∑ j, toTensorRepMat 1 k j *
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T.coord (splitIndexValue.symm (i, j)) := by
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erw [Finset.sum_eq_single_of_mem k]
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simp only [IndexValue, map_one, Matrix.one_apply_eq, one_mul]
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exact Finset.mem_univ k
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intro j _ hjk
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simp [hjk, IndexValue]
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exact Or.inl (Matrix.one_apply_ne' hjk)
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/-!
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## Definition of the Lorentz group action on Real Lorentz Tensors.
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-/
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/-- Action of the Lorentz group on `X`-indexed Real Lorentz Tensors. -/
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@[simps!]
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instance lorentzAction : MulAction (LorentzGroup d) (RealLorentzTensor d X) where
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smul Λ T := {color := T.color,
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coord := fun i => ∑ j, toTensorRepMat Λ i j * T.coord j}
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one_smul T := by
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refine ext rfl ?_
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funext i
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simp only [HSMul.hSMul, map_one]
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erw [Finset.sum_eq_single_of_mem i]
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simp only [Matrix.one_apply_eq, one_mul, IndexValue]
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rfl
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exact Finset.mem_univ i
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exact fun j _ hij => mul_eq_zero.mpr (Or.inl (Matrix.one_apply_ne' hij))
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mul_smul Λ Λ' T := by
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refine ext rfl ?_
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simp only [HSMul.hSMul]
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funext i
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have h1 : ∑ j : IndexValue d T.color, toTensorRepMat (Λ * Λ') i j
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* T.coord j = ∑ j : IndexValue d T.color, ∑ (k : IndexValue d T.color),
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(∏ x, ((colorMatrix (T.color x) Λ (i x) (k x)) *
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(colorMatrix (T.color x) Λ' (k x) (j x)))) * T.coord j := by
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refine Finset.sum_congr rfl (fun j _ => ?_)
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rw [toTensorRepMat_mul', Finset.sum_mul]
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rw [h1]
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rw [Finset.sum_comm]
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refine Finset.sum_congr rfl (fun j _ => ?_)
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rw [Finset.mul_sum]
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refine Finset.sum_congr rfl (fun k _ => ?_)
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simp only [toTensorRepMat, IndexValue]
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rw [← mul_assoc]
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congr
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rw [Finset.prod_mul_distrib]
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rfl
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lemma lorentzAction_smul_coord' {d : ℕ} {X : Type} [Fintype X] [DecidableEq X] (Λ : ↑(𝓛 d))
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(T : RealLorentzTensor d X) (i : IndexValue d T.color) :
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(Λ • T).coord i = ∑ j : IndexValue d T.color, toTensorRepMat Λ i j * T.coord j := by
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rfl
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/-!
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## Properties of the Lorentz action.
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-/
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/-- The action on an empty Lorentz tensor is trivial. -/
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lemma lorentzAction_on_isEmpty [IsEmpty X] (Λ : LorentzGroup d) (T : RealLorentzTensor d X) :
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Λ • T = T := by
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refine ext rfl ?_
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funext i
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erw [lorentzAction_smul_coord]
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simp only [Finset.univ_unique, Finset.univ_eq_empty, Finset.prod_empty, one_mul,
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Finset.sum_singleton, toTensorRepMat_apply]
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simp only [IndexValue, Unique.eq_default, Finset.univ_unique, Finset.sum_const,
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Finset.card_singleton, one_smul]
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/-- The Lorentz action commutes with `mapIso`. -/
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lemma lorentzAction_mapIso (f : X ≃ Y) (Λ : LorentzGroup d) (T : RealLorentzTensor d X) :
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mapIso d f (Λ • T) = Λ • (mapIso d f T) := by
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refine ext rfl ?_
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funext i
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rw [mapIso_apply_coord]
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rw [lorentzAction_smul_coord', lorentzAction_smul_coord']
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let is : IndexValue d T.color ≃ IndexValue d ((mapIso d f) T).color :=
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indexValueIso d f ((Equiv.comp_symm_eq f ((mapIso d f) T).color T.color).mp rfl)
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rw [← Equiv.sum_comp is]
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refine Finset.sum_congr rfl (fun j _ => ?_)
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rw [mapIso_apply_coord]
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refine Mathlib.Tactic.Ring.mul_congr ?_ ?_ rfl
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· simp only [IndexValue, toTensorRepMat, MonoidHom.coe_mk, OneHom.coe_mk, mapIso_apply_color,
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indexValueIso_refl]
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rw [← Equiv.prod_comp f]
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apply Finset.prod_congr rfl (fun x _ => ?_)
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have h1 : (T.color (f.symm (f x))) = T.color x := by
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simp only [Equiv.symm_apply_apply]
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rw [colorMatrix_cast h1]
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apply congrArg
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simp only [is]
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erw [indexValueIso_eq_symm, indexValueIso_symm_apply']
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simp only [colorsIndexCast, Function.comp_apply, mapIso_apply_color, Equiv.cast_refl,
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Equiv.refl_symm, Equiv.refl_apply, Equiv.cast_apply]
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symm
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refine cast_eq_iff_heq.mpr ?_
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congr
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exact Equiv.symm_apply_apply f x
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· apply congrArg
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exact (Equiv.apply_eq_iff_eq_symm_apply (indexValueIso d f (mapIso.proof_1 d f T))).mp rfl
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/-!
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## The Lorentz action on marked tensors.
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-/
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@[simps!]
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instance : MulAction (LorentzGroup d) (Marked d X n) := lorentzAction
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/-- Action of the Lorentz group on just marked indices. -/
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@[simps!]
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def markedLorentzAction : MulAction (LorentzGroup d) (Marked d X n) where
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smul Λ T := {
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color := T.color,
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coord := fun i => ∑ j, toTensorRepMat Λ (splitIndexValue i).2 j *
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T.coord (splitIndexValue.symm ((splitIndexValue i).1, j))}
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one_smul T := by
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refine ext rfl ?_
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funext i
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simp only [HSMul.hSMul, map_one]
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erw [Finset.sum_eq_single_of_mem (splitIndexValue i).2]
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erw [Matrix.one_apply_eq (splitIndexValue i).2]
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simp only [IndexValue, one_mul, indexValueIso_refl, Equiv.refl_apply]
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apply congrArg
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exact Equiv.symm_apply_apply splitIndexValue i
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exact Finset.mem_univ (splitIndexValue i).2
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exact fun j _ hij => mul_eq_zero.mpr (Or.inl (Matrix.one_apply_ne' hij))
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mul_smul Λ Λ' T := by
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refine ext rfl ?_
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simp only [HSMul.hSMul]
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funext i
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have h1 : ∑ (j : T.MarkedIndexValue), toTensorRepMat (Λ * Λ') (splitIndexValue i).2 j
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* T.coord (splitIndexValue.symm ((splitIndexValue i).1, j)) =
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∑ (j : T.MarkedIndexValue), ∑ (k : T.MarkedIndexValue),
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(∏ x, ((colorMatrix (T.markedColor x) Λ ((splitIndexValue i).2 x) (k x)) *
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(colorMatrix (T.markedColor x) Λ' (k x) (j x)))) *
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T.coord (splitIndexValue.symm ((splitIndexValue i).1, j)) := by
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refine Finset.sum_congr rfl (fun j _ => ?_)
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rw [toTensorRepMat_mul', Finset.sum_mul]
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rfl
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erw [h1]
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rw [Finset.sum_comm]
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refine Finset.sum_congr rfl (fun j _ => ?_)
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rw [Finset.mul_sum]
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refine Finset.sum_congr rfl (fun k _ => ?_)
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simp only [toTensorRepMat, IndexValue]
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rw [← mul_assoc]
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congr
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rw [Finset.prod_mul_distrib]
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rfl
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/-- Action of the Lorentz group on just unmarked indices. -/
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@[simps!]
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def unmarkedLorentzAction : MulAction (LorentzGroup d) (Marked d X n) where
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smul Λ T := {
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color := T.color,
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coord := fun i => ∑ j, toTensorRepMat Λ (splitIndexValue i).1 j *
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T.coord (splitIndexValue.symm (j, (splitIndexValue i).2))}
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one_smul T := by
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refine ext rfl ?_
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funext i
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simp only [HSMul.hSMul, map_one]
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erw [Finset.sum_eq_single_of_mem (splitIndexValue i).1]
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erw [Matrix.one_apply_eq (splitIndexValue i).1]
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simp only [IndexValue, one_mul, indexValueIso_refl, Equiv.refl_apply]
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apply congrArg
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exact Equiv.symm_apply_apply splitIndexValue i
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exact Finset.mem_univ (splitIndexValue i).1
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exact fun j _ hij => mul_eq_zero.mpr (Or.inl (Matrix.one_apply_ne' hij))
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mul_smul Λ Λ' T := by
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refine ext rfl ?_
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simp only [HSMul.hSMul]
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funext i
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have h1 : ∑ (j : T.UnmarkedIndexValue), toTensorRepMat (Λ * Λ') (splitIndexValue i).1 j
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* T.coord (splitIndexValue.symm (j, (splitIndexValue i).2)) =
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∑ (j : T.UnmarkedIndexValue), ∑ (k : T.UnmarkedIndexValue),
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(∏ x, ((colorMatrix (T.unmarkedColor x) Λ ((splitIndexValue i).1 x) (k x)) *
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(colorMatrix (T.unmarkedColor x) Λ' (k x) (j x)))) *
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T.coord (splitIndexValue.symm (j, (splitIndexValue i).2)) := by
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refine Finset.sum_congr rfl (fun j _ => ?_)
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rw [toTensorRepMat_mul', Finset.sum_mul]
|
||
rfl
|
||
erw [h1]
|
||
rw [Finset.sum_comm]
|
||
refine Finset.sum_congr rfl (fun j _ => ?_)
|
||
rw [Finset.mul_sum]
|
||
refine Finset.sum_congr rfl (fun k _ => ?_)
|
||
simp only [toTensorRepMat, IndexValue]
|
||
rw [← mul_assoc]
|
||
congr
|
||
rw [Finset.prod_mul_distrib]
|
||
rfl
|
||
|
||
/-- Notation for `markedLorentzAction.smul`. -/
|
||
scoped[RealLorentzTensor] infixr:73 " •ₘ " => markedLorentzAction.smul
|
||
|
||
/-- Notation for `unmarkedLorentzAction.smul`. -/
|
||
scoped[RealLorentzTensor] infixr:73 " •ᵤₘ " => unmarkedLorentzAction.smul
|
||
|
||
/-- Acting on unmarked and then marked indices is equivalent to acting on all indices. -/
|
||
lemma marked_unmarked_action_eq_action (T : Marked d X n) : Λ •ₘ (Λ •ᵤₘ T) = Λ • T := by
|
||
refine ext rfl ?_
|
||
funext i
|
||
change ∑ j, toTensorRepMat Λ (splitIndexValue i).2 j *
|
||
(∑ k, toTensorRepMat Λ (splitIndexValue i).1 k * T.coord (splitIndexValue.symm (k, j))) = _
|
||
trans ∑ j, ∑ k, (toTensorRepMat Λ (splitIndexValue i).2 j *
|
||
toTensorRepMat Λ (splitIndexValue i).1 k) * T.coord (splitIndexValue.symm (k, j))
|
||
apply Finset.sum_congr rfl (fun j _ => ?_)
|
||
rw [Finset.mul_sum]
|
||
apply Finset.sum_congr rfl (fun k _ => ?_)
|
||
exact Eq.symm (mul_assoc _ _ _)
|
||
trans ∑ j, ∑ k, (toTensorRepMat Λ i (splitIndexValue.symm (k, j))
|
||
* T.coord (splitIndexValue.symm (k, j)))
|
||
apply Finset.sum_congr rfl (fun j _ => (Finset.sum_congr rfl (fun k _ => ?_)))
|
||
rw [mul_comm (toTensorRepMat _ _ _), toTensorRepMat_of_splitIndexValue']
|
||
simp only [IndexValue, Finset.mem_univ, Prod.mk.eta, Equiv.symm_apply_apply]
|
||
trans ∑ p, (toTensorRepMat Λ i p * T.coord p)
|
||
rw [← Equiv.sum_comp splitIndexValue.symm, Fintype.sum_prod_type, Finset.sum_comm]
|
||
rfl
|
||
rfl
|
||
|
||
/-- Acting on marked and then unmarked indices is equivalent to acting on all indices. -/
|
||
lemma unmarked_marked_action_eq_action (T : Marked d X n) : Λ •ᵤₘ (Λ •ₘ T) = Λ • T := by
|
||
refine ext rfl ?_
|
||
funext i
|
||
change ∑ j, toTensorRepMat Λ (splitIndexValue i).1 j *
|
||
(∑ k, toTensorRepMat Λ (splitIndexValue i).2 k * T.coord (splitIndexValue.symm (j, k))) = _
|
||
trans ∑ j, ∑ k, (toTensorRepMat Λ (splitIndexValue i).1 j *
|
||
toTensorRepMat Λ (splitIndexValue i).2 k) * T.coord (splitIndexValue.symm (j, k))
|
||
apply Finset.sum_congr rfl (fun j _ => ?_)
|
||
rw [Finset.mul_sum]
|
||
apply Finset.sum_congr rfl (fun k _ => ?_)
|
||
exact Eq.symm (mul_assoc _ _ _)
|
||
trans ∑ j, ∑ k, (toTensorRepMat Λ i (splitIndexValue.symm (j, k))
|
||
* T.coord (splitIndexValue.symm (j, k)))
|
||
apply Finset.sum_congr rfl (fun j _ => (Finset.sum_congr rfl (fun k _ => ?_)))
|
||
rw [toTensorRepMat_of_splitIndexValue']
|
||
simp only [IndexValue, Finset.mem_univ, Prod.mk.eta, Equiv.symm_apply_apply]
|
||
trans ∑ p, (toTensorRepMat Λ i p * T.coord p)
|
||
rw [← Equiv.sum_comp splitIndexValue.symm, Fintype.sum_prod_type]
|
||
rfl
|
||
rfl
|
||
|
||
/-- The marked and unmarked actions commute. -/
|
||
lemma marked_unmarked_action_comm (T : Marked d X n) : Λ •ᵤₘ (Λ •ₘ T) = Λ •ₘ (Λ •ᵤₘ T) := by
|
||
rw [unmarked_marked_action_eq_action, marked_unmarked_action_eq_action]
|
||
|
||
/-! TODO: Show that the Lorentz action commutes with contraction. -/
|
||
/-! TODO: Show that the Lorentz action commutes with rising and lowering indices. -/
|
||
end RealLorentzTensor
|