2024-07-15 16:57:06 -04:00
|
|
|
|
/-
|
|
|
|
|
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
|
|
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
|
|
|
Authors: Joseph Tooby-Smith
|
|
|
|
|
-/
|
|
|
|
|
import HepLean.SpaceTime.LorentzTensor.Real.Basic
|
2024-07-16 09:45:03 -04:00
|
|
|
|
import HepLean.SpaceTime.LorentzGroup.Basic
|
2024-07-15 16:57:06 -04:00
|
|
|
|
/-!
|
|
|
|
|
|
|
|
|
|
# Lorentz group action on Real Lorentz Tensors
|
|
|
|
|
|
|
|
|
|
We define the action of the Lorentz group on Real Lorentz Tensors.
|
|
|
|
|
|
2024-07-16 09:45:03 -04:00
|
|
|
|
The Lorentz action is currently only defined for finite and decidable types `X`.
|
|
|
|
|
|
2024-07-15 16:57:06 -04:00
|
|
|
|
-/
|
|
|
|
|
|
|
|
|
|
namespace RealLorentzTensor
|
|
|
|
|
|
2024-07-16 16:58:42 -04:00
|
|
|
|
variable {d : ℕ} {X Y : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
|
2024-07-17 13:53:36 -04:00
|
|
|
|
(T : RealLorentzTensor d X) (c : X → Colors) (Λ Λ' : LorentzGroup d) {μ : Colors}
|
2024-07-16 09:45:03 -04:00
|
|
|
|
|
2024-07-17 13:53:36 -04:00
|
|
|
|
open LorentzGroup BigOperators Marked
|
2024-07-16 09:45:03 -04:00
|
|
|
|
|
|
|
|
|
/-- Monoid homomorphism from the Lorentz group to matrices indexed by `ColorsIndex d μ` for a
|
|
|
|
|
color `μ`.
|
|
|
|
|
|
2024-07-16 11:40:00 -04:00
|
|
|
|
This can be thought of as the representation of the Lorentz group for that color index. -/
|
2024-07-16 09:45:03 -04:00
|
|
|
|
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
|
|
|
|
|
| .down => fun i j => (LorentzGroup.transpose Λ⁻¹).1 i j
|
|
|
|
|
map_one' := by
|
|
|
|
|
match μ with
|
|
|
|
|
| .up =>
|
|
|
|
|
simp only [lorentzGroupIsGroup_one_coe]
|
|
|
|
|
ext i j
|
|
|
|
|
simp only [OfNat.ofNat, One.one, ColorsIndex]
|
|
|
|
|
congr
|
|
|
|
|
| .down =>
|
|
|
|
|
simp only [transpose, inv_one, lorentzGroupIsGroup_one_coe, Matrix.transpose_one]
|
|
|
|
|
ext i j
|
|
|
|
|
simp only [OfNat.ofNat, One.one, ColorsIndex]
|
|
|
|
|
congr
|
|
|
|
|
map_mul' Λ Λ' := by
|
|
|
|
|
match μ with
|
|
|
|
|
| .up =>
|
|
|
|
|
ext i j
|
|
|
|
|
simp only [lorentzGroupIsGroup_mul_coe]
|
|
|
|
|
| .down =>
|
|
|
|
|
ext i j
|
|
|
|
|
simp only [transpose, mul_inv_rev, lorentzGroupIsGroup_inv, lorentzGroupIsGroup_mul_coe,
|
|
|
|
|
Matrix.transpose_mul, Matrix.transpose_apply]
|
|
|
|
|
rfl
|
|
|
|
|
|
2024-07-16 16:58:42 -04:00
|
|
|
|
lemma colorMatrix_cast {μ ν : Colors} (h : μ = ν) (Λ : LorentzGroup d) :
|
|
|
|
|
colorMatrix μ Λ =
|
2024-07-17 13:53:36 -04:00
|
|
|
|
Matrix.reindex (colorsIndexCast h).symm (colorsIndexCast h).symm (colorMatrix ν Λ) := by
|
2024-07-16 16:58:42 -04:00
|
|
|
|
subst h
|
|
|
|
|
rfl
|
|
|
|
|
|
2024-07-17 13:53:36 -04:00
|
|
|
|
lemma colorMatrix_dual_cast {μ : Colors} (Λ : LorentzGroup d) :
|
|
|
|
|
colorMatrix (τ μ) Λ = Matrix.reindex (colorsIndexDualCastSelf) (colorsIndexDualCastSelf)
|
|
|
|
|
(colorMatrix μ (LorentzGroup.transpose Λ⁻¹)) := by
|
|
|
|
|
match μ with
|
|
|
|
|
| .up => rfl
|
|
|
|
|
| .down =>
|
|
|
|
|
ext i j
|
|
|
|
|
simp only [τ, colorMatrix, MonoidHom.coe_mk, OneHom.coe_mk, colorsIndexDualCastSelf, transpose,
|
|
|
|
|
lorentzGroupIsGroup_inv, Matrix.transpose_apply, minkowskiMetric.dual_transpose,
|
|
|
|
|
minkowskiMetric.dual_dual, Matrix.reindex_apply, Equiv.coe_fn_symm_mk, Matrix.submatrix_apply]
|
|
|
|
|
lemma colorMatrix_transpose {μ : Colors} (Λ : LorentzGroup d) :
|
|
|
|
|
colorMatrix μ (LorentzGroup.transpose Λ) = (colorMatrix μ Λ).transpose := by
|
|
|
|
|
match μ with
|
|
|
|
|
| .up => rfl
|
|
|
|
|
| .down =>
|
|
|
|
|
ext i j
|
|
|
|
|
simp only [colorMatrix, transpose, lorentzGroupIsGroup_inv, Matrix.transpose_apply,
|
|
|
|
|
MonoidHom.coe_mk, OneHom.coe_mk, minkowskiMetric.dual_transpose]
|
|
|
|
|
|
|
|
|
|
/-!
|
|
|
|
|
|
|
|
|
|
## Lorentz group to tensor representation matrices.
|
|
|
|
|
|
|
|
|
|
-/
|
|
|
|
|
|
2024-07-17 16:12:30 -04:00
|
|
|
|
/-- The matrix representation of the Lorentz group given a color of index. -/
|
2024-07-16 16:58:42 -04:00
|
|
|
|
@[simps!]
|
|
|
|
|
def toTensorRepMat {c : X → Colors} :
|
|
|
|
|
LorentzGroup d →* Matrix (IndexValue d c) (IndexValue d c) ℝ where
|
|
|
|
|
toFun Λ := fun i j => ∏ x, colorMatrix (c x) Λ (i x) (j x)
|
|
|
|
|
map_one' := by
|
|
|
|
|
ext i j
|
|
|
|
|
by_cases hij : i = j
|
|
|
|
|
· subst hij
|
|
|
|
|
simp only [map_one, Matrix.one_apply_eq, Finset.prod_const_one]
|
|
|
|
|
· obtain ⟨x, hijx⟩ := Function.ne_iff.mp hij
|
|
|
|
|
simp only [map_one]
|
|
|
|
|
rw [@Finset.prod_eq_zero _ _ _ _ _ x]
|
|
|
|
|
exact Eq.symm (Matrix.one_apply_ne' fun a => hij (id (Eq.symm a)))
|
|
|
|
|
exact Finset.mem_univ x
|
|
|
|
|
exact Matrix.one_apply_ne' (id (Ne.symm hijx))
|
|
|
|
|
map_mul' Λ Λ' := by
|
|
|
|
|
ext i j
|
|
|
|
|
rw [Matrix.mul_apply]
|
|
|
|
|
trans ∑ (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]
|
|
|
|
|
rfl
|
|
|
|
|
rw [h1]
|
|
|
|
|
simp only [map_mul]
|
2024-07-17 16:12:30 -04:00
|
|
|
|
rfl
|
2024-07-16 16:58:42 -04:00
|
|
|
|
refine Finset.sum_congr rfl (fun k _ => ?_)
|
|
|
|
|
rw [Finset.prod_mul_distrib]
|
|
|
|
|
|
|
|
|
|
lemma toTensorRepMat_mul' (i j : IndexValue d c) :
|
|
|
|
|
toTensorRepMat (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
|
2024-07-17 16:12:30 -04:00
|
|
|
|
simp [Matrix.mul_apply, IndexValue]
|
2024-07-16 16:58:42 -04:00
|
|
|
|
refine Finset.sum_congr rfl (fun k _ => ?_)
|
|
|
|
|
rw [Finset.prod_mul_distrib]
|
|
|
|
|
rfl
|
|
|
|
|
|
2024-07-17 13:53:36 -04:00
|
|
|
|
lemma toTensorRepMat_of_indexValueSumEquiv {cX : X → Colors} {cY : Y → Colors}
|
|
|
|
|
(i j : IndexValue d (Sum.elim cX cY)) :
|
|
|
|
|
toTensorRepMat Λ i j = toTensorRepMat Λ (indexValueSumEquiv i).1 (indexValueSumEquiv j).1 *
|
2024-07-17 16:12:30 -04:00
|
|
|
|
toTensorRepMat Λ (indexValueSumEquiv i).2 (indexValueSumEquiv j).2 :=
|
|
|
|
|
Fintype.prod_sum_type fun x => (colorMatrix (Sum.elim cX cY x)) Λ (i x) (j x)
|
2024-07-16 09:45:03 -04:00
|
|
|
|
|
2024-07-17 13:53:36 -04:00
|
|
|
|
lemma toTensorRepMat_of_indexValueSumEquiv' {cX : X → Colors} {cY : Y → Colors}
|
|
|
|
|
(i j : IndexValue d cX) (k l : IndexValue d cY) :
|
|
|
|
|
toTensorRepMat Λ i j * toTensorRepMat Λ k l =
|
2024-07-17 16:12:30 -04:00
|
|
|
|
toTensorRepMat Λ (indexValueSumEquiv.symm (i, k)) (indexValueSumEquiv.symm (j, l)) :=
|
|
|
|
|
(Fintype.prod_sum_type fun x => (colorMatrix (Sum.elim cX cY x)) Λ
|
|
|
|
|
(indexValueSumEquiv.symm (i, k) x) (indexValueSumEquiv.symm (j, l) x)).symm
|
2024-07-16 16:58:42 -04:00
|
|
|
|
|
2024-07-17 13:53:36 -04:00
|
|
|
|
/-!
|
|
|
|
|
|
|
|
|
|
## Tensor representation matrices and marked tensors.
|
|
|
|
|
|
|
|
|
|
-/
|
|
|
|
|
|
|
|
|
|
lemma toTensorRepMat_of_splitIndexValue' (T : Marked d X n)
|
|
|
|
|
(i j : T.UnmarkedIndexValue) (k l : T.MarkedIndexValue) :
|
|
|
|
|
toTensorRepMat Λ i j * toTensorRepMat Λ k l =
|
2024-07-17 16:12:30 -04:00
|
|
|
|
toTensorRepMat Λ (splitIndexValue.symm (i, k)) (splitIndexValue.symm (j, l)) :=
|
|
|
|
|
(Fintype.prod_sum_type fun x =>
|
|
|
|
|
(colorMatrix (T.color x)) Λ (splitIndexValue.symm (i, k) x) (splitIndexValue.symm (j, l) x)).symm
|
2024-07-16 16:58:42 -04:00
|
|
|
|
|
2024-07-17 13:53:36 -04:00
|
|
|
|
lemma toTensorRepMat_oneMarkedIndexValue_dual (T : Marked d X 1) (S : Marked d Y 1)
|
|
|
|
|
(h : T.markedColor 0 = τ (S.markedColor 0)) (x : ColorsIndex d (T.markedColor 0))
|
|
|
|
|
(k : S.MarkedIndexValue) :
|
|
|
|
|
toTensorRepMat Λ (oneMarkedIndexValue $ colorsIndexDualCast h x) k =
|
|
|
|
|
toTensorRepMat Λ⁻¹ (oneMarkedIndexValue
|
|
|
|
|
$ (colorsIndexDualCast h).symm $ oneMarkedIndexValue.symm k)
|
|
|
|
|
(oneMarkedIndexValue x) := by
|
|
|
|
|
rw [toTensorRepMat_apply, toTensorRepMat_apply]
|
|
|
|
|
erw [Finset.prod_singleton, Finset.prod_singleton]
|
2024-07-17 15:15:44 -04:00
|
|
|
|
simp only [Fin.zero_eta, Fin.isValue, lorentzGroupIsGroup_inv]
|
2024-07-17 13:53:36 -04:00
|
|
|
|
rw [colorMatrix_cast h, colorMatrix_dual_cast]
|
|
|
|
|
rw [Matrix.reindex_apply, Matrix.reindex_apply]
|
2024-07-17 15:15:44 -04:00
|
|
|
|
simp only [Fin.isValue, lorentzGroupIsGroup_inv, minkowskiMetric.dual_dual, Subtype.coe_eta,
|
|
|
|
|
Equiv.symm_symm, Matrix.submatrix_apply]
|
2024-07-17 13:53:36 -04:00
|
|
|
|
rw [colorMatrix_transpose]
|
2024-07-17 15:15:44 -04:00
|
|
|
|
simp only [Fin.isValue, Matrix.transpose_apply]
|
2024-07-17 13:53:36 -04:00
|
|
|
|
apply congrArg
|
|
|
|
|
simp only [Fin.isValue, oneMarkedIndexValue, colorsIndexDualCast, Equiv.coe_fn_symm_mk,
|
|
|
|
|
Equiv.symm_trans_apply, Equiv.symm_symm, Equiv.coe_fn_mk, Equiv.apply_symm_apply,
|
|
|
|
|
Equiv.symm_apply_apply]
|
|
|
|
|
|
|
|
|
|
lemma toTensorRepMap_sum_dual (T : Marked d X 1) (S : Marked d Y 1)
|
|
|
|
|
(h : T.markedColor 0 = τ (S.markedColor 0)) (j : T.MarkedIndexValue) (k : S.MarkedIndexValue) :
|
|
|
|
|
∑ x, toTensorRepMat Λ (oneMarkedIndexValue $ colorsIndexDualCast h x) k
|
|
|
|
|
* toTensorRepMat Λ (oneMarkedIndexValue x) j =
|
|
|
|
|
toTensorRepMat 1
|
|
|
|
|
(oneMarkedIndexValue $ (colorsIndexDualCast h).symm $ oneMarkedIndexValue.symm k) j := by
|
|
|
|
|
trans ∑ x, toTensorRepMat Λ⁻¹ (oneMarkedIndexValue$ (colorsIndexDualCast h).symm $
|
|
|
|
|
oneMarkedIndexValue.symm k) (oneMarkedIndexValue x) * toTensorRepMat Λ (oneMarkedIndexValue x) j
|
|
|
|
|
apply Finset.sum_congr rfl (fun x _ => ?_)
|
|
|
|
|
rw [toTensorRepMat_oneMarkedIndexValue_dual]
|
|
|
|
|
rw [← Equiv.sum_comp oneMarkedIndexValue.symm]
|
|
|
|
|
change ∑ i, toTensorRepMat Λ⁻¹ (oneMarkedIndexValue $ (colorsIndexDualCast h).symm $
|
|
|
|
|
oneMarkedIndexValue.symm k) i * toTensorRepMat Λ i j = _
|
|
|
|
|
rw [← Matrix.mul_apply, ← toTensorRepMat.map_mul, inv_mul_self Λ]
|
|
|
|
|
|
|
|
|
|
lemma toTensorRepMat_one_coord_sum (T : Marked d X n) (i : T.UnmarkedIndexValue)
|
|
|
|
|
(k : T.MarkedIndexValue) : T.coord (splitIndexValue.symm (i, k)) = ∑ j, toTensorRepMat 1 k j *
|
|
|
|
|
T.coord (splitIndexValue.symm (i, j)) := by
|
|
|
|
|
erw [Finset.sum_eq_single_of_mem k]
|
|
|
|
|
simp only [IndexValue, map_one, Matrix.one_apply_eq, one_mul]
|
|
|
|
|
exact Finset.mem_univ k
|
|
|
|
|
intro j _ hjk
|
2024-07-17 16:12:30 -04:00
|
|
|
|
simp [hjk, IndexValue]
|
2024-07-17 13:53:36 -04:00
|
|
|
|
exact Or.inl (Matrix.one_apply_ne' hjk)
|
|
|
|
|
|
2024-07-16 16:58:42 -04:00
|
|
|
|
/-!
|
|
|
|
|
|
|
|
|
|
## Definition of the Lorentz group action on Real Lorentz Tensors.
|
|
|
|
|
|
|
|
|
|
-/
|
2024-07-16 09:45:03 -04:00
|
|
|
|
|
2024-07-16 11:40:00 -04:00
|
|
|
|
/-- Action of the Lorentz group on `X`-indexed Real Lorentz Tensors. -/
|
|
|
|
|
@[simps!]
|
|
|
|
|
instance lorentzAction : MulAction (LorentzGroup d) (RealLorentzTensor d X) where
|
2024-07-16 09:45:03 -04:00
|
|
|
|
smul Λ T := {color := T.color,
|
2024-07-16 16:58:42 -04:00
|
|
|
|
coord := fun i => ∑ j, toTensorRepMat Λ i j * T.coord j}
|
2024-07-16 09:45:03 -04:00
|
|
|
|
one_smul T := by
|
2024-07-17 13:53:36 -04:00
|
|
|
|
refine ext rfl ?_
|
2024-07-16 09:45:03 -04:00
|
|
|
|
funext i
|
|
|
|
|
simp only [HSMul.hSMul, map_one]
|
|
|
|
|
erw [Finset.sum_eq_single_of_mem i]
|
2024-07-16 16:58:42 -04:00
|
|
|
|
simp only [Matrix.one_apply_eq, one_mul, IndexValue]
|
2024-07-16 09:45:03 -04:00
|
|
|
|
rfl
|
|
|
|
|
exact Finset.mem_univ i
|
2024-07-17 13:53:36 -04:00
|
|
|
|
exact fun j _ hij => mul_eq_zero.mpr (Or.inl (Matrix.one_apply_ne' hij))
|
2024-07-16 09:45:03 -04:00
|
|
|
|
mul_smul Λ Λ' T := by
|
2024-07-17 13:53:36 -04:00
|
|
|
|
refine ext rfl ?_
|
2024-07-16 09:45:03 -04:00
|
|
|
|
simp only [HSMul.hSMul]
|
|
|
|
|
funext i
|
2024-07-16 16:58:42 -04:00
|
|
|
|
have h1 : ∑ j : IndexValue d T.color, toTensorRepMat (Λ * Λ') i j
|
2024-07-16 09:45:03 -04:00
|
|
|
|
* T.coord j = ∑ j : IndexValue d T.color, ∑ (k : IndexValue d T.color),
|
|
|
|
|
(∏ x, ((colorMatrix (T.color x) Λ (i x) (k x)) *
|
|
|
|
|
(colorMatrix (T.color x) Λ' (k x) (j x)))) * T.coord j := by
|
|
|
|
|
refine Finset.sum_congr rfl (fun j _ => ?_)
|
2024-07-16 16:58:42 -04:00
|
|
|
|
rw [toTensorRepMat_mul', Finset.sum_mul]
|
2024-07-16 09:45:03 -04:00
|
|
|
|
rw [h1]
|
|
|
|
|
rw [Finset.sum_comm]
|
|
|
|
|
refine Finset.sum_congr rfl (fun j _ => ?_)
|
|
|
|
|
rw [Finset.mul_sum]
|
|
|
|
|
refine Finset.sum_congr rfl (fun k _ => ?_)
|
2024-07-16 16:58:42 -04:00
|
|
|
|
simp only [toTensorRepMat, IndexValue]
|
2024-07-16 09:45:03 -04:00
|
|
|
|
rw [← mul_assoc]
|
|
|
|
|
congr
|
|
|
|
|
rw [Finset.prod_mul_distrib]
|
|
|
|
|
rfl
|
|
|
|
|
|
2024-07-17 13:53:36 -04:00
|
|
|
|
lemma lorentzAction_smul_coord' {d : ℕ} {X : Type} [Fintype X] [DecidableEq X] (Λ : ↑(𝓛 d))
|
|
|
|
|
(T : RealLorentzTensor d X) (i : IndexValue d T.color) :
|
|
|
|
|
(Λ • T).coord i = ∑ j : IndexValue d T.color, toTensorRepMat Λ i j * T.coord j := by
|
2024-07-16 16:58:42 -04:00
|
|
|
|
rfl
|
|
|
|
|
|
|
|
|
|
/-!
|
|
|
|
|
|
|
|
|
|
## Properties of the Lorentz action.
|
|
|
|
|
|
|
|
|
|
-/
|
|
|
|
|
|
2024-07-16 11:59:29 -04:00
|
|
|
|
/-- The action on an empty Lorentz tensor is trivial. -/
|
|
|
|
|
lemma lorentzAction_on_isEmpty [IsEmpty X] (Λ : LorentzGroup d) (T : RealLorentzTensor d X) :
|
|
|
|
|
Λ • T = T := by
|
2024-07-17 13:53:36 -04:00
|
|
|
|
refine ext rfl ?_
|
2024-07-16 11:59:29 -04:00
|
|
|
|
funext i
|
|
|
|
|
erw [lorentzAction_smul_coord]
|
|
|
|
|
simp only [Finset.univ_unique, Finset.univ_eq_empty, Finset.prod_empty, one_mul,
|
2024-07-16 16:58:42 -04:00
|
|
|
|
Finset.sum_singleton, toTensorRepMat_apply]
|
2024-07-17 16:12:30 -04:00
|
|
|
|
simp only [IndexValue, Unique.eq_default, Finset.univ_unique, Finset.sum_const,
|
|
|
|
|
Finset.card_singleton, one_smul]
|
2024-07-16 16:58:42 -04:00
|
|
|
|
|
2024-07-17 13:53:36 -04:00
|
|
|
|
/-- The Lorentz action commutes with `mapIso`. -/
|
|
|
|
|
lemma lorentzAction_mapIso (f : X ≃ Y) (Λ : LorentzGroup d) (T : RealLorentzTensor d X) :
|
|
|
|
|
mapIso d f (Λ • T) = Λ • (mapIso d f T) := by
|
|
|
|
|
refine ext rfl ?_
|
2024-07-16 16:58:42 -04:00
|
|
|
|
funext i
|
2024-07-17 13:53:36 -04:00
|
|
|
|
rw [mapIso_apply_coord]
|
|
|
|
|
rw [lorentzAction_smul_coord', lorentzAction_smul_coord']
|
|
|
|
|
let is : IndexValue d T.color ≃ IndexValue d ((mapIso d f) T).color :=
|
2024-07-17 16:12:30 -04:00
|
|
|
|
indexValueIso d f ((Equiv.comp_symm_eq f ((mapIso d f) T).color T.color).mp rfl)
|
2024-07-17 13:53:36 -04:00
|
|
|
|
rw [← Equiv.sum_comp is]
|
2024-07-16 16:58:42 -04:00
|
|
|
|
refine Finset.sum_congr rfl (fun j _ => ?_)
|
2024-07-17 13:53:36 -04:00
|
|
|
|
rw [mapIso_apply_coord]
|
|
|
|
|
refine Mathlib.Tactic.Ring.mul_congr ?_ ?_ rfl
|
|
|
|
|
· simp only [IndexValue, toTensorRepMat, MonoidHom.coe_mk, OneHom.coe_mk, mapIso_apply_color,
|
|
|
|
|
indexValueIso_refl]
|
|
|
|
|
rw [← Equiv.prod_comp f]
|
|
|
|
|
apply Finset.prod_congr rfl (fun x _ => ?_)
|
|
|
|
|
have h1 : (T.color (f.symm (f x))) = T.color x := by
|
|
|
|
|
simp only [Equiv.symm_apply_apply]
|
|
|
|
|
rw [colorMatrix_cast h1]
|
|
|
|
|
apply congrArg
|
|
|
|
|
simp only [is]
|
|
|
|
|
erw [indexValueIso_eq_symm, indexValueIso_symm_apply']
|
|
|
|
|
simp only [colorsIndexCast, Function.comp_apply, mapIso_apply_color, Equiv.cast_refl,
|
|
|
|
|
Equiv.refl_symm, Equiv.refl_apply, Equiv.cast_apply]
|
|
|
|
|
symm
|
|
|
|
|
refine cast_eq_iff_heq.mpr ?_
|
|
|
|
|
congr
|
|
|
|
|
exact Equiv.symm_apply_apply f x
|
|
|
|
|
· apply congrArg
|
2024-07-17 16:12:30 -04:00
|
|
|
|
exact (Equiv.apply_eq_iff_eq_symm_apply (indexValueIso d f (mapIso.proof_1 d f T))).mp rfl
|
2024-07-17 13:53:36 -04:00
|
|
|
|
|
|
|
|
|
/-!
|
|
|
|
|
|
|
|
|
|
## The Lorentz action on marked tensors.
|
|
|
|
|
|
|
|
|
|
-/
|
|
|
|
|
|
|
|
|
|
@[simps!]
|
|
|
|
|
instance : MulAction (LorentzGroup d) (Marked d X n) := lorentzAction
|
|
|
|
|
|
|
|
|
|
/-- Action of the Lorentz group on just marked indices. -/
|
|
|
|
|
@[simps!]
|
|
|
|
|
def markedLorentzAction : MulAction (LorentzGroup d) (Marked d X n) where
|
|
|
|
|
smul Λ T := {
|
|
|
|
|
color := T.color,
|
|
|
|
|
coord := fun i => ∑ j, toTensorRepMat Λ (splitIndexValue i).2 j *
|
|
|
|
|
T.coord (splitIndexValue.symm ((splitIndexValue i).1, j))}
|
|
|
|
|
one_smul T := by
|
|
|
|
|
refine ext rfl ?_
|
|
|
|
|
funext i
|
|
|
|
|
simp only [HSMul.hSMul, map_one]
|
|
|
|
|
erw [Finset.sum_eq_single_of_mem (splitIndexValue i).2]
|
|
|
|
|
erw [Matrix.one_apply_eq (splitIndexValue i).2]
|
|
|
|
|
simp only [IndexValue, one_mul, indexValueIso_refl, Equiv.refl_apply]
|
|
|
|
|
apply congrArg
|
|
|
|
|
exact Equiv.symm_apply_apply splitIndexValue i
|
|
|
|
|
exact Finset.mem_univ (splitIndexValue i).2
|
|
|
|
|
exact fun j _ hij => mul_eq_zero.mpr (Or.inl (Matrix.one_apply_ne' hij))
|
|
|
|
|
mul_smul Λ Λ' T := by
|
|
|
|
|
refine ext rfl ?_
|
|
|
|
|
simp only [HSMul.hSMul]
|
|
|
|
|
funext i
|
|
|
|
|
have h1 : ∑ (j : T.MarkedIndexValue), toTensorRepMat (Λ * Λ') (splitIndexValue i).2 j
|
|
|
|
|
* T.coord (splitIndexValue.symm ((splitIndexValue i).1, j)) =
|
|
|
|
|
∑ (j : T.MarkedIndexValue), ∑ (k : T.MarkedIndexValue),
|
|
|
|
|
(∏ x, ((colorMatrix (T.markedColor x) Λ ((splitIndexValue i).2 x) (k x)) *
|
|
|
|
|
(colorMatrix (T.markedColor x) Λ' (k x) (j x)))) *
|
|
|
|
|
T.coord (splitIndexValue.symm ((splitIndexValue i).1, j)) := by
|
|
|
|
|
refine Finset.sum_congr rfl (fun j _ => ?_)
|
|
|
|
|
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
|
|
|
|
|
|
|
|
|
|
/-- Action of the Lorentz group on just unmarked indices. -/
|
|
|
|
|
@[simps!]
|
|
|
|
|
def unmarkedLorentzAction : MulAction (LorentzGroup d) (Marked d X n) where
|
|
|
|
|
smul Λ T := {
|
|
|
|
|
color := T.color,
|
|
|
|
|
coord := fun i => ∑ j, toTensorRepMat Λ (splitIndexValue i).1 j *
|
|
|
|
|
T.coord (splitIndexValue.symm (j, (splitIndexValue i).2))}
|
|
|
|
|
one_smul T := by
|
|
|
|
|
refine ext rfl ?_
|
|
|
|
|
funext i
|
|
|
|
|
simp only [HSMul.hSMul, map_one]
|
|
|
|
|
erw [Finset.sum_eq_single_of_mem (splitIndexValue i).1]
|
|
|
|
|
erw [Matrix.one_apply_eq (splitIndexValue i).1]
|
|
|
|
|
simp only [IndexValue, one_mul, indexValueIso_refl, Equiv.refl_apply]
|
|
|
|
|
apply congrArg
|
|
|
|
|
exact Equiv.symm_apply_apply splitIndexValue i
|
|
|
|
|
exact Finset.mem_univ (splitIndexValue i).1
|
|
|
|
|
exact fun j _ hij => mul_eq_zero.mpr (Or.inl (Matrix.one_apply_ne' hij))
|
|
|
|
|
mul_smul Λ Λ' T := by
|
|
|
|
|
refine ext rfl ?_
|
|
|
|
|
simp only [HSMul.hSMul]
|
|
|
|
|
funext i
|
|
|
|
|
have h1 : ∑ (j : T.UnmarkedIndexValue), toTensorRepMat (Λ * Λ') (splitIndexValue i).1 j
|
|
|
|
|
* T.coord (splitIndexValue.symm (j, (splitIndexValue i).2)) =
|
|
|
|
|
∑ (j : T.UnmarkedIndexValue), ∑ (k : T.UnmarkedIndexValue),
|
|
|
|
|
(∏ x, ((colorMatrix (T.unmarkedColor x) Λ ((splitIndexValue i).1 x) (k x)) *
|
|
|
|
|
(colorMatrix (T.unmarkedColor x) Λ' (k x) (j x)))) *
|
|
|
|
|
T.coord (splitIndexValue.symm (j, (splitIndexValue i).2)) := by
|
|
|
|
|
refine Finset.sum_congr rfl (fun j _ => ?_)
|
|
|
|
|
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
|
2024-07-16 16:58:42 -04:00
|
|
|
|
|
2024-07-17 16:12:30 -04:00
|
|
|
|
/-- Notation for `markedLorentzAction.smul`. -/
|
2024-07-17 13:53:36 -04:00
|
|
|
|
scoped[RealLorentzTensor] infixr:73 " •ₘ " => markedLorentzAction.smul
|
2024-07-17 16:12:30 -04:00
|
|
|
|
|
|
|
|
|
/-- Notation for `unmarkedLorentzAction.smul`. -/
|
2024-07-17 13:53:36 -04:00
|
|
|
|
scoped[RealLorentzTensor] infixr:73 " •ᵤₘ " => unmarkedLorentzAction.smul
|
2024-07-16 16:58:42 -04:00
|
|
|
|
|
2024-07-17 13:53:36 -04:00
|
|
|
|
/-- 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 ?_
|
2024-07-16 16:58:42 -04:00
|
|
|
|
funext i
|
2024-07-17 13:53:36 -04:00
|
|
|
|
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
|
2024-07-16 16:58:42 -04:00
|
|
|
|
rfl
|
|
|
|
|
|
2024-07-17 13:53:36 -04:00
|
|
|
|
/-- 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
|
2024-07-16 16:58:42 -04:00
|
|
|
|
|
2024-07-17 13:53:36 -04:00
|
|
|
|
/-- 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]
|
2024-07-16 11:59:29 -04:00
|
|
|
|
|
|
|
|
|
/-! TODO: Show that the Lorentz action commutes with contraction. -/
|
|
|
|
|
/-! TODO: Show that the Lorentz action commutes with rising and lowering indices. -/
|
2024-07-15 16:57:06 -04:00
|
|
|
|
end RealLorentzTensor
|