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feat: General construction of tensors

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jstoobysmith 2024-07-26 14:43:20 -04:00
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HepLean/SpaceTime/LorentzTensor/Basic.lean
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@ -34,7 +34,11 @@ open TensorProduct
variable {R : Type} [CommSemiring R]
/-- An initial structure specifying a tensor system (e.g. a system in which you can
define real Lorentz tensors). -/
structure PreTensorStructure (R : Type) [CommSemiring R] where
/-- The allowed colors of indices.
For example for a real Lorentz tensor these are `{up, down}`. -/
Color : Type
ColorModule : Color → Type
τ : Color → Color
@ -48,9 +52,10 @@ namespace PreTensorStructure
variable (𝓣 : PreTensorStructure R)
variable {d : ℕ} {X Y Y' Z : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
[Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z]
{c c₂ : X → 𝓣.Color} {d : Y → 𝓣.Color} {b : Z → 𝓣.Color} {d' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
variable {d : ℕ} {X Y Y' Z W : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
[Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z] [Fintype W] [DecidableEq W]
{c c₂ : X → 𝓣.Color} {d : Y → 𝓣.Color} {b : Z → 𝓣.Color}
{bw : W → 𝓣.Color} {d' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
instance : AddCommMonoid (𝓣.ColorModule μ) := 𝓣.colorModule_addCommMonoid μ
@ -79,6 +84,27 @@ def colorModuleCast (h : μ = ν) : 𝓣.ColorModule μ ≃ₗ[R] 𝓣.ColorModu
subst h
rfl
lemma tensorProd_piTensorProd_ext {M : Type} [AddCommMonoid M] [Module R M] {f g : 𝓣.Tensor c ⊗[R] 𝓣.Tensor d →ₗ[R] M}
(h : ∀ p q, f (PiTensorProduct.tprod R p ⊗ₜ[R] PiTensorProduct.tprod R q)
= g (PiTensorProduct.tprod R p ⊗ₜ[R] PiTensorProduct.tprod R q)) : f = g := by
apply TensorProduct.ext'
refine fun x ↦
PiTensorProduct.induction_on' x ?_ (by
intro a b hx hy y
simp [map_add, add_tmul, hx, hy])
intro rx fx
refine fun y ↦
PiTensorProduct.induction_on' y ?_ (by
intro a b hx hy
simp at hx hy
simp [map_add, tmul_add, hx, hy])
intro ry fy
simp
apply congrArg
simp [TensorProduct.smul_tmul]
apply congrArg
exact h fx fy
/-!
## Mapping isomorphisms
@ -170,6 +196,259 @@ lemma mapIso_tprod {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (
/-!
## Pure tensors
This section is needed since: `PiTensorProduct.tmulEquiv` is not defined for dependent types.
Hence we need to construct a version of it here.
-/
abbrev PureTensor (c : X → 𝓣.Color) := (x : X) → 𝓣.ColorModule (c x)
def elimPureTensor (p : 𝓣.PureTensor c) (q : 𝓣.PureTensor d) : 𝓣.PureTensor (Sum.elim c d) :=
fun x =>
match x with
| Sum.inl x => p x
| Sum.inr x => q x
@[simp]
lemma elimPureTensor_update_right (p : 𝓣.PureTensor c) (q : 𝓣.PureTensor d)
(y : Y) (r : 𝓣.ColorModule (d y)) : 𝓣.elimPureTensor p (Function.update q y r) =
Function.update (𝓣.elimPureTensor p q) (Sum.inr y) r := by
funext x
match x with
| Sum.inl x => rfl
| Sum.inr x =>
change Function.update q y r x = _
simp only [Function.update, Sum.inr.injEq, Sum.elim_inr]
split_ifs
rename_i h
subst h
simp_all only
rfl
@[simp]
lemma elimPureTensor_update_left (p : 𝓣.PureTensor c) (q : 𝓣.PureTensor d)
(x : X) (r : 𝓣.ColorModule (c x)) : 𝓣.elimPureTensor (Function.update p x r) q =
Function.update (𝓣.elimPureTensor p q) (Sum.inl x) r := by
funext y
match y with
| Sum.inl y =>
change (Function.update p x r) y = _
simp only [Function.update, Sum.inl.injEq, Sum.elim_inl]
split_ifs
rename_i h
subst h
simp_all only
rfl
| Sum.inr y => rfl
def inlPureTensor (p : 𝓣.PureTensor (Sum.elim c d)) : 𝓣.PureTensor c := fun xy => p (Sum.inl xy)
def inrPureTensor (p : 𝓣.PureTensor (Sum.elim c d)) : 𝓣.PureTensor d := fun xy => p (Sum.inr xy)
@[simp]
lemma inlPureTensor_update_left [DecidableEq (X ⊕ Y)] (f : 𝓣.PureTensor (Sum.elim c d)) (x : X)
(v1 : 𝓣.ColorModule (Sum.elim c d (Sum.inl x))) :
𝓣.inlPureTensor (Function.update f (Sum.inl x) v1) =Function.update (𝓣.inlPureTensor f) x v1 := by
funext y
simp [inlPureTensor, Function.update, Sum.inl.injEq, Sum.elim_inl]
split
next h =>
subst h
simp_all only
next h => simp_all only
@[simp]
lemma inrPureTensor_update_left [DecidableEq (X ⊕ Y)] (f : 𝓣.PureTensor (Sum.elim c d)) (x : X)
(v1 : 𝓣.ColorModule (Sum.elim c d (Sum.inl x))) :
𝓣.inrPureTensor (Function.update f (Sum.inl x) v1) = (𝓣.inrPureTensor f) := by
funext x
simp [inrPureTensor, Function.update]
@[simp]
lemma inrPureTensor_update_right [DecidableEq (X ⊕ Y)] (f : 𝓣.PureTensor (Sum.elim c d)) (y : Y)
(v1 : 𝓣.ColorModule (Sum.elim c d (Sum.inr y))) :
𝓣.inrPureTensor (Function.update f (Sum.inr y) v1) =Function.update (𝓣.inrPureTensor f) y v1 := by
funext y
simp [inrPureTensor, Function.update, Sum.inl.injEq, Sum.elim_inl]
split
next h =>
subst h
simp_all only
next h => simp_all only
@[simp]
lemma inlPureTensor_update_right [DecidableEq (X ⊕ Y)] (f : 𝓣.PureTensor (Sum.elim c d)) (y : Y)
(v1 : 𝓣.ColorModule (Sum.elim c d (Sum.inr y))) :
𝓣.inlPureTensor (Function.update f (Sum.inr y) v1) = (𝓣.inlPureTensor f) := by
funext x
simp [inlPureTensor, Function.update]
def elimPureTensorMulLin : MultilinearMap R (fun i => 𝓣.ColorModule (c i))
(MultilinearMap R (fun x => 𝓣.ColorModule (d x)) (𝓣.Tensor (Sum.elim c d))) where
toFun p := {
toFun := fun q => PiTensorProduct.tprod R (𝓣.elimPureTensor p q)
map_add' := fun m x v1 v2 => by
simp [Sum.elim_inl, Sum.elim_inr]
map_smul' := fun m x r v => by
simp [Sum.elim_inl, Sum.elim_inr]}
map_add' p x v1 v2 := by
apply MultilinearMap.ext
intro y
simp
map_smul' p x r v := by
apply MultilinearMap.ext
intro y
simp
/-!
## tensorator
-/
/-! TODO: Replace with dependent type version of `MultilinearMap.domCoprod` when in Mathlib. -/
def domCoprod : MultilinearMap R (fun x => 𝓣.ColorModule (Sum.elim c d x)) (𝓣.Tensor c ⊗[R] 𝓣.Tensor d) where
toFun f := (PiTensorProduct.tprod R (𝓣.inlPureTensor f)) ⊗ₜ
(PiTensorProduct.tprod R (𝓣.inrPureTensor f))
map_add' f xy v1 v2:= by
match xy with
| Sum.inl x => simp [← TensorProduct.add_tmul]
| Sum.inr y => simp [← TensorProduct.tmul_add]
map_smul' f xy r p := by
match xy with
| Sum.inl x => simp [TensorProduct.tmul_smul, TensorProduct.smul_tmul]
| Sum.inr y => simp [TensorProduct.tmul_smul, TensorProduct.smul_tmul]
def tensoratorSymm : 𝓣.Tensor c ⊗[R] 𝓣.Tensor d →ₗ[R] 𝓣.Tensor (Sum.elim c d) := by
refine TensorProduct.lift {
toFun := fun a ↦
PiTensorProduct.lift <|
PiTensorProduct.lift (𝓣.elimPureTensorMulLin) a,
map_add' := fun a b ↦ by simp
map_smul' := fun r a ↦ by simp}
/-! TODO: Replace with dependent type version of `PiTensorProduct.tmulEquiv` when in Mathlib. -/
def tensorator : 𝓣.Tensor (Sum.elim c d) →ₗ[R] 𝓣.Tensor c ⊗[R] 𝓣.Tensor d :=
PiTensorProduct.lift 𝓣.domCoprod
def tensoratorEquiv (c : X → 𝓣.Color) (d : Y → 𝓣.Color) : 𝓣.Tensor c ⊗[R] 𝓣.Tensor d ≃ₗ[R] 𝓣.Tensor (Sum.elim c d) :=
LinearEquiv.ofLinear (𝓣.tensoratorSymm) (𝓣.tensorator)
(by
apply PiTensorProduct.ext
apply MultilinearMap.ext
intro p
simp [tensorator, tensoratorSymm, domCoprod]
change (PiTensorProduct.lift (_)) ((PiTensorProduct.tprod R) _) =
LinearMap.id ((PiTensorProduct.tprod R) p)
rw [PiTensorProduct.lift.tprod]
simp [elimPureTensorMulLin, elimPureTensor]
change (PiTensorProduct.tprod R) _ = _
apply congrArg
funext x
match x with
| Sum.inl x => rfl
| Sum.inr x => rfl)
(by
apply tensorProd_piTensorProd_ext
intro p q
simp [tensorator, tensoratorSymm]
change (PiTensorProduct.lift 𝓣.domCoprod) ((PiTensorProduct.lift (𝓣.elimPureTensorMulLin p)) ((PiTensorProduct.tprod R) q)) =_
rw [PiTensorProduct.lift.tprod]
simp [elimPureTensorMulLin]
rfl)
@[simp]
lemma tensoratorEquiv_tmul_tprod (p : 𝓣.PureTensor c) (q : 𝓣.PureTensor d) :
(𝓣.tensoratorEquiv c d) ((PiTensorProduct.tprod R) p ⊗ₜ[R] (PiTensorProduct.tprod R) q) =
(PiTensorProduct.tprod R) (𝓣.elimPureTensor p q) := by
simp [tensoratorEquiv, tensorator, tensoratorSymm, domCoprod]
change (PiTensorProduct.lift (𝓣.elimPureTensorMulLin p)) ((PiTensorProduct.tprod R) q) = _
rw [PiTensorProduct.lift.tprod]
simp [elimPureTensorMulLin]
lemma tensoratorEquiv_mapIso_cond {e : X ≃ Y} {e' : Z ≃ Y} {e'' : W ≃ X}
(h : c = 𝓣.τ ∘ d ∘ e) (h' : b = d ∘ e') (h'' : bW = c ∘ e'') :
Sum.elim bW b = Sum.elim c d ∘ ⇑(e''.sumCongr e') := by
subst h h' h''
funext x
match x with
| Sum.inl x => rfl
| Sum.inr x => rfl
@[simp]
lemma tensoratorEquiv_mapIso (e : X ≃ Y) (e' : Z ≃ Y) (e'' : W ≃ X)
(h : c = 𝓣.τ ∘ d ∘ e) (h' : b = d ∘ e') (h'' : bW = c ∘ e'') :
(TensorProduct.congr (𝓣.mapIso e'' h'') (𝓣.mapIso e' h')) ≪≫ₗ (𝓣.tensoratorEquiv c d)
= (𝓣.tensoratorEquiv bW b)
≪≫ₗ (𝓣.mapIso (Equiv.sumCongr e'' e') (𝓣.tensoratorEquiv_mapIso_cond h h' h'' )) := by
apply LinearEquiv.toLinearMap_inj.mp
apply tensorProd_piTensorProd_ext
intro p q
simp
apply congrArg
funext x
match x with
| Sum.inl x => rfl
| Sum.inr x => rfl
@[simp]
lemma tensoratorEquiv_mapIso_apply (e : X ≃ Y) (e' : Z ≃ Y) (e'' : W ≃ X)
(h : c = 𝓣.τ ∘ d ∘ e) (h' : b = d ∘ e') (h'' : bW = c ∘ e'')
(x : 𝓣.Tensor bW ⊗[R] 𝓣.Tensor b) :
(𝓣.tensoratorEquiv c d) ((TensorProduct.congr (𝓣.mapIso e'' h'') (𝓣.mapIso e' h')) x) =
(𝓣.mapIso (Equiv.sumCongr e'' e') (𝓣.tensoratorEquiv_mapIso_cond h h' h'' )) ((𝓣.tensoratorEquiv bW b) x) := by
trans ((TensorProduct.congr (𝓣.mapIso e'' h'') (𝓣.mapIso e' h')) ≪≫ₗ (𝓣.tensoratorEquiv c d)) x
rfl
rw [tensoratorEquiv_mapIso]
rfl
exact e
exact h
lemma tensoratorEquiv_mapIso_tmul (e : X ≃ Y) (e' : Z ≃ Y) (e'' : W ≃ X)
(h : c = 𝓣.τ ∘ d ∘ e) (h' : b = d ∘ e') (h'' : bW = c ∘ e'')
(x : 𝓣.Tensor bW) (y : 𝓣.Tensor b) :
(𝓣.tensoratorEquiv c d) ((𝓣.mapIso e'' h'' x) ⊗ₜ[R] (𝓣.mapIso e' h' y)) =
(𝓣.mapIso (Equiv.sumCongr e'' e') (𝓣.tensoratorEquiv_mapIso_cond h h' h'' )) ((𝓣.tensoratorEquiv bW b) (x ⊗ₜ y)) := by
rw [← tensoratorEquiv_mapIso_apply]
rfl
exact e
exact h
/-!
## Splitting tensors into tensor products
-/
def decompEmbedSet (f : Y ↪ X) :
X ≃ {x // x ∈ (Finset.image f Finset.univ)ᶜ} ⊕ Y :=
(Equiv.Set.sumCompl (Set.range ⇑f)).symm.trans <|
(Equiv.sumComm _ _).trans <|
Equiv.sumCongr ((Equiv.subtypeEquivRight (by simp))) <|
(Function.Embedding.toEquivRange f).symm
def decompEmbedColorLeft (c : X → 𝓣.Color) (f : Y ↪ X) : {x // x ∈ (Finset.image f Finset.univ)ᶜ} → 𝓣.Color :=
(c ∘ (decompEmbedSet f).symm) ∘ Sum.inl
def decompEmbedColorRight (c : X → 𝓣.Color) (f : Y ↪ X) : Y → 𝓣.Color :=
(c ∘ (decompEmbedSet f).symm) ∘ Sum.inr
lemma decompEmbed_cond (c : X → 𝓣.Color) (f : Y ↪ X) : c =
(Sum.elim (𝓣.decompEmbedColorLeft c f) (𝓣.decompEmbedColorRight c f)) ∘ decompEmbedSet f := by
simpa [decompEmbedColorLeft, decompEmbedColorRight] using (Equiv.comp_symm_eq _ _ _).mp rfl
/-- Decomposes a tensor into a tensor product based on an embedding. -/
def decompEmbed (f : Y ↪ X) :
𝓣.Tensor c ≃ₗ[R] 𝓣.Tensor (𝓣.decompEmbedColorLeft c f) ⊗[R] 𝓣.Tensor (c ∘ f) :=
(𝓣.mapIso (decompEmbedSet f) (𝓣.decompEmbed_cond c f)) ≪≫ₗ
(𝓣.tensoratorEquiv (𝓣.decompEmbedColorLeft c f) (𝓣.decompEmbedColorRight c f)).symm
/-!
## Contraction
-/
@ -191,7 +470,6 @@ lemma mkPiAlgebra_equiv (e : X ≃ Y) :
MultilinearMap.mkPiAlgebra_apply, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
PiTensorProduct.reindex_tprod, Equiv.prod_comp]
def contrAll' : 𝓣.Tensor c ⊗[R] 𝓣.Tensor (𝓣.τ ∘ c) →ₗ[R] R :=
(PiTensorProduct.lift (MultilinearMap.mkPiAlgebra R X R)) ∘ₗ
(PiTensorProduct.map (fun x => 𝓣.contrDual (c x))) ∘ₗ
@ -377,7 +655,6 @@ def rep : Representation R G (𝓣.Tensor c) where
local infixl:78 " • " => 𝓣.rep
@[simp]
lemma repColorModule_colorModuleCast_apply (h : μ = ν) (g : G) (x : 𝓣.ColorModule μ) :
(𝓣.repColorModule ν g) ((𝓣.colorModuleCast h) x) = (𝓣.colorModuleCast h) ((𝓣.repColorModule μ g) x) := by
subst h
@ -389,7 +666,7 @@ lemma repColorModule_colorModuleCast (h : μ = ν) (g : G) :
(𝓣.colorModuleCast h).toLinearMap ∘ₗ (𝓣.repColorModule μ g) := by
apply LinearMap.ext
intro x
simp
simp [repColorModule_colorModuleCast_apply]
@[simp]
@ -400,7 +677,7 @@ lemma rep_mapIso (e : X ≃ Y) (h : c = d ∘ e) (g : G) :
intro x
simp
erw [mapIso_tprod]
simp [rep]
simp [rep, repColorModule_colorModuleCast_apply]
change (PiTensorProduct.map fun x => (𝓣.repColorModule (d x)) g)
((PiTensorProduct.tprod R) fun i => (𝓣.colorModuleCast _) (x (e.symm i))) =
(𝓣.mapIso e h) ((PiTensorProduct.map fun x => (𝓣.repColorModule (c x)) g) ((PiTensorProduct.tprod R) x))
@ -409,20 +686,103 @@ lemma rep_mapIso (e : X ≃ Y) (h : c = d ∘ e) (g : G) :
apply congrArg
funext i
subst h
simp
simp [repColorModule_colorModuleCast_apply]
@[simp]
lemma rep_mapIso_apply (e : X ≃ Y) (h : c = d ∘ e) (g : G) (x : 𝓣.Tensor c) :
(𝓣.rep g) ((𝓣.mapIso e h) x) = (𝓣.mapIso e h) ((𝓣.rep g) x) := by
g • (𝓣.mapIso e h x) = (𝓣.mapIso e h) (g • x) := by
trans ((𝓣.rep g) ∘ₗ (𝓣.mapIso e h).toLinearMap) x
rfl
simp
@[simp]
lemma rep_tprod (g : G) (f : (i : X) → 𝓣.ColorModule (c i)) :
g • (PiTensorProduct.tprod R f) = PiTensorProduct.tprod R (fun x =>
𝓣.repColorModule (c x) g (f x)) := by
simp [rep]
change (PiTensorProduct.map fun x => (𝓣.repColorModule (c x)) g) ((PiTensorProduct.tprod R) f) = _
rw [PiTensorProduct.map_tprod]
/-!
## Group acting on tensor products
-/
lemma rep_tensoratorEquiv (g : G) :
(𝓣.tensoratorEquiv c d) ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g)) = 𝓣.rep g ∘ₗ
(𝓣.tensoratorEquiv c d).toLinearMap := by
apply tensorProd_piTensorProd_ext
intro p q
simp
apply congrArg
funext x
match x with
| Sum.inl x => rfl
| Sum.inr x => rfl
lemma rep_tensoratorEquiv_apply (g : G) (x : 𝓣.Tensor c ⊗[R] 𝓣.Tensor d) :
(𝓣.tensoratorEquiv c d) ((TensorProduct.map (𝓣.rep g) (𝓣.rep g)) x) = (𝓣.rep g) ((𝓣.tensoratorEquiv c d) x) := by
trans ((𝓣.tensoratorEquiv c d) ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g))) x
rfl
rw [rep_tensoratorEquiv]
rfl
lemma rep_tensoratorEquiv_tmul (g : G) (x : 𝓣.Tensor c) (y : 𝓣.Tensor d) :
(𝓣.tensoratorEquiv c d) ((g • x) ⊗ₜ[R] (g • y)) = g • ((𝓣.tensoratorEquiv c d) (x ⊗ₜ[R] y)) := by
nth_rewrite 1 [← rep_tensoratorEquiv_apply]
rfl
/-!
## Group acting on contraction
-/
@[simp]
lemma contrAll_rep {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e) (g : G) :
𝓣.contrAll e h ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g)) = 𝓣.contrAll e h := by
apply TensorProduct.ext'
refine fun x ↦ PiTensorProduct.induction_on' x ?_ (by
intro a b hx hy y
simp [map_add, add_tmul, hx, hy])
intro rx fx
refine fun y ↦ PiTensorProduct.induction_on' y ?_ (by
intro a b hx hy
simp at hx hy
simp [map_add, tmul_add, hx, hy])
intro ry fy
simp [contrAll, TensorProduct.smul_tmul]
apply congrArg
apply congrArg
simp [contrAll']
apply congrArg
simp [pairProd]
change (PiTensorProduct.map _) ((PiTensorProduct.map₂ _ _) _) =
(PiTensorProduct.map _) ((PiTensorProduct.map₂ _ _) _)
rw [PiTensorProduct.map₂_tprod_tprod, PiTensorProduct.map₂_tprod_tprod, PiTensorProduct.map_tprod,
PiTensorProduct.map_tprod]
simp
apply congrArg
funext x
rw [← repColorModule_colorModuleCast_apply]
nth_rewrite 2 [← 𝓣.contrDual_inv (c x) g]
rfl
@[simp]
lemma contrAll_rep_apply {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e)
(g : G) (x : 𝓣.Tensor c ⊗ 𝓣.Tensor d) :
𝓣.contrAll e h (TensorProduct.map (𝓣.rep g) (𝓣.rep g) x) = 𝓣.contrAll e h x := by
change (𝓣.contrAll e h ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g))) x = _
rw [contrAll_rep]
@[simp]
lemma contrAll_rep_tmul {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e)
(g : G) (x : 𝓣.Tensor c) (y : 𝓣.Tensor d) :
𝓣.contrAll e h ((g • x) ⊗ₜ[R] (g • y)) = 𝓣.contrAll e h (x ⊗ₜ[R] y) := by
nth_rewrite 2 [← contrAll_rep_apply]
rfl
end GroupTensorStructure