refactor: Spelling and typos
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jstoobysmith
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25 changed files with 37 additions and 37 deletions
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@ -12,7 +12,7 @@ import Mathlib.Algebra.BigOperators.Group.Finset
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/-- The type `CreateAnnihilate` is the type containing two elements `create` and `annihilate`.
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This type is used to specify if an operator is a creation, or annihilation, operator
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or the sum thereof or intergral thereover etc. -/
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or the sum thereof or integral thereover etc. -/
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inductive CreateAnnihilate where
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| create : CreateAnnihilate
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| annihilate : CreateAnnihilate
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@ -83,17 +83,17 @@ def euclidInner : F.HalfEdgeMomenta →ₗ[ℝ] F.HalfEdgeMomenta →ₗ[ℝ]
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Corresponding to that spanned by its total outflowing momentum. -/
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def EdgeMomenta : Type := F.𝓔 → ℝ
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/-- The edge momenta form an additive commuative group. -/
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/-- The edge momenta form an additive commutative group. -/
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instance : AddCommGroup F.EdgeMomenta := Pi.addCommGroup
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/-- The edge momenta form a module over `ℝ`. -/
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instance : Module ℝ F.EdgeMomenta := Pi.module _ _ _
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/-- The type which associates to each ege a `1`-dimensional vector space.
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/-- The type which associates to each edge a `1`-dimensional vector space.
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Corresponding to that spanned by its total inflowing momentum. -/
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def VertexMomenta : Type := F.𝓥 → ℝ
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/-- The vertex momenta carries the structure of an additive commuative group. -/
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/-- The vertex momenta carries the structure of an additive commutative group. -/
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instance : AddCommGroup F.VertexMomenta := Pi.addCommGroup
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/-- The vertex momenta carries the structure of a module over `ℝ`. -/
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@ -106,7 +106,7 @@ def EdgeVertexMomentaMap : Fin 2 → Type := fun i =>
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| 1 => F.VertexMomenta
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/-- The target of the map `EdgeVertexMomentaMap` is either the type of edge momenta
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or vertex momenta and thus carries the structure of an additive commuative group. -/
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or vertex momenta and thus carries the structure of an additive commutative group. -/
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instance (i : Fin 2) : AddCommGroup (EdgeVertexMomentaMap F i) :=
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match i with
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| 0 => instAddCommGroupEdgeMomenta F
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@ -40,7 +40,7 @@ def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
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This corresponds to the condition that two annihilation operators always super-commute.
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- `[ofCrAnOpF φ, ofCrAnOpF φ']ₛca` for `φ` and `φ'` operators with different statistics.
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This corresponds to the condition that two operators with different statistics always
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super-commute. In otherwords, fermions and bosons always super-commute.
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super-commute. In other words, fermions and bosons always super-commute.
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- `[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca`. This corresponds to the condition,
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when combined with the conditions above, that the super-commutor is in the center of the
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of the algebra.
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@ -218,7 +218,7 @@ lemma ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_mem_center (φ ψ : 𝓕.CrAnFieldOp)
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/-!
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## The kernal of ι
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## The kernel of ι
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-/
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lemma ι_eq_zero_iff_mem_ideal (x : FieldOpFreeAlgebra 𝓕) :
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@ -282,7 +282,7 @@ lemma ofCrAnOp_superCommute_normalOrder_ofFieldOpList_sum (φ : 𝓕.CrAnFieldOp
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/--
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The commutor of the annihilation part of a field operator with a normal ordered list of field
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operators can be decomponsed into the sum of the commutators of the annihilation part with each
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operators can be decomposed into the sum of the commutators of the annihilation part with each
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element of the list of field operators, i.e.
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`[anPart φ, 𝓝(φ₀…φₙ)]ₛ= ∑ i, 𝓢(φ, φ₀…φᵢ₋₁) • [anPart φ, φᵢ]ₛ * 𝓝(φ₀…φᵢ₋₁φᵢ₊₁…φₙ)`.
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-/
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@ -104,7 +104,7 @@ is equal the product of
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- `φsΛ.timeContract`
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- `s • [anPart φ, ofFieldOp φs[k]]ₛ` where `s` is the sign associated with moving `φ` through
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uncontracted fields in `φ₀…φₖ₋₁`
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- the normal ordering `[φsΛ]ᵘᶜ` with the field corresonding to `k` removed.
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- the normal ordering `[φsΛ]ᵘᶜ` with the field corresponding to `k` removed.
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The proof of this result relies on
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- `timeContract_insert_some_of_not_lt`
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@ -34,7 +34,7 @@ This result follows from
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those `𝓣(φsΛ.staticWickTerm)` for which `φsΛ` has a contracted pair which are not
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equal time to zero.
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- `staticContract_eq_timeContract_of_eqTimeOnly` to rewrite the static contract
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in the reminaing `𝓣(φsΛ.staticWickTerm)` as a time contract.
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in the remaining `𝓣(φsΛ.staticWickTerm)` as a time contract.
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- `timeOrder_timeContract_mul_of_eqTimeOnly_left` to move the time contracts out of the time
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ordering.
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-/
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@ -88,7 +88,7 @@ As some intuition, if `f` corresponds to a Weyl-fermion field, then
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- `position f e x`, `e` would correspond to a Lorentz index `α`, and `position f e x` would,
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once represented in the operator algebra, be proportional to the operator
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`∑ s, ∫ d^3p/(…) (x_α(p,s) a(p, s) e^{-i p x} + y_α(p,s) a^†(p, s) e^{-i p x})`.
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- `outAsymp f e p`, `e` would corresond to a spin `s`, and `outAsymp f e p` would,
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- `outAsymp f e p`, `e` would correspond to a spin `s`, and `outAsymp f e p` would,
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once represented in the operator algebra, be proportional to the
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annihilation operator `a^†(p, s)`.
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@ -83,7 +83,7 @@ As some intuition, if `f` corresponds to a Weyl-fermion field, it would contribu
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- an element corresponding to the creation parts of position operators for each each Lorentz
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index `α`:
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`∑ s, ∫ d^3p/(…) (x_α(p,s) a(p, s) e^{-i p x})`.
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- an element corresponding to anihilation parts of position operator,
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- an element corresponding to annihilation parts of position operator,
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for each each Lorentz index `α`:
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`∑ s, ∫ d^3p/(…) (y_α(p,s) a^†(p, s) e^{-i p x})`.
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- an element corresponding to outgoing asymptotic operators for each spin `s`: `a^†(p, s)`.
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@ -98,7 +98,7 @@ def crAnFieldOpToFieldOp : 𝓕.CrAnFieldOp → 𝓕.FieldOp := Sigma.fst
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lemma crAnFieldOpToFieldOp_prod (s : 𝓕.FieldOp) (t : 𝓕.fieldOpToCrAnType s) :
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𝓕.crAnFieldOpToFieldOp ⟨s, t⟩ = s := rfl
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/-- For a field specficiation `𝓕`, `𝓕.crAnFieldOpToCreateAnnihilate` is the map from
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/-- For a field specification `𝓕`, `𝓕.crAnFieldOpToCreateAnnihilate` is the map from
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`𝓕.CrAnFieldOp` to `CreateAnnihilate` taking `φ` to `create` if
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- `φ` corresponds to an incoming asymptotic field operator or the creation part of a position based
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field operator.
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@ -126,7 +126,7 @@ def singletonEquiv {φ : 𝓕.FieldOp} : CrAnSection [φ] ≃
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simp only [head]
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rfl
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/-- An equivalence seperating the head of a creation and annihilation section
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/-- An equivalence separating the head of a creation and annihilation section
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from the tail. -/
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def consEquiv {φ : 𝓕.FieldOp} {φs : List 𝓕.FieldOp} : CrAnSection (φ :: φs) ≃
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𝓕.fieldOpToCrAnType φ × CrAnSection φs where
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@ -32,7 +32,7 @@ def timeOrderRel : 𝓕.FieldOp → 𝓕.FieldOp → Prop
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| FieldOp.inAsymp _, FieldOp.position _ => False
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| FieldOp.inAsymp _, FieldOp.inAsymp _ => True
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/-- The relation `timeOrderRel` is decidable, but not computablly so due to
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/-- The relation `timeOrderRel` is decidable, but not computable so due to
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`Real.decidableLE`. -/
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noncomputable instance : (φ φ' : 𝓕.FieldOp) → Decidable (timeOrderRel φ φ')
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| FieldOp.outAsymp _, _ => isTrue True.intro
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@ -206,7 +206,7 @@ it is needed that the operator with the greatest time is to the left.
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-/
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def crAnTimeOrderRel (a b : 𝓕.CrAnFieldOp) : Prop := 𝓕.timeOrderRel a.1 b.1
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/-- The relation `crAnTimeOrderRel` is decidable, but not computablly so due to
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/-- The relation `crAnTimeOrderRel` is decidable, but not computable so due to
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`Real.decidableLE`. -/
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noncomputable instance (φ φ' : 𝓕.CrAnFieldOp) : Decidable (crAnTimeOrderRel φ φ') :=
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inferInstanceAs (Decidable (𝓕.timeOrderRel φ.1 φ'.1))
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@ -508,7 +508,7 @@ lemma sum_crAnSections_timeOrder {φs : List 𝓕.FieldOp} [AddCommMonoid M]
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def normTimeOrderRel (a b : 𝓕.CrAnFieldOp) : Prop :=
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crAnTimeOrderRel a b ∧ (crAnTimeOrderRel b a → normalOrderRel a b)
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/-- The relation `normTimeOrderRel` is decidable, but not computablly so due to
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/-- The relation `normTimeOrderRel` is decidable, but not computable so due to
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`Real.decidableLE`. -/
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noncomputable instance (φ φ' : 𝓕.CrAnFieldOp) : Decidable (normTimeOrderRel φ φ') :=
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instDecidableAnd
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@ -26,7 +26,7 @@ namespace FieldStatistic
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variable {𝓕 : Type}
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/-- The type `FieldStatistic` carries an instance of a commuative group in which
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/-- The type `FieldStatistic` carries an instance of a commutative group in which
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- `bosonic * bosonic = bosonic`
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- `bosonic * fermionic = fermionic`
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- `fermionic * bosonic = fermionic`
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@ -21,7 +21,7 @@ open FieldStatistic
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/-- Given a Wick contraction `c : WickContraction n` and `i1 i2 : Fin n` the finite set
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of elements of `Fin n` between `i1` and `i2` which are either uncontracted
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or are contracted but are contracted with an element occuring after `i1`.
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or are contracted but are contracted with an element occurring after `i1`.
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I.e. the elements of `Fin n` between `i1` and `i2` which are not contracted with before `i1`.
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One should assume `i1 < i2` otherwise this finite set is empty. -/
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def signFinset (c : WickContraction n) (i1 i2 : Fin n) : Finset (Fin n) :=
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@ -506,7 +506,7 @@ lemma hasEqTimeEquiv_ext_sigma {φs : List 𝓕.FieldOp} {x1 x2 :
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simp only [ne_eq, congr_refl] at h2
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simp [h2]
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/-- The equivalence which seperates a Wick contraction which has an equal time contraction
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/-- The equivalence which separates a Wick contraction which has an equal time contraction
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into a non-empty contraction only between equal-time fields and a Wick contraction which
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does not have equal time contractions. -/
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def hasEqTimeEquiv (φs : List 𝓕.FieldOp) :
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