refactor: More spellings
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jstoobysmith
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10 changed files with 21 additions and 21 deletions
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@ -19,8 +19,8 @@ open FieldStatistic
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variable (𝓕 : FieldSpecification)
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/-- The set contains the super-commutors equal to zero in the operator algebra.
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This contains e.g. the super-commutor of two creation operators. -/
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/-- The set contains the super-commutators equal to zero in the operator algebra.
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This contains e.g. the super-commutator of two creation operators. -/
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def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
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{ x |
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(∃ (φ1 φ2 φ3 : 𝓕.CrAnFieldOp),
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@ -42,7 +42,7 @@ def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
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This corresponds to the condition that two operators with different statistics always
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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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when combined with the conditions above, that the super-commutator is in the center of the
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of the algebra.
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-/
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abbrev FieldOpAlgebra : Type := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.Quotient
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@ -229,8 +229,8 @@ The proof of this result ultimately goes as follows
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a `ofCrAnList φsn` where `φsn` is the normal ordering of `φ₀…φₙ`.
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- `superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum` is used to rewrite the super commutator of `φ`
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(considered as a list with one element) with
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`ofCrAnList φsn` as a sum of supercommutors, one for each element of `φsn`.
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- The fact that super-commutors are in the center of `𝓕.FieldOpAlgebra` is used to rearrange terms.
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`ofCrAnList φsn` as a sum of super commutators, one for each element of `φsn`.
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- The fact that super-commutators are in the center of `𝓕.FieldOpAlgebra` is used to rearrange terms.
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- Properties of ordered lists, and `normalOrderSign_eraseIdx` are then used to complete the proof.
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-/
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lemma ofCrAnOp_superCommute_normalOrder_ofCrAnList_sum (φ : 𝓕.CrAnFieldOp)
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@ -281,7 +281,7 @@ lemma ofCrAnOp_superCommute_normalOrder_ofFieldOpList_sum (φ : 𝓕.CrAnFieldOp
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rw [← Finset.sum_mul, ← map_sum, ← map_sum, ← ofFieldOp_eq_sum, ← ofFieldOpList_eq_sum]
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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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The commutator of the annihilation part of a field operator with a normal ordered list of field
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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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@ -49,7 +49,7 @@ lemma ι_superCommuteF_eq_of_equiv_right (a b1 b2 : 𝓕.FieldOpFreeAlgebra) (h
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simp only [LinearMap.mem_ker, ← map_sub]
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exact ι_superCommuteF_right_zero_of_mem_ideal a (b1 - b2) h
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/-- The super commutor on the `FieldOpAlgebra` defined as a linear map `[a,_]ₛ`. -/
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/-- The super commutator on the `FieldOpAlgebra` defined as a linear map `[a,_]ₛ`. -/
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noncomputable def superCommuteRight (a : 𝓕.FieldOpFreeAlgebra) :
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FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
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toFun := Quotient.lift (ι.toLinearMap ∘ₗ superCommuteF a)
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@ -367,7 +367,7 @@ lemma superCommute_anPart_ofFieldOp (φ φ' : 𝓕.FieldOp) :
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## Mul equal superCommute
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Lemmas which rewrite a multiplication of two elements of the algebra as their commuted
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multiplication with a sign plus the super commutor.
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multiplication with a sign plus the super commutator.
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-/
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@ -446,7 +446,7 @@ lemma anPart_mul_anPart_swap (φ φ' : 𝓕.FieldOp) :
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/-!
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## Symmetry of the super commutor.
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## Symmetry of the super commutator.
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-/
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@ -17,7 +17,7 @@ namespace FieldOpFreeAlgebra
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/-!
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## The super commutor on the FieldOpFreeAlgebra.
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## The super commutator on the FieldOpFreeAlgebra.
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-/
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@ -40,7 +40,7 @@ scoped[FieldSpecification.FieldOpFreeAlgebra] notation "[" φs "," φs' "]ₛca"
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/-!
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## The super commutor of different types of elements
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## The super commutator of different types of elements
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-/
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@ -257,7 +257,7 @@ lemma superCommuteF_anPartF_ofFieldOpF (φ φ' : 𝓕.FieldOp) :
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## Mul equal superCommuteF
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Lemmas which rewrite a multiplication of two elements of the algebra as their commuted
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multiplication with a sign plus the super commutor.
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multiplication with a sign plus the super commutator.
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-/
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lemma ofCrAnListF_mul_ofCrAnListF_eq_superCommuteF (φs φs' : List 𝓕.CrAnFieldOp) :
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@ -328,7 +328,7 @@ lemma ofCrAnListF_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnField
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/-!
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## Symmetry of the super commutor.
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## Symmetry of the super commutator.
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-/
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@ -359,7 +359,7 @@ lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_symm (φ φ' : 𝓕.CrAnFieldOp) :
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/-!
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## Splitting the super commutor on lists into sums.
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## Splitting the super commutator on lists into sums.
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-/
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