refactor: Update supercommute notation
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@ -66,29 +66,29 @@ def fieldOpToCreateAnnihilateTypeCongr : {i j : 𝓕.FieldOp} → i = j →
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For a field specification `𝓕`, the (sigma) type `𝓕.CrAnFieldOp`
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corresponds to the type of creation and annihilation parts of field operators.
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It formally defined to consist of the following elements:
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- for each incoming asymptotic field operator `φ` in `𝓕.FieldOp` an element
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- For each incoming asymptotic field operator `φ` in `𝓕.FieldOp` an element
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written as `⟨φ, ()⟩` in `𝓕.CrAnFieldOp`, corresponding to the creation part of `φ`.
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Here `φ` has no annihilation part. (Here `()` is the unique element of `Unit`.)
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- for each position field operator `φ` in `𝓕.FieldOp` an element of `𝓕.CrAnFieldOp`
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- For each position field operator `φ` in `𝓕.FieldOp` an element of `𝓕.CrAnFieldOp`
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written as `⟨φ, .create⟩`, corresponding to the creation part of `φ`.
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- for each position field operator `φ` in `𝓕.FieldOp` an element of `𝓕.CrAnFieldOp`
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- For each position field operator `φ` in `𝓕.FieldOp` an element of `𝓕.CrAnFieldOp`
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written as `⟨φ, .annihilate⟩`, corresponding to the annihilation part of `φ`.
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- for each outgoing asymptotic field operator `φ` in `𝓕.FieldOp` an element
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- For each outgoing asymptotic field operator `φ` in `𝓕.FieldOp` an element
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written as `⟨φ, ()⟩` in `𝓕.CrAnFieldOp`, corresponding to the annihilation part of `φ`.
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Here `φ` has no creation part. (Here `()` is the unique element of `Unit`.)
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As an example, if `f` corresponds to a Weyl-fermion field, it would contribute
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the following elements to `𝓕.CrAnFieldOp`
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- an element corresponding to incoming asymptotic operators for each spin `s`: `a(p, s)`.
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- an element corresponding to the creation parts of position operators for each each Lorentz
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index `a`:
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- For each spin `s`, an element corresponding to an incoming asymptotic operator: `a(p, s)`.
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- For each each Lorentz
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index `a`, an element corresponding to the creation part of a position operator:
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`∑ s, ∫ d³p/(…) (xₐ (p,s) a(p, s) e ^ (-i p x))`.
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- an element corresponding to annihilation parts of position operator,
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for each each Lorentz index `a`:
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- For each each Lorentz
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index `a`,an element corresponding to annihilation part of a position operator:
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`∑ s, ∫ d³p/(…) (yₐ(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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- For each spin `s`, element corresponding to an outgoing asymptotic operator: `a†(p, s)`.
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-/
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def CrAnFieldOp : Type := Σ (s : 𝓕.FieldOp), 𝓕.fieldOpToCrAnType s
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