refactor: Update supercommute notation

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jstoobysmith 2025-02-13 10:44:15 +00:00
parent d2ce55ddd0
commit 82fae67ba3
12 changed files with 197 additions and 188 deletions

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