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refactor: More notes on Wick's thoerem

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jstoobysmith 2025-02-04 09:58:30 +00:00
parent 20df8ece6c
commit 034f6c8c91
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15
HepLean/PerturbationTheory/FieldOpAlgebra/Basic.lean
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@ -32,8 +32,19 @@ def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
∨ (∃ (φ φ' : 𝓕.CrAnFieldOp) (_ : ¬ (𝓕 |>ₛ φ) = (𝓕 |>ₛ φ')),
x = [ofCrAnOpF φ, ofCrAnOpF φ']ₛca)}
/-- The algebra spanned by cr and an parts of fields, with appropriate super-commutors
set to zero. -/
/-- For a field specification `𝓕`, the algebra `𝓕.FieldOpAlgebra` is defined as the quotient
of the free algebra `𝓕.FieldOpFreeAlgebra` by the ideal generated by elements of the form
- `[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca`, which (with the other conditions)
gives us that super-commutors are in the center.
- `[ofCrAnOpF φc, ofCrAnOpF φc']ₛca` for `φc` and `φc'` creation operators. I.e two creation
operators always super-commute.
- `[ofCrAnOpF φa, ofCrAnOpF φa']ₛca` for `φa` and `φa'` annihilation operators. I.e two
annihilation operators always super-commute.
- `[ofCrAnOpF φ, ofCrAnOpF φ']ₛca` for `φ` and `φ'` field operators with different statistics.
I.e. Fermions super-commute with bosons.
The algebra `𝓕.FieldOpAlgebra` is the most general (in the correct sense) algebra
satisfying these properties.
-/
abbrev FieldOpAlgebra : Type := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.Quotient
namespace FieldOpAlgebra