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doc: Edits to Wick theorem docs

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jstoobysmith 2025-02-10 10:21:57 +00:00
parent 26d2c24c83
commit 4096010e70
32 changed files with 255 additions and 134 deletions
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5
HepLean/PerturbationTheory/WickContraction/Sign/Basic.lean
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@ -30,8 +30,9 @@ def signFinset (c : WickContraction n) (i1 i2 : Fin n) : Finset (Fin n) :=
/-- For a list `φs` of `𝓕.FieldOp`, and a Wick contraction `φsΛ` of `φs`,
the complex number `φsΛ.sign` is defined to be the sign (`1` or `-1`) corresponding
to the number of `fermionic`-`fermionic` exchanges that must done to put
contracted pairs with `φsΛ` next to one another, starting from the contracted pair
to the number of `fermionic`-`fermionic` exchanges that must be done to put
contracted pairs within `φsΛ` next to one another, starting recursively
from the contracted pair
whose first element occurs at the left-most position. -/
def sign (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length) : ℂ :=
∏ (a : φsΛ.1), 𝓢(𝓕 |>ₛ φs[φsΛ.sndFieldOfContract a],

4
HepLean/PerturbationTheory/WickContraction/Sign/InsertNone.lean
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@ -239,7 +239,7 @@ lemma signInsertNone_eq_filterset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
· exact hG
/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a graded compliant Wick contraction `φsΛ` of `φs`,
an `i ≤ φs.length` and a `φ` in `𝓕.FieldOp`, the following relation holds
an `i ≤ φs.length`, and a `φ` in `𝓕.FieldOp`, then
`(φsΛ ↩Λ φ i none).sign = s * φsΛ.sign`
where `s` is the sign got by moving `φ` through the elements of `φ₀…φᵢ₋₁` which
are contracted with some element.
@ -255,7 +255,7 @@ lemma sign_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
exact hG
/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a graded compliant Wick contraction `φsΛ` of `φs`,
and a `φ` in `𝓕.FieldOp`, the following relation holds `(φsΛ ↩Λ φ 0 none).sign = φsΛ.sign`.
and a `φ` in `𝓕.FieldOp`, then `(φsΛ ↩Λ φ 0 none).sign = φsΛ.sign`.
This is a direct corollary of `sign_insert_none`. -/
lemma sign_insert_none_zero (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)

2
HepLean/PerturbationTheory/WickContraction/Sign/Join.lean
View file

@ -422,7 +422,7 @@ lemma join_sign_induction {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs
`(join φsΛ φsucΛ).sign = φsΛ.sign * φsucΛ.sign`.
In `φsΛ.sign` the sign is determined by starting with the contracted pair
whose first element occurs at the left-most position. This lemma manifests that
whose first element occurs at the left-most position. This lemma manifests that this
choice does not matter, and that contracted pairs can be brought together in any order. -/
lemma join_sign {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (hc : φsΛ.GradingCompliant) :