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docs: Docs for Wick contractions

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jstoobysmith 2025-02-07 06:58:41 +00:00
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9
HepLean/PerturbationTheory/WickContraction/Sign/Basic.lean
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@ -28,10 +28,11 @@ def signFinset (c : WickContraction n) (i1 i2 : Fin n) : Finset (Fin n) :=
Finset.univ.filter (fun i => i1 < i ∧ i < i2 ∧
(c.getDual? i = none ∨ ∀ (h : (c.getDual? i).isSome), i1 < (c.getDual? i).get h))
/-- Given a Wick contraction `φsΛ` associated with a list of states `φs`
the sign associated with `φsΛ` is the sign corresponding to the number
of fermionic-fermionic exchanges one must do to put elements in contracted pairs
of `φsΛ` next to each other. -/
/-- 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
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],
𝓕 |>ₛ ⟨φs.get, φsΛ.signFinset (φsΛ.fstFieldOfContract a) (φsΛ.sndFieldOfContract a)⟩)