52 lines
2 KiB
Text
52 lines
2 KiB
Text
/-
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Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.PerturbationTheory.WickContraction.InsertAndContract
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/-!
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# Sign associated with a contraction
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-/
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open FieldSpecification
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variable {𝓕 : FieldSpecification}
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namespace WickContraction
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variable {n : ℕ} (c : WickContraction n)
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open HepLean.List
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open FieldStatistic
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/-- Given a Wick contraction `c : WickContraction n` and `i1 i2 : Fin n` the finite set
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of elements of `Fin n` between `i1` and `i2` which are either uncontracted
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or are contracted but are contracted with an element occurring after `i1`.
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In other words, the elements of `Fin n` between `i1` and `i2` which are not
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contracted with before `i1`.
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One should assume `i1 < i2` otherwise this finite set is empty. -/
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def signFinset (c : WickContraction n) (i1 i2 : Fin n) : Finset (Fin n) :=
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Finset.univ.filter (fun i => i1 < i ∧ i < i2 ∧
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(c.getDual? i = none ∨ ∀ (h : (c.getDual? i).isSome), i1 < (c.getDual? i).get h))
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/-- For a list `φs` of `𝓕.FieldOp`, and a Wick contraction `φsΛ` of `φs`,
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the complex number `φsΛ.sign` is defined to be the sign (`1` or `-1`) corresponding
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to the number of `fermionic`-`fermionic` exchanges that must be done to put
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contracted pairs within `φsΛ` next to one another, starting recursively
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from the contracted pair
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whose first element occurs at the left-most position. -/
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def sign (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length) : ℂ :=
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∏ (a : φsΛ.1), 𝓢(𝓕 |>ₛ φs[φsΛ.sndFieldOfContract a],
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𝓕 |>ₛ ⟨φs.get, φsΛ.signFinset (φsΛ.fstFieldOfContract a) (φsΛ.sndFieldOfContract a)⟩)
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lemma sign_empty (φs : List 𝓕.FieldOp) :
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sign φs empty = 1 := by
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rw [sign]
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simp [empty]
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lemma sign_congr {φs φs' : List 𝓕.FieldOp} (h : φs = φs') (φsΛ : WickContraction φs.length) :
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sign φs' (congr (by simp [h]) φsΛ) = sign φs φsΛ := by
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subst h
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rfl
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end WickContraction
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