docs: Docs for Wick contractions
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13 changed files with 123 additions and 64 deletions
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@ -28,10 +28,11 @@ 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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/-- Given a Wick contraction `φsΛ` associated with a list of states `φs`
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the sign associated with `φsΛ` is the sign corresponding to the number
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of fermionic-fermionic exchanges one must do to put elements in contracted pairs
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of `φsΛ` next to each other. -/
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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 done to put
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contracted pairs with `φsΛ` next to one another, starting 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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@ -238,10 +238,14 @@ lemma signInsertNone_eq_filterset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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simp [h]
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· exact hG
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/-- For `φsΛ` a grading compliant Wick contraction, and `i : Fin φs.length.succ` we have
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/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a graded compliant Wick contraction `φsΛ` of `φs`,
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an `i ≤ φs.length` and a `φ` in `𝓕.FieldOp`, the following relation holds
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`(φsΛ ↩Λ φ i none).sign = s * φsΛ.sign`
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where `s` is the sign got by moving `φ` through the elements of `φ₀…φᵢ₋₁` which
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are contracted. -/
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are contracted with some element.
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The proof of this result involves a careful consideration of the contributions of different
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`FieldOp`s in `φs` to the sign of `φsΛ ↩Λ φ i none`. -/
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lemma sign_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (hG : GradingCompliant φs φsΛ) :
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(φsΛ ↩Λ φ i none).sign = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, Finset.univ.filter
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@ -250,6 +254,11 @@ lemma sign_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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rw [signInsertNone_eq_filterset]
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exact hG
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/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a graded compliant Wick contraction `φsΛ` of `φs`,
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and a `φ` in `𝓕.FieldOp`, the following relation holds
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`(φsΛ ↩Λ φ 0 none).sign = φsΛ.sign`.
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This is a direct corollary of `sign_insert_none`. -/
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lemma sign_insert_none_zero (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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(φsΛ : WickContraction φs.length) : (φsΛ ↩Λ φ 0 none).sign = φsΛ.sign := by
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rw [sign_insert_none_eq_signInsertNone_mul_sign]
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@ -855,13 +855,15 @@ lemma signInsertSome_mul_filter_contracted_of_not_lt (φ : 𝓕.FieldOp) (φs :
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exact fun h => lt_of_le_of_ne h (Fin.succAbove_ne i ((φsΛ.getDual? j).get hj))
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/--
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For `k < i`, the sign of `φsΛ ↩Λ φ i (some k)` is equal to the product of
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- the sign associated with moving `φ` through the `φsΛ`-uncontracted fields in `φ₀…φₖ`,
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- the sign associated with moving `φ` through the fields in `φ₀…φᵢ₋₁`,
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For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
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`𝓕.FieldOp`, a `i ≤ φs.length` and a `k` in `φsΛ.uncontracted` such that `k<i`,
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the sign of `φsΛ ↩Λ φ i (some k)` is equal to the product of
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- the sign associated with moving `φ` through the `φsΛ`-uncontracted `FieldOp` in `φ₀…φₖ`,
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- the sign associated with moving `φ` through all `FieldOp` in `φ₀…φᵢ₋₁`,
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- the sign of `φsΛ`.
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The proof of this result involves a careful consideration of the contributions of different
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fields in `φs` to the sign of `φsΛ ↩Λ φ i (some k)`.
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`FieldOp` in `φs` to the sign of `φsΛ ↩Λ φ i (some k)`.
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-/
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lemma sign_insert_some_of_lt (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (k : φsΛ.uncontracted)
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@ -878,13 +880,15 @@ lemma sign_insert_some_of_lt (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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simp
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/--
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For `i ≤ k`, the sign of `φsΛ ↩Λ φ i (some k)` is equal to the product of
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- the sign associated with moving `φ` through the `φsΛ`-uncontracted fields in `φ₀…φₖ₋₁`,
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- the sign associated with moving `φ` through the fields in `φ₀…φᵢ₋₁`,
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For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
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`𝓕.FieldOp`, a `i ≤ φs.length` and a `k` in `φsΛ.uncontracted` such that `i ≤ k`,
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the sign of `φsΛ ↩Λ φ i (some k)` is equal to the product of
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- the sign associated with moving `φ` through the `φsΛ`-uncontracted `FieldOp` in `φ₀…φₖ₋₁`,
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- the sign associated with moving `φ` through all the `FieldOp` in `φ₀…φᵢ₋₁`,
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- the sign of `φsΛ`.
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The proof of this result involves a careful consideration of the contributions of different
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fields in `φs` to the sign of `φsΛ ↩Λ φ i (some k)`.
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`FieldOp` in `φs` to the sign of `φsΛ ↩Λ φ i (some k)`.
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-/
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lemma sign_insert_some_of_not_lt (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (k : φsΛ.uncontracted)
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@ -900,6 +904,15 @@ lemma sign_insert_some_of_not_lt (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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rw [mul_comm, ← mul_assoc]
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simp
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/--
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For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
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`𝓕.FieldOp`, and a `k` in `φsΛ.uncontracted`,
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the sign of `φsΛ ↩Λ φ 0 (some k)` is equal to the product of
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- the sign associated with moving `φ` through the `φsΛ`-uncontracted `FieldOp` in `φ₀…φₖ₋₁`,
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- the sign of `φsΛ`.
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This is a direct corollary of `sign_insert_some_of_not_lt`.
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-/
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lemma sign_insert_some_zero (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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(φsΛ : WickContraction φs.length) (k : φsΛ.uncontracted)
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(hn : GradingCompliant φs φsΛ ∧ (𝓕|>ₛφ) = 𝓕|>ₛφs[k.1]) :
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@ -417,11 +417,13 @@ lemma join_sign_induction {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs
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apply sign_congr
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exact join_uncontractedListGet (singleton hij) φsucΛ'
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/-- Let `φsΛ` be a grading compliant Wick contraction for `φs` and
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`φsucΛ` a Wick contraction for `[φsΛ]ᵘᶜ`. Then `(join φsΛ φsucΛ).sign = φsΛ.sign * φsucΛ.sign`.
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/-- For a list `φs` of `𝓕.FieldOp`, a grading compliant Wick contraction `φsΛ` of `φs`,
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and a Wick contraction `φsucΛ` of `[φsΛ]ᵘᶜ`, the following relation holds
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`(join φsΛ φsucΛ).sign = φsΛ.sign * φsucΛ.sign`.
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This lemma manifests the fact that it does not matter which order contracted pairs are brought
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together when defining the sign of a contraction. -/
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In `φsΛ.sign` the sign is determined by starting with the contracted pair
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whose first element occurs at the left-most position. This lemma manifests that
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choice does not matter, and that contracted pairs can be brought together in any order. -/
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lemma join_sign {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
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(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (hc : φsΛ.GradingCompliant) :
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(join φsΛ φsucΛ).sign = φsΛ.sign * φsucΛ.sign := by
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