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feat: cardinality of Wick contractions

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jstoobysmith 2025-02-04 11:50:07 +00:00
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27
HepLean/PerturbationTheory/WickContraction/Sign/InsertSome.lean
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@ -223,7 +223,7 @@ related to inserting a field `φ` at position `i` and contracting it with `j :
- If `j < i`, for each field `φₐ` in `φⱼ₊₁…φᵢ₋₁` without a dual at position `< j`
the sign `𝓢(φₐ, φᵢ)`.
- Else, for each field `φₐ` in `φᵢ…φⱼ₋₁` of `φs` without dual at position `< i` the sign
`𝓢(φₐ, φⱼ)`. -/
`𝓢(φₐ, φⱼ)`. -/
def signInsertSomeCoef (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length)
(i : Fin φs.length.succ) (j : φsΛ.uncontracted) : ℂ :=
let a : (φsΛ ↩Λ φ i (some j)).1 := congrLift (insertIdx_length_fin φ φs i).symm
@ -600,13 +600,13 @@ lemma signInsertSomeCoef_eq_finset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
simp [hφj]
/--
The following two signs are equal for `i.succAbove k < i`. The sign `signInsertSome φ φs φsΛ i k` which is constructed
as follows:
The following two signs are equal for `i.succAbove k < i`. The sign `signInsertSome φ φs φsΛ i k`
which is constructed as follows:
1a. For each contracted pair `{a1, a2}` in `φsΛ` with `a1 < a2` the sign
`𝓢(φ, φₐ₂)` if `a₁ < i ≤ a₂` and `a₁ < k`.
1b. For each contracted pair `{a1, a2}` in `φsΛ` with `a1 < a2` the sign
1b. For each contracted pair `{a1, a2}` in `φsΛ` with `a1 < a2` the sign
`𝓢(φⱼ, φₐ₂)` if `a₁ < k < a₂` and `i < a₁`.
1c. For each field `φₐ` in `φₖ₊₁…φᵢ₋₁` without a dual at position `< k`
1c. For each field `φₐ` in `φₖ₊₁…φᵢ₋₁` without a dual at position `< k`
the sign `𝓢(φₐ, φᵢ)`.
and the sign constructed as follows:
2a. For each uncontracted field `φₐ` in `φ₀…φₖ` in `φsΛ` the sign `𝓢(φ, φₐ)`
@ -621,11 +621,11 @@ The outline of why this is true can be got by considering contributions of field
For contracted fields `{a₁, a₂}` in `φsΛ` with `a₁ < a₂` we have the following cases:
- `φₐ₁` `φₐ₂` `a₁ < a₂ < k < i`, `a₁ -> 2b`, `a₂ -> 2b`,
- `φₐ₁` `φₐ₂` `a₁ < k < a₂ < i`, `a₁ -> 2b`, `a₂ -> 2b`,
- `φₐ₁` `φₐ₂` `a₁ < k < i ≤ a₂`, `a₁ -> 2b`, `a₂ -> 1a`
- `φₐ₁` `φₐ₂` `a₁ < k < i ≤ a₂`, `a₁ -> 2b`, `a₂ -> 1a`
- `φₐ₁` `φₐ₂` `k < a₁ < a₂ < i`, `a₁ -> 2b`, `a₂ -> 2b`, `a₁ -> 1c`, `a₂ -> 1c`
- `φₐ₁` `φₐ₂` `k < a₁ < i ≤ a₂ `,`a₁ -> 2b`, `a₁ -> 1c`
- `φₐ₁` `φₐ₂` `k < i ≤ a₁ < a₂ `, No contributions.
-/
- `φₐ₁` `φₐ₂` `k < i ≤ a₁ < a₂ `, No contributions.
-/
lemma signInsertSome_mul_filter_contracted_of_lt (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (k : φsΛ.uncontracted)
(hk : i.succAbove k < i) (hg : GradingCompliant φs φsΛ ∧ (𝓕 |>ₛ φ) = 𝓕 |>ₛ φs[k.1]) :
@ -727,16 +727,15 @@ lemma signInsertSome_mul_filter_contracted_of_lt (φ : 𝓕.FieldOp) (φs : List
or_true, imp_self]
omega
/--
The following two signs are equal for `i < i.succAbove k`.
The sign `signInsertSome φ φs φsΛ i k` which is constructed
as follows:
1a. For each contracted pair `{a1, a2}` in `φsΛ` with `a1 < a2` the sign
`𝓢(φ, φₐ₂)` if `a₁ < i ≤ a₂` and `a₁ < k`.
1b. For each contracted pair `{a1, a2}` in `φsΛ` with `a1 < a2` the sign
1b. For each contracted pair `{a1, a2}` in `φsΛ` with `a1 < a2` the sign
`𝓢(φⱼ, φₐ₂)` if `a₁ < k < a₂` and `i < a₁`.
1c. For each field `φₐ` in `φᵢ…φₖ₋₁` of `φs` without dual at position `< i` the sign
1c. For each field `φₐ` in `φᵢ…φₖ₋₁` of `φs` without dual at position `< i` the sign
`𝓢(φₐ, φⱼ)`.
and the sign constructed as follows:
2a. For each uncontracted field `φₐ` in `φ₀…φₖ₋₁` in `φsΛ` the sign `𝓢(φ, φₐ)`
@ -751,11 +750,11 @@ The outline of why this is true can be got by considering contributions of field
For contracted fields `{a₁, a₂}` in `φsΛ` with `a₁ < a₂` we have the following cases:
- `φₐ₁` `φₐ₂` `a₁ < a₂ < i ≤ k`, `a₁ -> 2b`, `a₂ -> 2b`
- `φₐ₁` `φₐ₂` `a₁ < i ≤ a₂ < k`, `a₁ -> 2b`, `a₂ -> 1a`
- `φₐ₁` `φₐ₂` `a₁ < i ≤ k < a₂`, `a₁ -> 2b`, `a₂ -> 1a`
- `φₐ₁` `φₐ₂` `i ≤ a₁ < a₂ < k`, `a₂ -> 1c`, `a₁ -> 1c`
- `φₐ₁` `φₐ₂` `a₁ < i ≤ k < a₂`, `a₁ -> 2b`, `a₂ -> 1a`
- `φₐ₁` `φₐ₂` `i ≤ a₁ < a₂ < k`, `a₂ -> 1c`, `a₁ -> 1c`
- `φₐ₁` `φₐ₂` `i ≤ a₁ < k < a₂ `, `a₁ -> 1c`, `a₁ -> 1b`
- `φₐ₁` `φₐ₂` `i ≤ k ≤ a₁ < a₂ `, No contributions
-/
-/
lemma signInsertSome_mul_filter_contracted_of_not_lt (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (k : φsΛ.uncontracted)
(hk : ¬ i.succAbove k < i) (hg : GradingCompliant φs φsΛ ∧ (𝓕 |>ₛ φ) = 𝓕 |>ₛ φs[k.1]) :