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refactor: More docs for Wick's theorems

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jstoobysmith 2025-02-10 10:40:07 +00:00
parent 4096010e70
commit b30a49d7db
8 changed files with 38 additions and 38 deletions
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6
HepLean/PerturbationTheory/WickContraction/TimeCond.lean
View file

@ -97,7 +97,7 @@ lemma empty_mem {φs : List 𝓕.FieldOp} : empty (n := φs.length).EqTimeOnly :
simp [empty]
/-- Let `φs` be a list of `𝓕.FieldOp` and `φsΛ` a `WickContraction` of `φs` with
in which every contraction involves two `FieldOp`s that have the same time. Then
in which every contraction involves two `𝓕FieldOp`s that have the same time, then
`φsΛ.staticContract = φsΛ.timeContract`. -/
lemma staticContract_eq_timeContract_of_eqTimeOnly (h : φsΛ.EqTimeOnly) :
φsΛ.staticContract = φsΛ.timeContract := by
@ -194,7 +194,7 @@ lemma timeOrder_timeContract_mul_of_eqTimeOnly_mid {φs : List 𝓕.FieldOp}
exact timeOrder_timeContract_mul_of_eqTimeOnly_mid_induction φsΛ hl a b φsΛ.1.card rfl
/-- Let `φs` be a list of `𝓕.FieldOp`, `φsΛ` a `WickContraction` of `φs` with
in which every contraction involves two `FieldOp`s that have the same time and
in which every contraction involves two `𝓕.FieldOp`s that have the same time and
`b` a general element in `𝓕.FieldOpAlgebra`. Then
`𝓣(φsΛ.timeContract.1 * b) = φsΛ.timeContract.1 * 𝓣(b)`.
@ -248,7 +248,7 @@ lemma timeOrder_timeContract_of_not_eqTimeOnly {φs : List 𝓕.FieldOp}
simp_all
/-- Let `φs` be a list of `𝓕.FieldOp` and `φsΛ` a `WickContraction` with
at least one contraction between `FieldOp` that do not have the same time. Then
at least one contraction between `𝓕.FieldOp` that do not have the same time. Then
`𝓣(φsΛ.staticContract.1) = 0`. -/
lemma timeOrder_staticContract_of_not_mem {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
(hl : ¬ φsΛ.EqTimeOnly) : 𝓣(φsΛ.staticContract.1) = 0 := by

8
HepLean/PerturbationTheory/WickContraction/TimeContract.lean
View file

@ -21,7 +21,7 @@ open FieldOpAlgebra
/-- For a list `φs` of `𝓕.FieldOp` and a Wick contraction `φsΛ` the
element of the center of `𝓕.FieldOpAlgebra`, `φsΛ.timeContract` is defined as the product
of `timeContract φs[j] φs[k]` over contracted pairs `{j, k}` (both indices of `φs`) in `φsΛ`
of `timeContract φs[j] φs[k]` over contracted pairs `{j, k}` in `φsΛ`
with `j < k`. -/
noncomputable def timeContract {φs : List 𝓕.FieldOp}
(φsΛ : WickContraction φs.length) :
@ -86,7 +86,7 @@ open FieldStatistic
- `[anPart φ, φs[k]]ₛ`
- `φsΛ.timeContract`
- two copies of the exchange sign of `φ` with the uncontracted fields in `φ₀…φₖ₋₁`.
These two exchange signs cancle each other out but are included for convenience.
These two exchange signs cancel each other out but are included for convenience.
The proof of this result ultimately a consequence of definitions and
`timeContract_of_timeOrderRel`. -/
@ -125,15 +125,13 @@ lemma timeContract_insert_some_of_lt
/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
`𝓕.FieldOp`, a `i ≤ φs.length` and a `k` in `φsΛ.uncontracted` such that `k < i`, with the
condition that `φs[k]` does not have has greater or equal time to `φ`, then
condition that `φs[k]` does not have time greater or equal to `φ`, then
`(φsΛ ↩Λ φ i (some k)).timeContract` is equal to the product of
- `[anPart φ, φs[k]]ₛ`
- `φsΛ.timeContract`
- the exchange sign of `φ` with the uncontracted fields in `φ₀…φₖ₋₁`.
- the exchange sign of `φ` with the uncontracted fields in `φ₀…φₖ`.
Most of the contributes to the exchange signs cancle.
The proof of this result ultimately a consequence of definitions and
`timeContract_of_not_timeOrderRel_expand`. -/
lemma timeContract_insert_some_of_not_lt