refactor: More docs for Wick's theorems
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8 changed files with 38 additions and 38 deletions
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@ -31,8 +31,8 @@ noncomputable section
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def staticWickTerm {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) : 𝓕.FieldOpAlgebra :=
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φsΛ.sign • φsΛ.staticContract * 𝓝(ofFieldOpList [φsΛ]ᵘᶜ)
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/-- For the empty list `[]` of `𝓕.FieldOp`, the `staticWickTerm` of the empty Wick contraction
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`empty` of `[]` (its only Wick contraction) is `1`. -/
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/-- For the empty list `[]` of `𝓕.FieldOp`, the `staticWickTerm` of the Wick contraction
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corresponding to the empty set `∅` (the only Wick contraction of `[]`) is `1`. -/
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@[simp]
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lemma staticWickTerm_empty_nil :
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staticWickTerm (empty (n := ([] : List 𝓕.FieldOp).length)) = 1 := by
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@ -116,9 +116,9 @@ lemma staticWickTerm_insert_zero_some (φ : 𝓕.FieldOp) (φs : List 𝓕.Field
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For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, the following relation
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holds
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`φ * φsΛ.staticWickTerm = ∑ k, (φsΛ ↩Λ φ i k).wickTerm`
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`φ * φsΛ.staticWickTerm = ∑ k, (φsΛ ↩Λ φ 0 k).staticWickTerm`
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where the sum is over all `k` in `Option φsΛ.uncontracted` (so either `none` or `some k`).
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where the sum is over all `k` in `Option φsΛ.uncontracted`, so `k` is either `none` or `some k`.
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The proof of proceeds as follows:
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- `ofFieldOp_mul_normalOrder_ofFieldOpList_eq_sum` is used to expand `φ 𝓝([φsΛ]ᵘᶜ)` as
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@ -22,7 +22,7 @@ namespace FieldOpAlgebra
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/--
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For a list `φs` of `𝓕.FieldOp`, the static version of Wick's theorem states that
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`φs =∑ φsΛ, φsΛ.staticWickTerm`
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`φs = ∑ φsΛ, φsΛ.staticWickTerm`
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where the sum is over all Wick contraction `φsΛ`.
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@ -24,7 +24,7 @@ namespace FieldOpAlgebra
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open FieldStatistic
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/-- For a field specification `𝓕`, and `φ` and `ψ` elements of `𝓕.FieldOp`, the element of
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`𝓕.FieldOpAlgebra`, `timeContract φ ψ` is defined to be `𝓣(φψ)- 𝓝(φψ)`. -/
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`𝓕.FieldOpAlgebra`, `timeContract φ ψ` is defined to be `𝓣(φψ) - 𝓝(φψ)`. -/
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def timeContract (φ ψ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra :=
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𝓣(ofFieldOp φ * ofFieldOp ψ) - 𝓝(ofFieldOp φ * ofFieldOp ψ)
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@ -33,8 +33,8 @@ noncomputable section
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def wickTerm {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) : 𝓕.FieldOpAlgebra :=
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φsΛ.sign • φsΛ.timeContract * 𝓝(ofFieldOpList [φsΛ]ᵘᶜ)
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/-- For the empty list `[]` of `𝓕.FieldOp`, the `wickTerm` of the empty Wick contraction
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`empty` of `[]` (its only Wick contraction) is `1`. -/
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/-- For the empty list `[]` of `𝓕.FieldOp`, the `wickTerm` of the Wick contraction
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corresponding to the empty set `∅` (the only Wick contraction of `[]`) is `1`. -/
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@[simp]
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lemma wickTerm_empty_nil :
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wickTerm (empty (n := ([] : List 𝓕.FieldOp).length)) = 1 := by
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@ -95,9 +95,9 @@ lemma wickTerm_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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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`, `i ≤ φs.length` and a `k` in `φsΛ.uncontracted`,
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such that all `FieldOp` in `φ₀…φᵢ₋₁` have time strictly less then `φ` and
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such that all `𝓕.FieldOp` in `φ₀…φᵢ₋₁` have time strictly less then `φ` and
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`φ` has a time greater then or equal to all `FieldOp` in `φ₀…φₙ`, then
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`(φsΛ ↩Λ φ i (some k)).wickTerm`
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`(φsΛ ↩Λ φ i (some k)).staticWickTerm`
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is equal the product of
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- the sign `𝓢(φ, φ₀…φᵢ₋₁) `
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- the sign `φsΛ.sign`
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@ -175,17 +175,18 @@ lemma wickTerm_insert_some (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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exact hg'
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/--
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Let `φsΛ` be a Wick contraction for `φs = φ₀φ₁…φₙ`. Let `φ` be a field with time
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greater then or equal to all the `FieldOp` in `φs`. Let `i ≤ φs.length` be such that
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all `FieldOp` in `φ₀…φᵢ₋₁` have time strictly less then `φ`. Then
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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 `i ≤ φs.length`
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such that all `𝓕.FieldOp` in `φ₀…φᵢ₋₁` have time strictly less then `φ` and
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`φ` has a time greater then or equal to all `FieldOp` in `φ₀…φₙ`, then
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`φ * φsΛ.wickTerm = 𝓢(φ, φ₀…φᵢ₋₁) • ∑ k, (φsΛ ↩Λ φ i k).wickTerm`
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where the sum is over all `k` in `Option φsΛ.uncontracted` (so either `none` or `some k`).
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where the sum is over all `k` in `Option φsΛ.uncontracted`, so `k` is either `none` or `some k`.
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The proof of proceeds as follows:
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- `ofFieldOp_mul_normalOrder_ofFieldOpList_eq_sum` is used to expand `φ 𝓝([φsΛ]ᵘᶜ)` as
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a sum over `k` in `Option φsΛ.uncontracted` of terms involving `[anPart φ, φs[k]]ₛ`..
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a sum over `k` in `Option φsΛ.uncontracted` of terms involving `[anPart φ, φs[k]]ₛ`.
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- Then `wickTerm_insert_none` and `wickTerm_insert_some` are used to equate terms.
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-/
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lemma mul_wickTerm_eq_sum (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (i : Fin φs.length.succ)
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@ -75,11 +75,11 @@ The inductive step works as follows:
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For the LHS:
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1. `timeOrder_eq_maxTimeField_mul_finset` is used to write
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`𝓣(φ₀…φₙ)` as ` 𝓢(φᵢ,φ₀…φᵢ₋₁) • φᵢ * 𝓣(φ₀…φᵢ₋₁φᵢ₊₁φₙ)` where `φᵢ` is
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`𝓣(φ₀…φₙ)` as `𝓢(φᵢ,φ₀…φᵢ₋₁) • φᵢ * 𝓣(φ₀…φᵢ₋₁φᵢ₊₁φₙ)` where `φᵢ` is
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the maximal time field in `φ₀…φₙ`
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2. The induction hypothesis is then used on `𝓣(φ₀…φᵢ₋₁φᵢ₊₁φₙ)` to expand it as a sum over
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Wick contractions of `φ₀…φᵢ₋₁φᵢ₊₁φₙ`.
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3. This gives terms of the form `φᵢ * φsΛ.timeContract` on which
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3. This gives terms of the form `φᵢ * φsΛ.wickTerm` on which
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`mul_wickTerm_eq_sum` is used where `φsΛ` is a Wick contraction of `φ₀…φᵢ₋₁φᵢ₊₁φ`,
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to rewrite terms as a sum over optional uncontracted elements of `φsΛ`
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@ -31,9 +31,10 @@ This result follows from
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- `static_wick_theorem` to rewrite `𝓣(φs)` on the left hand side as a sum of
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`𝓣(φsΛ.staticWickTerm)`.
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- `EqTimeOnly.timeOrder_staticContract_of_not_mem` and `timeOrder_timeOrder_mid` to set to
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zero those terms in which the contracted elements do not have equal time.
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- `staticContract_eq_timeContract_of_eqTimeOnly` to rewrite the static contract as a time contract
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for those terms which have equal time.
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those `𝓣(φsΛ.staticWickTerm)` for which `φsΛ` has a contracted pair which are not
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equal time to zero.
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- `staticContract_eq_timeContract_of_eqTimeOnly` to rewrite the static contract
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in the reminaing `𝓣(φsΛ.staticWickTerm)` as a time contract.
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- `timeOrder_timeContract_mul_of_eqTimeOnly_left` to move the time contracts out of the time
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ordering.
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-/
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@ -111,20 +112,20 @@ For a list `φs` of `𝓕.FieldOp`, then `𝓣(φs)` is equal to the sum of
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- `∑ φsΛ, φsΛ.wickTerm` where the sum is over all Wick contraction `φsΛ` which have
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no contractions of equal time.
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- `∑ φsΛ, sign φs ↑φsΛ • (φsΛ.1).timeContract ∑ φssucΛ, φssucΛ.wickTerm`, where
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- `∑ φsΛ, φsΛ.sign • φsΛ.timeContract * (∑ φssucΛ, φssucΛ.wickTerm)`, where
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the first sum is over all Wick contraction `φsΛ` which only have equal time contractions
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and the second sum is over all Wick contraction `φssucΛ` of the uncontracted elements of `φsΛ`
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which do not have any equal time contractions.
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The proof of this result relies on `wicks_theorem` to rewrite `𝓣(φs)` as a sum over
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all Wick contractions.
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The sum over all Wick contractions is then split additively into two parts using based on having or
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not having equal time contractions.
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The sum over Wick contractions which do have equal time contractions is turned into two sums
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one over the Wick contractions which only have equal time contractions and the other over the
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uncontracted elements of the Wick contraction which do not have equal time contractions using
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`join`.
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The properties of `join_sign_timeContract` is then used to equate terms.
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The proof of proceeds as follows
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- `wicks_theorem` is used to rewrite `𝓣(φs)` as a sum over all Wick contractions.
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- The sum over all Wick contractions is then split additively into two parts using based on having
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or not having an equal time contractions.
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- Using `join`, the sum `∑ φsΛ, _` over Wick contractions which do have equal time contractions
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is split into two sums `∑ φsΛ, ∑ φsucΛ, _`, the first over non-zero elements
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which only have equal time contractions and the second over Wick contractions `φsucΛ` of
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`[φsΛ]ᵘᶜ` which do not have equal time contractions.
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- `join_sign_timeContract` is then used to equate terms.
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-/
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lemma timeOrder_haveEqTime_split (φs : List 𝓕.FieldOp) :
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𝓣(ofFieldOpList φs) = (∑ (φsΛ : {φsΛ : WickContraction φs.length // ¬ HaveEqTime φsΛ}),
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@ -97,7 +97,7 @@ lemma empty_mem {φs : List 𝓕.FieldOp} : empty (n := φs.length).EqTimeOnly :
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simp [empty]
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/-- Let `φs` be a list of `𝓕.FieldOp` and `φsΛ` a `WickContraction` of `φs` with
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in which every contraction involves two `FieldOp`s that have the same time. Then
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in which every contraction involves two `𝓕FieldOp`s that have the same time, then
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`φsΛ.staticContract = φsΛ.timeContract`. -/
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lemma staticContract_eq_timeContract_of_eqTimeOnly (h : φsΛ.EqTimeOnly) :
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φsΛ.staticContract = φsΛ.timeContract := by
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@ -194,7 +194,7 @@ lemma timeOrder_timeContract_mul_of_eqTimeOnly_mid {φs : List 𝓕.FieldOp}
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exact timeOrder_timeContract_mul_of_eqTimeOnly_mid_induction φsΛ hl a b φsΛ.1.card rfl
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/-- Let `φs` be a list of `𝓕.FieldOp`, `φsΛ` a `WickContraction` of `φs` with
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in which every contraction involves two `FieldOp`s that have the same time and
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in which every contraction involves two `𝓕.FieldOp`s that have the same time and
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`b` a general element in `𝓕.FieldOpAlgebra`. Then
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`𝓣(φsΛ.timeContract.1 * b) = φsΛ.timeContract.1 * 𝓣(b)`.
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@ -248,7 +248,7 @@ lemma timeOrder_timeContract_of_not_eqTimeOnly {φs : List 𝓕.FieldOp}
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simp_all
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/-- Let `φs` be a list of `𝓕.FieldOp` and `φsΛ` a `WickContraction` with
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at least one contraction between `FieldOp` that do not have the same time. Then
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at least one contraction between `𝓕.FieldOp` that do not have the same time. Then
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`𝓣(φsΛ.staticContract.1) = 0`. -/
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lemma timeOrder_staticContract_of_not_mem {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
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(hl : ¬ φsΛ.EqTimeOnly) : 𝓣(φsΛ.staticContract.1) = 0 := by
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@ -21,7 +21,7 @@ open FieldOpAlgebra
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/-- For a list `φs` of `𝓕.FieldOp` and a Wick contraction `φsΛ` the
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element of the center of `𝓕.FieldOpAlgebra`, `φsΛ.timeContract` is defined as the product
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of `timeContract φs[j] φs[k]` over contracted pairs `{j, k}` (both indices of `φs`) in `φsΛ`
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of `timeContract φs[j] φs[k]` over contracted pairs `{j, k}` in `φsΛ`
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with `j < k`. -/
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noncomputable def timeContract {φs : List 𝓕.FieldOp}
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(φsΛ : WickContraction φs.length) :
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@ -86,7 +86,7 @@ open FieldStatistic
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- `[anPart φ, φs[k]]ₛ`
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- `φsΛ.timeContract`
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- two copies of the exchange sign of `φ` with the uncontracted fields in `φ₀…φₖ₋₁`.
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These two exchange signs cancle each other out but are included for convenience.
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These two exchange signs cancel each other out but are included for convenience.
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The proof of this result ultimately a consequence of definitions and
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`timeContract_of_timeOrderRel`. -/
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@ -125,15 +125,13 @@ lemma timeContract_insert_some_of_lt
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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`, with the
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condition that `φs[k]` does not have has greater or equal time to `φ`, then
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condition that `φs[k]` does not have time greater or equal to `φ`, then
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`(φsΛ ↩Λ φ i (some k)).timeContract` is equal to the product of
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- `[anPart φ, φs[k]]ₛ`
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- `φsΛ.timeContract`
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- the exchange sign of `φ` with the uncontracted fields in `φ₀…φₖ₋₁`.
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- the exchange sign of `φ` with the uncontracted fields in `φ₀…φₖ`.
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Most of the contributes to the exchange signs cancle.
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The proof of this result ultimately a consequence of definitions and
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`timeContract_of_not_timeOrderRel_expand`. -/
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lemma timeContract_insert_some_of_not_lt
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