docs: Notes for normal-ordered Wicks
This commit is contained in:
jstoobysmith
parent
c9607c459f
commit
d472604aec
15 changed files with 324 additions and 131 deletions
|
|
@ -347,7 +347,7 @@ noncomputable def contractStateAtIndex (φ : 𝓕.FieldOp) (φs : List 𝓕.Fiel
|
|||
/--
|
||||
For a field specification `𝓕`, a `φ` in `𝓕.FieldOp` and a list `φs` of `𝓕.FieldOp`
|
||||
the following relation holds in the algebra `𝓕.FieldOpAlgebra`,
|
||||
`φ * 𝓝(φ₀φ₁…φₙ) = 𝓝(φφ₀φ₁…φₙ) + ∑ i, (𝓢(φ,φ₀φ₁…φᵢ₋₁) • [anPartF φ, φᵢ]ₛ) * 𝓝(φ₀φ₁…φᵢ₋₁φᵢ₊₁…φₙ)`.
|
||||
`φ * 𝓝(φ₀φ₁…φₙ) = 𝓝(φφ₀φ₁…φₙ) + ∑ i, (𝓢(φ,φ₀φ₁…φᵢ₋₁) • [anPart φ, φᵢ]ₛ) * 𝓝(φ₀φ₁…φᵢ₋₁φᵢ₊₁…φₙ)`.
|
||||
|
||||
The proof of ultimetly goes as follows:
|
||||
- `ofFieldOp_eq_crPart_add_anPart` is used to split `φ` into its creation and annihilation parts.
|
||||
|
|
|
|||
|
|
@ -101,7 +101,7 @@ lemma normalOrder_uncontracted_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp
|
|||
/--
|
||||
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`, then
|
||||
`𝓝([φsΛ ↩Λ φ i none]ᵘᶜ)` is equal to the normal ordering of `[φsΛ]ᵘᶜ` with the `FieldOp`
|
||||
`𝓝([φsΛ ↩Λ φ i (some k)]ᵘᶜ)` is equal to the normal ordering of `[φsΛ]ᵘᶜ` with the `FieldOp`
|
||||
corresponding to `k` removed.
|
||||
|
||||
The proof of this result ultimately a consequence of definitions.
|
||||
|
|
|
|||
|
|
@ -22,12 +22,17 @@ open FieldOpAlgebra
|
|||
open FieldStatistic
|
||||
noncomputable section
|
||||
|
||||
/-- For a Wick contraction `φsΛ`, we define `staticWickTerm φsΛ` to be the element of
|
||||
`𝓕.FieldOpAlgebra` given by `φsΛ.sign • φsΛ.staticContract * 𝓝([φsΛ]ᵘᶜ)`. -/
|
||||
/-- For a list `φs` of `𝓕.FieldOp`, and a Wick contraction `φsΛ` of `φs`, the element
|
||||
of `𝓕.FieldOpAlgebra`, `φsΛ.staticWickTerm` is defined as
|
||||
|
||||
`φsΛ.sign • φsΛ.staticContract * 𝓝([φsΛ]ᵘᶜ)`.
|
||||
|
||||
This is term which appears in the static version Wick's theorem. -/
|
||||
def staticWickTerm {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) : 𝓕.FieldOpAlgebra :=
|
||||
φsΛ.sign • φsΛ.staticContract * 𝓝(ofFieldOpList [φsΛ]ᵘᶜ)
|
||||
|
||||
/-- The static Wick term for the empty contraction of the empty list is `1`. -/
|
||||
/-- For the empty list `[]` of `𝓕.FieldOp`, the `staticWickTerm` of the empty Wick contraction
|
||||
`empty` of `[]` (its only Wick contraction) is `1`. -/
|
||||
@[simp]
|
||||
lemma staticWickTerm_empty_nil :
|
||||
staticWickTerm (empty (n := ([] : List 𝓕.FieldOp).length)) = 1 := by
|
||||
|
|
@ -35,7 +40,9 @@ lemma staticWickTerm_empty_nil :
|
|||
simp [sign, empty, staticContract]
|
||||
|
||||
/--
|
||||
Let `φsΛ` be a Wick Contraction for `φs = φ₀φ₁…φₙ`. Then the following holds
|
||||
For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, and an element `φ` of
|
||||
`𝓕.FieldOp`, the following relation holds
|
||||
|
||||
`(φsΛ ↩Λ φ 0 none).staticWickTerm = φsΛ.sign • φsΛ.staticWickTerm * 𝓝(φ :: [φsΛ]ᵘᶜ)`
|
||||
|
||||
The proof of this result relies on
|
||||
|
|
@ -51,8 +58,9 @@ lemma staticWickTerm_insert_zero_none (φ : 𝓕.FieldOp) (φs : List 𝓕.Field
|
|||
simp only [staticContract_insert_none, insertAndContract_uncontractedList_none_zero,
|
||||
Algebra.smul_mul_assoc]
|
||||
|
||||
/-- Let `φsΛ` be a Wick contraction for `φs = φ₀φ₁…φₙ`. Then`(φsΛ ↩Λ φ 0 (some k)).wickTerm`
|
||||
is equal the product of
|
||||
/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
|
||||
`𝓕.FieldOp`, and a `k` in `φsΛ.uncontracted`, `(φsΛ ↩Λ φ 0 (some k)).wickTerm` is equal
|
||||
to the product of
|
||||
- the sign `𝓢(φ, φ₀…φᵢ₋₁) `
|
||||
- the sign `φsΛ.sign`
|
||||
- `φsΛ.staticContract`
|
||||
|
|
@ -60,9 +68,8 @@ is equal the product of
|
|||
uncontracted fields in `φ₀…φₖ₋₁`
|
||||
- the normal ordering `𝓝([φsΛ]ᵘᶜ.erase (uncontractedFieldOpEquiv φs φsΛ k))`.
|
||||
|
||||
The proof of this result relies on
|
||||
- `staticContract_insert_some_of_lt` to rewrite static
|
||||
contractions.
|
||||
The proof of this result ultimitley relies on
|
||||
- `staticContract_insert_some` to rewrite static contractions.
|
||||
- `normalOrder_uncontracted_some` to rewrite normal orderings.
|
||||
- `sign_insert_some_zero` to rewrite signs.
|
||||
-/
|
||||
|
|
@ -106,13 +113,16 @@ lemma staticWickTerm_insert_zero_some (φ : 𝓕.FieldOp) (φs : List 𝓕.Field
|
|||
simp
|
||||
|
||||
/--
|
||||
Let `φsΛ` be a Wick contraction for `φs = φ₀φ₁…φₙ`. Then
|
||||
For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, the following relation
|
||||
holds
|
||||
|
||||
`φ * φsΛ.staticWickTerm = ∑ k, (φsΛ ↩Λ φ i k).wickTerm`
|
||||
|
||||
where the sum is over all `k` in `Option φsΛ.uncontracted` (so either `none` or `some k`).
|
||||
|
||||
The proof of proceeds as follows:
|
||||
- `ofFieldOp_mul_normalOrder_ofFieldOpList_eq_sum` is used to expand `φ 𝓝([φsΛ]ᵘᶜ)` as
|
||||
a sum over `k` in `Option φsΛ.uncontracted` of terms involving `[φ, φs[k]]` etc.
|
||||
a sum over `k` in `Option φsΛ.uncontracted` of terms involving `[anPart φ, φs[k]]ₛ`.
|
||||
- Then `staticWickTerm_insert_zero_none` and `staticWickTerm_insert_zero_some` are
|
||||
used to equate terms.
|
||||
-/
|
||||
|
|
|
|||
|
|
@ -20,24 +20,41 @@ open FieldStatistic
|
|||
namespace FieldOpAlgebra
|
||||
|
||||
/--
|
||||
The static Wicks theorem states that
|
||||
`φ₀…φₙ` is equal to
|
||||
`∑ φsΛ, φsΛ.1.sign • φsΛ.1.staticContract * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)`
|
||||
over all Wick contraction `φsΛ`.
|
||||
This is compared to the ordinary Wicks theorem in which `staticContract` is replaced with
|
||||
`timeContract`.
|
||||
For a list `φs` of `𝓕.FieldOp`, the static version of Wick's theorem states that
|
||||
|
||||
`φs =∑ φsΛ, φsΛ.staticWickTerm`
|
||||
|
||||
where the sum is over all Wick contraction `φsΛ`.
|
||||
|
||||
The proof is via induction on `φs`.
|
||||
- The base case `φs = []` is handled by `staticWickTerm_empty_nil`.
|
||||
|
||||
The proof is via induction on `φs`. The base case `φs = []` is handled by `static_wick_theorem_nil`.
|
||||
The inductive step works as follows:
|
||||
- The proof considers `φ₀…φₙ` as `φ₀(φ₁…φₙ)` and use the induction hypothesis on `φ₁…φₙ`.
|
||||
- It also uses `ofFieldOp_mul_normalOrder_ofFieldOpList_eq_sum`
|
||||
|
||||
For the LHS:
|
||||
1. The proof considers `φ₀…φₙ` as `φ₀(φ₁…φₙ)` and use the induction hypothesis on `φ₁…φₙ`.
|
||||
2. This gives terms of the form `φ * φsΛ.staticWickTerm` on which
|
||||
`mul_staticWickTerm_eq_sum` is used where `φsΛ` is a Wick contraction of `φ₁…φₙ`,
|
||||
to rewrite terms as a sum over optional uncontracted elements of `φsΛ`
|
||||
|
||||
On the LHS we now have a sum over Wick contractions `φsΛ` of `φ₁…φₙ` (from 1) and optional
|
||||
uncontracted elements of `φsΛ` (from 2)
|
||||
|
||||
For the RHS:
|
||||
1. The sum over Wick contractions of `φ₀…φₙ` on the RHS
|
||||
is split via `insertLift_sum` into a sum over Wick contractions `φsΛ` of `φ₁…φₙ` and
|
||||
sum over optional uncontracted elements of `φsΛ`.
|
||||
|
||||
Both side now are sums over the same thing and their terms equate by the nature of the
|
||||
lemmas used.
|
||||
|
||||
-/
|
||||
theorem static_wick_theorem : (φs : List 𝓕.FieldOp) →
|
||||
ofFieldOpList φs = ∑ (φsΛ : WickContraction φs.length), φsΛ.staticWickTerm
|
||||
| [] => by
|
||||
simp only [ofFieldOpList, ofFieldOpListF_nil, map_one, List.length_nil]
|
||||
rw [sum_WickContraction_nil]
|
||||
simp
|
||||
rw [staticWickTerm_empty_nil]
|
||||
| φ :: φs => by
|
||||
rw [ofFieldOpList_cons, static_wick_theorem φs]
|
||||
rw [show (φ :: φs) = φs.insertIdx (⟨0, Nat.zero_lt_succ φs.length⟩ : Fin φs.length.succ) φ
|
||||
|
|
|
|||
|
|
@ -23,15 +23,19 @@ namespace FieldOpAlgebra
|
|||
|
||||
open FieldStatistic
|
||||
|
||||
/-- The time contraction of two FieldOp as an element of `𝓞.A` defined
|
||||
as their time ordering in the state algebra minus their normal ordering in the
|
||||
creation and annihlation algebra, both mapped to `𝓞.A`.. -/
|
||||
/-- For a field specification `𝓕`, and `φ` and `ψ` elements of `𝓕.FieldOp`, the element of
|
||||
`𝓕.FieldOpAlgebra`, `timeContract φ ψ` is defined to be `𝓣(φψ)- 𝓝(φψ)`. -/
|
||||
def timeContract (φ ψ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra :=
|
||||
𝓣(ofFieldOp φ * ofFieldOp ψ) - 𝓝(ofFieldOp φ * ofFieldOp ψ)
|
||||
|
||||
lemma timeContract_eq_smul (φ ψ : 𝓕.FieldOp) : timeContract φ ψ =
|
||||
𝓣(ofFieldOp φ * ofFieldOp ψ) + (-1 : ℂ) • 𝓝(ofFieldOp φ * ofFieldOp ψ) := by rfl
|
||||
|
||||
|
||||
/-- For a field specification `𝓕`, and `φ` and `ψ` elements of `𝓕.FieldOp`, if
|
||||
`φ` and `ψ` are time-ordered then
|
||||
|
||||
`timeContract φ ψ = [anPart φ, ofFieldOp ψ]ₛ`. -/
|
||||
lemma timeContract_of_timeOrderRel (φ ψ : 𝓕.FieldOp) (h : timeOrderRel φ ψ) :
|
||||
timeContract φ ψ = [anPart φ, ofFieldOp ψ]ₛ := by
|
||||
conv_rhs =>
|
||||
|
|
@ -55,6 +59,10 @@ lemma timeContract_of_not_timeOrderRel (φ ψ : 𝓕.FieldOp) (h : ¬ timeOrderR
|
|||
simp only [instCommGroup.eq_1, map_smul, map_add, smul_add]
|
||||
rw [smul_smul, smul_smul, mul_comm]
|
||||
|
||||
/-- For a field specification `𝓕`, and `φ` and `ψ` elements of `𝓕.FieldOp`, if
|
||||
`φ` and `ψ` are not time-ordered then
|
||||
|
||||
`timeContract φ ψ = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • [anPart ψ, ofFieldOp φ]ₛ`. -/
|
||||
lemma timeContract_of_not_timeOrderRel_expand (φ ψ : 𝓕.FieldOp) (h : ¬ timeOrderRel φ ψ) :
|
||||
timeContract φ ψ = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • [anPart ψ, ofFieldOp φ]ₛ := by
|
||||
rw [timeContract_of_not_timeOrderRel _ _ h]
|
||||
|
|
@ -62,6 +70,8 @@ lemma timeContract_of_not_timeOrderRel_expand (φ ψ : 𝓕.FieldOp) (h : ¬ tim
|
|||
have h1 := IsTotal.total (r := 𝓕.timeOrderRel) φ ψ
|
||||
simp_all
|
||||
|
||||
/-- For a field specification `𝓕`, and `φ` and `ψ` elements of `𝓕.FieldOp`, then
|
||||
`timeContract φ ψ` is in the center of `𝓕.FieldOpAlgebra`. -/
|
||||
lemma timeContract_mem_center (φ ψ : 𝓕.FieldOp) :
|
||||
timeContract φ ψ ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
|
||||
by_cases h : timeOrderRel φ ψ
|
||||
|
|
|
|||
|
|
@ -496,6 +496,8 @@ lemma timeOrder_superCommute_anPart_ofFieldOp_neq_time {φ ψ : 𝓕.FieldOp}
|
|||
apply timeOrder_superCommute_neq_time
|
||||
simp_all [crAnTimeOrderRel]
|
||||
|
||||
/-- For a field specification `𝓕`, and `a`, `b`, `c` in `𝓕.FieldOpAlgebra`, then
|
||||
`𝓣(a * b * c) = 𝓣(a * 𝓣(b) * c)`. -/
|
||||
lemma timeOrder_timeOrder_mid (a b c : 𝓕.FieldOpAlgebra) :
|
||||
𝓣(a * b * c) = 𝓣(a * 𝓣(b) * c) := by
|
||||
obtain ⟨a, rfl⟩ := ι_surjective a
|
||||
|
|
|
|||
|
|
@ -24,12 +24,17 @@ open FieldOpAlgebra
|
|||
open FieldStatistic
|
||||
noncomputable section
|
||||
|
||||
/-- For a Wick contraction `φsΛ`, we define `wickTerm φsΛ` to be the element of `𝓕.FieldOpAlgebra`
|
||||
given by `φsΛ.sign • φsΛ.timeContract * 𝓝([φsΛ]ᵘᶜ)`. -/
|
||||
/-- For a list `φs` of `𝓕.FieldOp`, and a Wick contraction `φsΛ` of `φs`, the element
|
||||
of `𝓕.FieldOpAlgebra`, `φsΛ.wickTerm` is defined as
|
||||
|
||||
`φsΛ.sign • φsΛ.timeContract * 𝓝([φsΛ]ᵘᶜ)`.
|
||||
|
||||
This is term which appears in the Wick's theorem. -/
|
||||
def wickTerm {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) : 𝓕.FieldOpAlgebra :=
|
||||
φsΛ.sign • φsΛ.timeContract * 𝓝(ofFieldOpList [φsΛ]ᵘᶜ)
|
||||
|
||||
/-- The Wick term for the empty contraction of the empty list is `1`. -/
|
||||
/-- For the empty list `[]` of `𝓕.FieldOp`, the `wickTerm` of the empty Wick contraction
|
||||
`empty` of `[]` (its only Wick contraction) is `1`. -/
|
||||
@[simp]
|
||||
lemma wickTerm_empty_nil :
|
||||
wickTerm (empty (n := ([] : List 𝓕.FieldOp).length)) = 1 := by
|
||||
|
|
@ -37,7 +42,9 @@ lemma wickTerm_empty_nil :
|
|||
simp [sign_empty]
|
||||
|
||||
/--
|
||||
Let `φsΛ` be a Wick Contraction for `φs = φ₀φ₁…φₙ`. Then the following holds
|
||||
For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
|
||||
`𝓕.FieldOp`, and `i ≤ φs.length` the following relation holds
|
||||
|
||||
`(φsΛ ↩Λ φ i none).wickTerm = 𝓢(φ, φ₀…φᵢ₋₁) φsΛ.sign • φsΛ.timeContract * 𝓝(φ :: [φsΛ]ᵘᶜ)`
|
||||
|
||||
The proof of this result relies on
|
||||
|
|
@ -86,16 +93,18 @@ lemma wickTerm_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
|
|||
simp only [ZeroMemClass.coe_zero, zero_mul, smul_zero]
|
||||
exact hg
|
||||
|
||||
/-- Let `φsΛ` be a Wick contraction for `φs = φ₀φ₁…φₙ`. Let `φ` be a field with time
|
||||
greater then or equal to all the fields in `φs`. Let `i` be a in `Fin φs.length.succ` such that
|
||||
all files in `φ₀…φᵢ₋₁` have time strictly less then `φ`. Then`(φsΛ ↩Λ φ i (some k)).wickTerm`
|
||||
/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
|
||||
`𝓕.FieldOp`, `i ≤ φs.length` and a `k` in `φsΛ.uncontracted`,
|
||||
such that all `FieldOp` in `φ₀…φᵢ₋₁` have time strictly less then `φ` and
|
||||
`φ` has a time greater then or equal to all `FieldOp` in `φ₀…φₙ`, then
|
||||
`(φsΛ ↩Λ φ i (some k)).wickTerm`
|
||||
is equal the product of
|
||||
- the sign `𝓢(φ, φ₀…φᵢ₋₁) `
|
||||
- the sign `φsΛ.sign`
|
||||
- `φsΛ.timeContract`
|
||||
- `s • [anPart φ, ofFieldOp φs[k]]ₛ` where `s` is the sign associated with moving `φ` through
|
||||
uncontracted fields in `φ₀…φₖ₋₁`
|
||||
- the normal ordering `𝓝([φsΛ]ᵘᶜ.erase (uncontractedFieldOpEquiv φs φsΛ k))`.
|
||||
- the normal ordering `[φsΛ]ᵘᶜ` with the field corresonding to `k` removed.
|
||||
|
||||
The proof of this result relies on
|
||||
- `timeContract_insert_some_of_not_lt`
|
||||
|
|
@ -167,14 +176,16 @@ lemma wickTerm_insert_some (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
|
|||
|
||||
/--
|
||||
Let `φsΛ` be a Wick contraction for `φs = φ₀φ₁…φₙ`. Let `φ` be a field with time
|
||||
greater then or equal to all the fields in `φs`. Let `i` be a in `Fin φs.length.succ` such that
|
||||
all files in `φ₀…φᵢ₋₁` have time strictly less then `φ`. Then
|
||||
greater then or equal to all the `FieldOp` in `φs`. Let `i ≤ φs.length` be such that
|
||||
all `FieldOp` in `φ₀…φᵢ₋₁` have time strictly less then `φ`. Then
|
||||
|
||||
`φ * φsΛ.wickTerm = 𝓢(φ, φ₀…φᵢ₋₁) • ∑ k, (φsΛ ↩Λ φ i k).wickTerm`
|
||||
|
||||
where the sum is over all `k` in `Option φsΛ.uncontracted` (so either `none` or `some k`).
|
||||
|
||||
The proof of proceeds as follows:
|
||||
- `ofFieldOp_mul_normalOrder_ofFieldOpList_eq_sum` is used to expand `φ 𝓝([φsΛ]ᵘᶜ)` as
|
||||
a sum over `k` in `Option φsΛ.uncontracted` of terms involving `[φ, φs[k]]` etc.
|
||||
a sum over `k` in `Option φsΛ.uncontracted` of terms involving `[anPart φ, φs[k]]ₛ`..
|
||||
- Then `wickTerm_insert_none` and `wickTerm_insert_some` are used to equate terms.
|
||||
-/
|
||||
lemma mul_wickTerm_eq_sum (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (i : Fin φs.length.succ)
|
||||
|
|
|
|||
|
|
@ -61,18 +61,38 @@ remark wicks_theorem_context := "
|
|||
The statement of these theorems for bosons is simplier then when fermions are involved, since
|
||||
one does not have to worry about the minus-signs picked up on exchanging fields."
|
||||
|
||||
/-- Wick's theorem states that for a list of fields `φs = φ₀…φₙ`
|
||||
`𝓣(φs) = ∑ φsΛ, (φsΛ.sign • φsΛ.timeContract) * 𝓝([φsΛ]ᵘᶜ)`
|
||||
where the sum is over all Wick contractions `φsΛ` of `φs`.
|
||||
/--
|
||||
For a list `φs` of `𝓕.FieldOp`, Wick's theorem states that
|
||||
|
||||
`𝓣(φs) = ∑ φsΛ, φsΛ.wickTerm`
|
||||
|
||||
where the sum is over all Wick contraction `φsΛ`.
|
||||
|
||||
The proof is via induction on `φs`.
|
||||
- The base case `φs = []` is handled by `wickTerm_empty_nil`.
|
||||
|
||||
The proof is via induction on `φs`. The base case `φs = []` is handled by `wicks_theorem_nil`.
|
||||
The inductive step works as follows:
|
||||
- The lemma `timeOrder_eq_maxTimeField_mul_finset` is used to write
|
||||
|
||||
For the LHS:
|
||||
1. `timeOrder_eq_maxTimeField_mul_finset` is used to write
|
||||
`𝓣(φ₀…φₙ)` as ` 𝓢(φᵢ,φ₀…φᵢ₋₁) • φᵢ * 𝓣(φ₀…φᵢ₋₁φᵢ₊₁φₙ)` where `φᵢ` is
|
||||
the maximal time field in `φ₀…φₙ`.
|
||||
- The induction hypothesis is used to expand `𝓣(φ₀…φᵢ₋₁φᵢ₊₁φₙ)` as a sum over Wick contractions of
|
||||
`φ₀…φᵢ₋₁φᵢ₊₁φₙ`.
|
||||
- Further the lemmas `wick_term_cons_eq_sum_wick_term` and `insertLift_sum` are used.
|
||||
the maximal time field in `φ₀…φₙ`
|
||||
2. The induction hypothesis is then used on `𝓣(φ₀…φᵢ₋₁φᵢ₊₁φₙ)` to expand it as a sum over
|
||||
Wick contractions of `φ₀…φᵢ₋₁φᵢ₊₁φₙ`.
|
||||
3. This gives terms of the form `φᵢ * φsΛ.timeContract` on which
|
||||
`mul_wickTerm_eq_sum` is used where `φsΛ` is a Wick contraction of `φ₀…φᵢ₋₁φᵢ₊₁φ`,
|
||||
to rewrite terms as a sum over optional uncontracted elements of `φsΛ`
|
||||
|
||||
On the LHS we now have a sum over Wick contractions `φsΛ` of `φ₀…φᵢ₋₁φᵢ₊₁φ` (from 2) and optional
|
||||
uncontracted elements of `φsΛ` (from 3)
|
||||
|
||||
For the RHS:
|
||||
1. The sum over Wick contractions of `φ₀…φₙ` on the RHS
|
||||
is split via `insertLift_sum` into a sum over Wick contractions `φsΛ` of `φ₀…φᵢ₋₁φᵢ₊₁φ` and
|
||||
sum over optional uncontracted elements of `φsΛ`.
|
||||
|
||||
Both side now are sums over the same thing and their terms equate by the nature of the
|
||||
lemmas used.
|
||||
-/
|
||||
theorem wicks_theorem : (φs : List 𝓕.FieldOp) → 𝓣(ofFieldOpList φs) =
|
||||
∑ (φsΛ : WickContraction φs.length), φsΛ.wickTerm
|
||||
|
|
@ -80,7 +100,7 @@ theorem wicks_theorem : (φs : List 𝓕.FieldOp) → 𝓣(ofFieldOpList φs) =
|
|||
rw [timeOrder_ofFieldOpList_nil]
|
||||
simp only [map_one, List.length_nil, Algebra.smul_mul_assoc]
|
||||
rw [sum_WickContraction_nil]
|
||||
simp
|
||||
simp only [wickTerm_empty_nil]
|
||||
| φ :: φs => by
|
||||
have ih := wicks_theorem (eraseMaxTimeField φ φs)
|
||||
conv_lhs => rw [timeOrder_eq_maxTimeField_mul_finset, ih, Finset.mul_sum]
|
||||
|
|
|
|||
|
|
@ -20,6 +20,23 @@ namespace FieldOpAlgebra
|
|||
open WickContraction
|
||||
open EqTimeOnly
|
||||
|
||||
/--
|
||||
For a list `φs` of `𝓕.FieldOp`, then
|
||||
|
||||
`𝓣(φs) = ∑ φsΛ, φsΛ.1.wickTerm • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ))`
|
||||
|
||||
where the sum is over all Wick contraction `φsΛ` which only have equal time contractions.
|
||||
|
||||
This result follows from
|
||||
- `static_wick_theorem` to rewrite `𝓣(φs)` on the left hand side as a sum of
|
||||
`𝓣(φsΛ.staticWickTerm)`.
|
||||
- `EqTimeOnly.timeOrder_staticContract_of_not_mem` and `timeOrder_timeOrder_mid` to set to
|
||||
zero those terms in which the contracted elements do not have equal time.
|
||||
- `staticContract_eq_timeContract_of_eqTimeOnly` to rewrite the static contract as a time contract
|
||||
for those terms which have equal time.
|
||||
- `timeOrder_timeContract_mul_of_eqTimeOnly_left` to move the time contracts out of the time
|
||||
ordering.
|
||||
-/
|
||||
lemma timeOrder_ofFieldOpList_eqTimeOnly (φs : List 𝓕.FieldOp) :
|
||||
𝓣(ofFieldOpList φs) = ∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs)}),
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)) := by
|
||||
|
|
@ -44,8 +61,9 @@ lemma timeOrder_ofFieldOpList_eqTimeOnly (φs : List 𝓕.FieldOp) :
|
|||
exact x.2
|
||||
exact x.2
|
||||
|
||||
|
||||
lemma timeOrder_ofFieldOpList_eq_eqTimeOnly_empty (φs : List 𝓕.FieldOp) :
|
||||
timeOrder (ofFieldOpList φs) = 𝓣(𝓝(ofFieldOpList φs)) +
|
||||
𝓣(ofFieldOpList φs) = 𝓣(𝓝(ofFieldOpList φs)) +
|
||||
∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs) ∧ φsΛ ≠ empty}),
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)) := by
|
||||
let e1 : {φsΛ : WickContraction φs.length // φsΛ.EqTimeOnly} ≃
|
||||
|
|
@ -71,6 +89,17 @@ lemma timeOrder_ofFieldOpList_eq_eqTimeOnly_empty (φs : List 𝓕.FieldOp) :
|
|||
rw [← e2.symm.sum_comp]
|
||||
rfl
|
||||
|
||||
/--
|
||||
For a list `φs` of `𝓕.FieldOp`, then
|
||||
|
||||
`𝓣(𝓝(φs)) = 𝓣(φs) - ∑ φsΛ, φsΛ.1.wickTerm • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ))`
|
||||
|
||||
where the sum is over all *non-empty* Wick contraction `φsΛ` which only
|
||||
have equal time contractions.
|
||||
|
||||
This result follows directly from
|
||||
- `timeOrder_ofFieldOpList_eqTimeOnly`
|
||||
-/
|
||||
lemma normalOrder_timeOrder_ofFieldOpList_eq_eqTimeOnly_empty (φs : List 𝓕.FieldOp) :
|
||||
𝓣(𝓝(ofFieldOpList φs)) = 𝓣(ofFieldOpList φs) -
|
||||
∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs) ∧ φsΛ ≠ empty}),
|
||||
|
|
@ -78,14 +107,33 @@ lemma normalOrder_timeOrder_ofFieldOpList_eq_eqTimeOnly_empty (φs : List 𝓕.F
|
|||
rw [timeOrder_ofFieldOpList_eq_eqTimeOnly_empty]
|
||||
simp
|
||||
|
||||
lemma normalOrder_timeOrder_ofFieldOpList_eq_haveEqTime_sum_not_haveEqTime (φs : List 𝓕.FieldOp) :
|
||||
𝓣(𝓝(ofFieldOpList φs)) = (∑ (φsΛ : {φsΛ : WickContraction φs.length // ¬ HaveEqTime φsΛ}),
|
||||
/--
|
||||
For a list `φs` of `𝓕.FieldOp`, then `𝓣(φs)` is equal to the sum of
|
||||
|
||||
- `∑ φsΛ, φsΛ.wickTerm` where the sum is over all Wick contraction `φsΛ` which have
|
||||
no contractions of equal time.
|
||||
- `∑ φsΛ, sign φs ↑φsΛ • (φsΛ.1).timeContract ∑ φssucΛ, φssucΛ.wickTerm`, where
|
||||
the first sum is over all Wick contraction `φsΛ` which only have equal time contractions
|
||||
and the second sum is over all Wick contraction `φssucΛ` of the uncontracted elements of `φsΛ`
|
||||
which do not have any equal time contractions.
|
||||
|
||||
The proof of this result relies on `wicks_theorem` to rewrite `𝓣(φs)` as a sum over
|
||||
all Wick contractions.
|
||||
The sum over all Wick contractions is then split additively into two parts using based on having or
|
||||
not having equal time contractions.
|
||||
The sum over Wick contractions which do have equal time contractions is turned into two sums
|
||||
one over the Wick contractions which only have equal time contractions and the other over the
|
||||
uncontracted elements of the Wick contraction which do not have equal time contractions using
|
||||
`join`.
|
||||
The properties of `join_sign_timeContract` is then used to equate terms.
|
||||
-/
|
||||
lemma timeOrder_haveEqTime_split (φs : List 𝓕.FieldOp) :
|
||||
𝓣(ofFieldOpList φs) = (∑ (φsΛ : {φsΛ : WickContraction φs.length // ¬ HaveEqTime φsΛ}),
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ))
|
||||
+ (∑ (φsΛ : {φsΛ : WickContraction φs.length // HaveEqTime φsΛ}),
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ))
|
||||
- ∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs) ∧ φsΛ ≠ empty}),
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)) := by
|
||||
rw [normalOrder_timeOrder_ofFieldOpList_eq_eqTimeOnly_empty]
|
||||
+ ∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs) ∧ φsΛ ≠ empty}),
|
||||
sign φs ↑φsΛ • (φsΛ.1).timeContract *
|
||||
(∑ φssucΛ : { φssucΛ : WickContraction [φsΛ.1]ᵘᶜ.length // ¬ φssucΛ.HaveEqTime },
|
||||
sign [φsΛ.1]ᵘᶜ φssucΛ • (φssucΛ.1).timeContract * normalOrder (ofFieldOpList [φssucΛ.1]ᵘᶜ)) := by
|
||||
rw [wicks_theorem]
|
||||
simp only [wickTerm]
|
||||
let e1 : WickContraction φs.length ≃ {φsΛ // HaveEqTime φsΛ} ⊕ {φsΛ // ¬ HaveEqTime φsΛ} := by
|
||||
|
|
@ -94,17 +142,9 @@ lemma normalOrder_timeOrder_ofFieldOpList_eq_haveEqTime_sum_not_haveEqTime (φs
|
|||
simp only [Equiv.symm_symm, Algebra.smul_mul_assoc, Fintype.sum_sum_type,
|
||||
Equiv.sumCompl_apply_inl, Equiv.sumCompl_apply_inr, ne_eq, sub_left_inj, e1]
|
||||
rw [add_comm]
|
||||
|
||||
lemma haveEqTime_wick_sum_eq_split (φs : List 𝓕.FieldOp) :
|
||||
(∑ (φsΛ : {φsΛ : WickContraction φs.length // HaveEqTime φsΛ}),
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)) =
|
||||
∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs) ∧ φsΛ ≠ empty}),
|
||||
(sign φs ↑φsΛ • (φsΛ.1).timeContract *
|
||||
∑ φssucΛ : { φssucΛ : WickContraction [φsΛ.1]ᵘᶜ.length // ¬φssucΛ.HaveEqTime },
|
||||
sign [φsΛ.1]ᵘᶜ φssucΛ •
|
||||
(φssucΛ.1).timeContract * normalOrder (ofFieldOpList [φssucΛ.1]ᵘᶜ)) := by
|
||||
congr 1
|
||||
let f : WickContraction φs.length → 𝓕.FieldOpAlgebra := fun φsΛ =>
|
||||
φsΛ.sign • φsΛ.timeContract.1 * 𝓝(ofFieldOpList [φsΛ]ᵘᶜ)
|
||||
φsΛ.sign • (φsΛ.timeContract.1 * 𝓝(ofFieldOpList [φsΛ]ᵘᶜ))
|
||||
change ∑ (φsΛ : {φsΛ : WickContraction φs.length // HaveEqTime φsΛ}), f φsΛ.1 = _
|
||||
rw [sum_haveEqTime]
|
||||
congr
|
||||
|
|
@ -112,12 +152,12 @@ lemma haveEqTime_wick_sum_eq_split (φs : List 𝓕.FieldOp) :
|
|||
simp only [f]
|
||||
conv_lhs =>
|
||||
enter [2, φsucΛ]
|
||||
enter [1]
|
||||
rw [← Algebra.smul_mul_assoc]
|
||||
rw [join_sign_timeContract φsΛ.1 φsucΛ.1]
|
||||
conv_lhs =>
|
||||
enter [2, φsucΛ]
|
||||
rw [mul_assoc]
|
||||
rw [← Finset.mul_sum]
|
||||
rw [← Finset.mul_sum, ← Algebra.smul_mul_assoc]
|
||||
congr
|
||||
funext φsΛ'
|
||||
simp only [ne_eq, Algebra.smul_mul_assoc]
|
||||
|
|
@ -132,10 +172,10 @@ lemma normalOrder_timeOrder_ofFieldOpList_eq_not_haveEqTime_sub_inductive (φs :
|
|||
(∑ φssucΛ : { φssucΛ : WickContraction [φsΛ.1]ᵘᶜ.length // ¬ φssucΛ.HaveEqTime },
|
||||
sign [φsΛ.1]ᵘᶜ φssucΛ • (φssucΛ.1).timeContract * normalOrder (ofFieldOpList [φssucΛ.1]ᵘᶜ) -
|
||||
𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ))) := by
|
||||
rw [normalOrder_timeOrder_ofFieldOpList_eq_haveEqTime_sum_not_haveEqTime]
|
||||
rw [normalOrder_timeOrder_ofFieldOpList_eq_eqTimeOnly_empty,
|
||||
timeOrder_haveEqTime_split]
|
||||
rw [add_sub_assoc]
|
||||
congr 1
|
||||
rw [haveEqTime_wick_sum_eq_split]
|
||||
simp only [ne_eq, Algebra.smul_mul_assoc]
|
||||
rw [← Finset.sum_sub_distrib]
|
||||
congr 1
|
||||
|
|
@ -177,12 +217,19 @@ lemma wicks_theorem_normal_order_empty : 𝓣(𝓝(ofFieldOpList [])) =
|
|||
rw [timeOrderF_ofCrAnListF]
|
||||
simp
|
||||
|
||||
/--
|
||||
Wicks theorem for normal ordering followed by time-ordering, states that
|
||||
`𝓣(𝓝(φ₀…φₙ))` is equal to
|
||||
`∑ φsΛ, φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)`
|
||||
over those Wick contraction `φsΛ` which do not have any equal time contractions.
|
||||
This is compared to the ordinary Wicks theorem which sums over all Wick contractions.
|
||||
/--For a list `φs` of `𝓕.FieldOp`, the normal-ordered version of Wick's theorem states that
|
||||
|
||||
`𝓣(𝓝(φs)) = ∑ φsΛ, φsΛ.wickTerm`
|
||||
|
||||
where the sum is over all Wick contraction `φsΛ` in which no two contracted elements
|
||||
have the same time.
|
||||
|
||||
The proof of proceeds by induction on `φs`, with the base case `[]` holding by following
|
||||
through definitions. and the inductive case holding as a result of
|
||||
- `timeOrder_haveEqTime_split`
|
||||
- `normalOrder_timeOrder_ofFieldOpList_eq_eqTimeOnly_empty`
|
||||
- and the induction hypothesis on `𝓣(𝓝([φsΛ.1]ᵘᶜ))` for contractions `φsΛ` of `φs` which only
|
||||
have equal time contractions and are non-empty.
|
||||
-/
|
||||
theorem wicks_theorem_normal_order : (φs : List 𝓕.FieldOp) →
|
||||
𝓣(𝓝(ofFieldOpList φs)) =
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue