Sign on inserting and contracting #
The main results of this file are sign_insert_some_of_lt and sign_insert_some_of_not_lt which
write the sign of (φsΛ ↩Λ φ i (some k)).sign in terms of the sign of φsΛ.
Sign insert some #
Given a Wick contraction φsΛ the sign defined in the following way,
related to inserting a field φ at position i and contracting it with j : φsΛ.uncontracted.
- For each contracted pair
{a1, a2}inφsΛwitha1 < a2the sign𝓢(φ, φₐ₂)ifa₁ < i ≤ a₂anda₁ < j. - For each contracted pair
{a1, a2}inφsΛwitha1 < a2the sign𝓢(φⱼ, φₐ₂)ifa₁ < j < a₂andi < a₁.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Given a Wick contraction φsΛ the sign defined in the following way,
related to inserting a field φ at position i and contracting it with j : φsΛ.uncontracted.
- If
j < i, for each fieldφₐinφⱼ₊₁…φᵢ₋₁without a dual at position< jthe sign𝓢(φₐ, φᵢ). - Else, for each field
φₐinφᵢ…φⱼ₋₁ofφswithout dual at position< ithe sign𝓢(φₐ, φⱼ).
Equations
- One or more equations did not get rendered due to their size.
Instances For
Given a Wick contraction φsΛ associated with a list of states φs
and an i : Fin φs.length.succ, the change in sign of the contraction associated with
inserting φ into φs at position i and contracting it with j : c.uncontracted.
Equations
- WickContraction.signInsertSome φ φs φsΛ i j = WickContraction.signInsertSomeCoef φ φs φsΛ i j * WickContraction.signInsertSomeProd φ φs φsΛ i j
Instances For
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
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
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)
2b. For each field in φₐ in φ₀…φᵢ₋₁ the sign 𝓢(φ, φₐ).
The outline of why this is true can be got by considering contributions of fields.
φₐ,i ≤ a. No contributions.φₖ,k -> 2a,k -> 2bφₐ,a ≤ kuncontracteda -> 2a,a -> 2b.φₐ,k < a < iuncontracteda -> 1c,a -> 2b.
For contracted fields {a₁, a₂} in φsΛ with a₁ < a₂ we have the following cases:
φₐ₁φₐ₂a₁ < a₂ < k < i,a₁ -> 2b,a₂ -> 2b,φₐ₁φₐ₂a₁ < k < a₂ < i,a₁ -> 2b,a₂ -> 2b,φₐ₁φₐ₂a₁ < k < i ≤ a₂,a₁ -> 2b,a₂ -> 1aφₐ₁φₐ₂k < a₁ < a₂ < i,a₁ -> 2b,a₂ -> 2b,a₁ -> 1c,a₂ -> 1cφₐ₁φₐ₂k < a₁ < i ≤ a₂,a₁ -> 2b,a₁ -> 1cφₐ₁φₐ₂k < i ≤ a₁ < a₂, No contributions.
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
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
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)
2b. For each field in φₐ in φ₀…φᵢ₋₁ the sign 𝓢(φ, φₐ).
The outline of why this is true can be got by considering contributions of fields.
φₐ,k < a. No contributions.φₖ, No Contributesφₐ,a < iuncontracteda -> 2a,a -> 2b.φₐ,i ≤ a < kuncontracteda -> 1c,a -> 2a.
For contracted fields {a₁, a₂} in φsΛ with a₁ < a₂ we have the following cases:
φₐ₁φₐ₂a₁ < a₂ < i ≤ k,a₁ -> 2b,a₂ -> 2bφₐ₁φₐ₂a₁ < i ≤ a₂ < k,a₁ -> 2b,a₂ -> 1aφₐ₁φₐ₂a₁ < i ≤ k < a₂,a₁ -> 2b,a₂ -> 1aφₐ₁φₐ₂i ≤ a₁ < a₂ < k,a₂ -> 1c,a₁ -> 1cφₐ₁φₐ₂i ≤ a₁ < k < a₂,a₁ -> 1c,a₁ -> 1bφₐ₁φₐ₂i ≤ k ≤ a₁ < a₂, No contributions
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,
the sign of φsΛ ↩Λ φ i (some k) is equal to the product of
- the sign associated with moving
φthrough theφsΛ-uncontractedFieldOpinφ₀…φₖ, - the sign associated with moving
φthrough allFieldOpinφ₀…φᵢ₋₁, - the sign of
φsΛ.
The proof of this result involves a careful consideration of the contributions of different
FieldOp in φs to the sign of φsΛ ↩Λ φ i (some k).
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 i ≤ k,
the sign of φsΛ ↩Λ φ i (some k) is equal to the product of
- the sign associated with moving
φthrough theφsΛ-uncontractedFieldOpinφ₀…φₖ₋₁, - the sign associated with moving
φthrough all theFieldOpinφ₀…φᵢ₋₁, - the sign of
φsΛ.
The proof of this result involves a careful consideration of the contributions of different
FieldOp in φs to the sign of φsΛ ↩Λ φ i (some k).
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,
the sign of φsΛ ↩Λ φ 0 (some k) is equal to the product of
- the sign associated with moving
φthrough theφsΛ-uncontractedFieldOpinφ₀…φₖ₋₁, - the sign of
φsΛ.
This is a direct corollary of sign_insert_some_of_not_lt.