Inserting an element into a contraction based on a list #
Inserting an element into a list #
Given a Wick contraction φsΛ for a list φs of 𝓕.FieldOp,
an element φ of 𝓕.FieldOp, an i ≤ φs.length and a k
in Option φsΛ.uncontracted i.e. is either none or
some element of φsΛ.uncontracted, the new Wick contraction
φsΛ.insertAndContract φ i k is defined by inserting φ into φs after
the first i-elements and moving the values representing the contracted pairs in φsΛ
accordingly.
If k is not none, but rather some k, to this contraction is added the contraction
of φ (at position i) with the new position of k after φ is added.
In other words, φsΛ.insertAndContract φ i k is formed by adding φ to φs at position i,
and contracting φ with the field originally at position k if k is not none.
It is a Wick contraction of the list φs.insertIdx φ i corresponding to φs with φ inserted at
position i.
The notation φsΛ ↩Λ φ i k is used to denote φsΛ.insertAndContract φ i k.
Equations
- WickContraction.insertAndContract φ φsΛ i k = (WickContraction.congr ⋯) (φsΛ.insertAndContractNat i k)
Instances For
Given a Wick contraction φsΛ for a list φs of 𝓕.FieldOp,
an element φ of 𝓕.FieldOp, an i ≤ φs.length and a k
in Option φsΛ.uncontracted i.e. is either none or
some element of φsΛ.uncontracted, the new Wick contraction
φsΛ.insertAndContract φ i k is defined by inserting φ into φs after
the first i-elements and moving the values representing the contracted pairs in φsΛ
accordingly.
If k is not none, but rather some k, to this contraction is added the contraction
of φ (at position i) with the new position of k after φ is added.
In other words, φsΛ.insertAndContract φ i k is formed by adding φ to φs at position i,
and contracting φ with the field originally at position k if k is not none.
It is a Wick contraction of the list φs.insertIdx φ i corresponding to φs with φ inserted at
position i.
The notation φsΛ ↩Λ φ i k is used to denote φsΛ.insertAndContract φ i k.
Equations
- One or more equations did not get rendered due to their size.
Instances For
insertAndContract and getDual? #
Given a finite set of Fin φs.length the finite set of (φs.insertIdx i φ).length
formed by mapping elements using i.succAboveEmb and finCongr.
Equations
Instances For
For a list φs of 𝓕.FieldOp, a Wick contraction φsΛ of φs, an element φ of
𝓕.FieldOp and a i ≤ φs.length then a sum over
Wick contractions of φs with φ inserted at i is equal to the sum over Wick contractions
φsΛ of just φs and the sum over optional uncontracted elements of the φsΛ.
In other words,
∑ (φsΛ : WickContraction (φs.insertIdx i φ).length), f φsΛ
where (φs.insertIdx i φ) is φs with φ inserted at position i. is equal to
∑ (φsΛ : WickContraction φs.length), ∑ k, f (φsΛ ↩Λ φ i k) .
where the sum over k is over all k in Option φsΛ.uncontracted.