Physlib.QFT.PerturbationTheory.Koszul.KoszulSignInsert

Let $\mathcal{F}$ be a type of fields, $q : \mathcal{F} \to \{\text{bosonic}, \text{fermionic}\}$ be a function assigning a field statistic to each field, and $\le$ be a decidable relation on $\mathcal{F}$. For a given field $\phi \in \mathcal{F}$ and a list of fields $L = [\phi_1, \phi_2, \dots, \phi_n]$, the function returns the Koszul sign factor $(-1)^k \in \mathbb{C}$ incurred by inserting $\phi$ into the list. The sign is determined by the number $k$ of fermionic fields $\phi_i$ in the list $L$ that are strictly smaller than $\phi$ (i.e., for which $\phi \le \phi_i$ is false) when $\phi$ itself is fermionic. If $\phi$ is bosonic, the result is always $1$.

Let $\mathcal{F}$ be a type of fields, $q : \mathcal{F} \to \{\text{bosonic}, \text{fermionic}\}$ be a function assigning a field statistic to each field, and $\le$ be a decidable relation on $\mathcal{F}$. If a field $\phi \in \mathcal{F}$ is bosonic (i.e., $q(\phi) = \text{bosonic}$), then for any list of fields $L$, the Koszul sign factor incurred by inserting $\phi$ into $L$ is equal to $1$.

Let $\mathcal{F}$ be a type of fields, $q : \mathcal{F} \to \{\text{bosonic}, \text{fermionic}\}$ be a function assigning a field statistic to each field, and $\le$ be a decidable relation on $\mathcal{F}$. For any field $\phi \in \mathcal{F}$ and any list of fields $L$, the Koszul sign factor $\text{koszulSignInsert}(q, \le, \phi, L)$ satisfies $$\text{koszulSignInsert}(q, \le, \phi, L)^2 = 1.$$

Let $\mathcal{F}$ be a set of fields, $q$ be a function assigning a field statistic (bosonic or fermionic) to each field, and $\le$ be a decidable relation on $\mathcal{F}$. For any field $\phi \in \mathcal{F}$ and any list of fields $L$, if $\phi \le \phi'$ holds for all $\phi' \in \mathcal{F}$, then the Koszul sign factor incurred by inserting $\phi$ into $L$ is $1$.

Let $\mathcal{F}$ be a type of fields, $q: \mathcal{F} \to \{\text{bosonic, fermionic}\}$ be a function assigning a field statistic to each field, and $\le$ be a decidable relation on $\mathcal{F}$. For any list of fields $L$ and any fields $\phi, \phi' \in \mathcal{F}$, if $\phi$ is a field such that $\phi'' \le \phi$ for all $\phi'' \in \mathcal{F}$, then the Koszul sign factor incurred by inserting $\phi'$ into the list $L$ is equal to the Koszul sign factor incurred by inserting $\phi'$ into the list $L \text{ ++ } [\phi]$. The Koszul sign factor $\text{koszulSignInsert}(q, \le, \phi', L)$ is $(-1)^k$ where $k$ is the number of fermionic fields $\psi$ in $L$ such that $\neg(\phi' \le \psi)$ (if $\phi'$ is fermionic), or $1$ (if $\phi'$ is bosonic).

Let $\mathcal{F}$ be a type of fields, $q : \mathcal{F} \to \{\text{bosonic}, \text{fermionic}\}$ be a function assigning a field statistic to each field, and $\le$ be a decidable relation on $\mathcal{F}$. For any field $\phi \in \mathcal{F}$ and any list of fields $L$, the Koszul sign factor incurred by inserting $\phi$ into $L$ is equal to the Koszul sign factor incurred by inserting $\phi$ into the sublist of $L$ consisting of all elements $\psi$ such that $\neg(\phi \le \psi)$ holds.

Let $\mathcal{F}$ be a type of fields, $q : \mathcal{F} \to \{\text{bosonic}, \text{fermionic}\}$ be a function assigning a field statistic to each field, and $\le$ be a total relation on $\mathcal{F}$. For any field $\phi \in \mathcal{F}$ and list of fields $L$, the Koszul sign factor incurred by inserting $\phi$ into $L$ is equal to the Koszul sign factor incurred by inserting $\phi$ into the list $\phi :: L$ (the list $L$ with $\phi$ prepended to the front).

Let $\mathcal{F}$ be a type of fields with statistics $q : \mathcal{F} \to \{\text{bosonic}, \text{fermionic}\}$ and a decidable relation $\le$. For any field $\phi \in \mathcal{F}$ and list of fields $L$, the Koszul sign factor $\text{koszulSignInsert}(q, \le, \phi, L)$ is given by: $$\text{koszulSignInsert}(q, \le, \phi, L) = \begin{cases} -1 & \text{if } q(\phi) = \text{fermionic and } \text{grade}(\{\psi \in L \mid \neg(\phi \le \psi)\}) = \text{fermionic} \\ 1 & \text{otherwise} \end{cases}$$ where the $\text{grade}$ of a list is fermionic if it contains an odd number of fermionic fields and bosonic otherwise.

Let $\mathcal{F}$ be a type of fields with field statistics $q: \mathcal{F} \to \{\text{bosonic, fermionic}\}$ and a decidable relation $\le$. For any field $\phi \in \mathcal{F}$ and any two lists of fields $L$ and $L'$ such that $L$ is a permutation of $L'$, the Koszul sign factor incurred by inserting $\phi$ into $L$ is equal to the Koszul sign factor incurred by inserting $\phi$ into $L'$, i.e., $$\text{koszulSignInsert}(q, \le, \phi, L) = \text{koszulSignInsert}(q, \le, \phi, L')$$

Let $\mathcal{F}$ be a type of fields with statistics $q: \mathcal{F} \to \{\text{bosonic, fermionic}\}$ and a decidable relation $\le$. For any field $\phi \in \mathcal{F}$ and any list of fields $L$, the Koszul sign factor incurred by inserting $\phi$ into $L$ is equal to the Koszul sign factor incurred by inserting $\phi$ into the list obtained by sorting $L$ according to the relation $\le$: $$\text{koszulSignInsert}(q, \le, \phi, L) = \text{koszulSignInsert}(q, \le, \phi, \text{sort}_{\le}(L))$$

Let $\mathcal{F}$ be a type of fields with field statistics $q: \mathcal{F} \to \{\text{bosonic, fermionic}\}$ and a total, transitive relation $\le$. For any field $\phi \in \mathcal{F}$ and list of fields $L$, the Koszul sign factor incurred by inserting $\phi$ into $L$ is equal to the exchange sign between the statistic of $\phi$ and the collective statistic (grade) of the elements in $L$ that would precede $\phi$ in an ordered arrangement. Specifically, $$\text{koszulSignInsert}(q, \le, \phi, L) = \mathcal{S}\left(q(\phi), \text{grade}\left(\text{sort}_{\le}(L)_{< \phi}\right)\right)$$ where $\text{sort}_{\le}(L)_{< \phi}$ denotes the sublist of elements in $L$ that are strictly smaller than $\phi$ according to the ordering $\le$, and $\mathcal{S}(\sigma_1, \sigma_2)$ is the exchange sign factor $(-1)$ if both arguments are fermionic and $1$ otherwise.

Let $\mathcal{F}$ be a type of fields with field statistics $q: \mathcal{F} \to \{\text{bosonic, fermionic}\}$ and a decidable relation $\le$. For any fields $i, j \in \mathcal{F}$, any list of fields $r$, and any index $n$ such that $n \le \text{length}(r)$, the Koszul sign factor incurred by inserting field $j$ into the list obtained by inserting $i$ into $r$ at position $n$ is equal to the Koszul sign factor incurred by inserting $j$ into the list where $i$ is prepended to $r$, i.e., $$\text{koszulSignInsert}(q, \le, j, \text{insertIdx}(r, n, i)) = \text{koszulSignInsert}(q, \le, j, i :: r)$$

Given a set of fields $\mathcal{F}$, an ordering relation $\le$ on $\mathcal{F}$, and a function $q$ defining the quantum statistics (bosonic or fermionic) of each field, the function $\text{koszulSignCons}(\phi_0, \phi_1)$ returns a complex number representing the sign factor for the pair. If $\phi_0 \le \phi_1$, the value is $1$. If $\phi_0 \not\le \phi_1$, the value is $-1$ if both $\phi_0$ and $\phi_1$ are fermionic, and $1$ otherwise. Mathematically, this corresponds to the sign change when $\phi_0$ is moved past $\phi_1$ in an ordered sequence of operators.

Let $\mathcal{F}$ be a set of fields equipped with an ordering relation $\le$ and a quantum statistics function $q$. For any fields $\phi_0, \phi_1 \in \mathcal{F}$, the Koszul exchange sign $\text{koszulSignCons}(\phi_0, \phi_1)$ is equal to $1$ if $\phi_0 \le \phi_1$. Otherwise, if $\phi_0 \not\le \phi_1$, it is equal to the exchange sign $\mathcal{S}(q(\phi_0), q(\phi_1))$.

Let $\mathcal{F}$ be a type of quantum fields, $\le$ be a decidable ordering relation on $\mathcal{F}$, and $q$ be a function assigning quantum statistics (bosonic or fermionic) to each field. For any fields $\phi_0, \phi_1 \in \mathcal{F}$ and any list of fields $L$, the Koszul sign factor incurred by inserting $\phi_0$ into the list starting with $\phi_1$ (denoted $\phi_1 :: L$) is equal to the product of the exchange sign factor between $\phi_0$ and $\phi_1$ and the Koszul sign factor for inserting $\phi_0$ into the remaining list $L$: $$\text{koszulSignInsert}(\phi_0, \phi_1 :: L) = \text{koszulSignCons}(\phi_0, \phi_1) \cdot \text{koszulSignInsert}(\phi_0, L)$$ Here, $\text{koszulSignCons}(\phi_0, \phi_1)$ is $1$ if $\phi_0 \le \phi_1$, and otherwise is $-1$ if both fields are fermionic and $1$ if at least one is bosonic.

Let $\mathcal{F}$ be a type of fields, $q : \mathcal{F} \to \{\text{bosonic}, \text{fermionic}\}$ be a function assigning a field statistic to each field, and $\le$ be a decidable relation on $\mathcal{F}$. For any field $\phi_0 \in \mathcal{F}$ and any list of fields $L$, if $\phi_0 \le \phi$ for every field $\phi$ in the list $L$, then the Koszul sign factor incurred by inserting $\phi_0$ into $L$ is $1$.

Let $\mathcal{F}$ be a type of fields, $q: \mathcal{F} \to \{\text{bosonic, fermionic}\}$ be the field statistics, and $\le$ be a transitive relation on $\mathcal{F}$. Let $\phi, \psi \in \mathcal{F}$ be two fields such that $\phi \le \psi$ and $\psi \le \phi$, and their statistics are identical ($q(\phi) = q(\psi)$). Then for any list of fields $L = [\phi_1, \phi_2, \dots, \phi_n]$, the Koszul sign factor incurred by inserting $\phi$ into $L$ is equal to the factor incurred by inserting $\psi$ into $L$: $$\text{koszulSignInsert}(q, \le, \phi, L) = \text{koszulSignInsert}(q, \le, \psi, L)$$

Let $\mathcal{F}$ be a type of fields, $q : \mathcal{F} \to \{\text{bosonic, fermionic}\}$ be a function assigning field statistics, and $\le$ be a transitive relation on $\mathcal{F}$. Suppose $\phi, \psi, \phi' \in \mathcal{F}$ are fields such that $\phi$ and $\psi$ are equivalent under the relation (i.e., $\phi \le \psi$ and $\psi \le \phi$) and possess the same field statistics ($q(\psi) = q(\phi)$). Then, for any list of fields $L$, the Koszul sign factor incurred by inserting $\phi'$ into the list $[\phi, \psi] \cup L$ is equal to the Koszul sign factor of inserting $\phi'$ into $L$: $$\text{koszulSignInsert}(q, \le, \phi', \phi :: \psi :: L) = \text{koszulSignInsert}(q, \le, \phi', L)$$