Physlib.QFT.PerturbationTheory.FieldSpecification.NormalOrder

Given a field specification $\mathcal{F}$, let $\mathcal{O}_{\mathcal{F}}$ denote the set of creation and annihilation operators (represented by `𝓕.CrAnFieldOp`). The normal ordering relation $\le_{NO}$ is a binary relation on $\mathcal{O}_{\mathcal{F}}$ defined such that for any two operators $\phi_0, \phi_1 \in \mathcal{O}_{\mathcal{F}}$, $\phi_0 \le_{NO} \phi_1$ holds if $\phi_0$ is a creation operator or $\phi_1$ is an annihilation operator. This relation provides the precedence for normal ordering, effectively treating creation operators as "less than" annihilation operators so that creation operators are positioned to the left in a sorted product.

Given a field specification $\mathcal{F}$, let $\mathcal{O}_{\mathcal{F}}$ denote the set of creation and annihilation operators. The normal ordering relation $\le_{NO}$ on $\mathcal{O}_{\mathcal{F}}$ is total. That is, for any two operators $\phi, \psi \in \mathcal{O}_{\mathcal{F}}$, it holds that $\phi \le_{NO} \psi$ or $\psi \le_{NO} \phi$.

Given a field specification $\mathcal{F}$, let $\mathcal{O}_{\mathcal{F}}$ denote the set of creation and annihilation field operators. The normal ordering relation $\le_{NO}$ on $\mathcal{O}_{\mathcal{F}}$ is transitive. That is, for any operators $\phi_1, \phi_2, \phi_3 \in \mathcal{O}_{\mathcal{F}}$, if $\phi_1 \le_{NO} \phi_2$ and $\phi_2 \le_{NO} \phi_3$, then $\phi_1 \le_{NO} \phi_3$.

For a given field specification $\mathcal{F}$ and any two field operators $\phi, \phi' \in \mathcal{O}_{\mathcal{F}}$ (representing creation and annihilation operators), the normal ordering relation $\phi \le_{NO} \phi'$ is decidable. This means that for any pair of operators, there is a deterministic algorithm to determine whether $\phi$ precedes or is equal to $\phi'$ according to the normal ordering convention.

Given a field specification $\mathcal{F}$ and a list of creation and annihilation operators $\phi_s = [\phi_1, \phi_2, \dots, \phi_n]$ (where each $\phi_i \in \mathcal{O}_{\mathcal{F}}$), the function `normalOrderSign` returns the phase factor $\eta \in \mathbb{C}$ (where $\eta = \pm 1$) associated with the permutation required to reach a normal-ordered arrangement. This sign is calculated based on the number of transpositions of two fermionic operators performed during an insertion sort according to the normal ordering relation $\le_{NO}$. In this relation, creation operators are ordered before annihilation operators.

Let $\mathcal{F}$ be a field specification. For any list of creation and annihilation operators $\phi_s = [\phi_1, \phi_2, \dots, \phi_n]$ where each $\phi_i \in \mathcal{O}_{\mathcal{F}}$, the normal ordering sign $\text{normalOrderSign}(\phi_s)$ (the phase factor in $\mathbb{C}$ associated with the permutation required to reach a normal-ordered arrangement) satisfies: $$\text{normalOrderSign}(\phi_s) \cdot \text{normalOrderSign}(\phi_s) = 1$$

Let $\mathcal{F}$ be a field specification and $\phi$ be a field operator in $\mathcal{F}$ that is a creation operator. For any list of field operators $\phi_s$, the Koszul sign $\text{koszulSignInsert}(\sigma, \le_{NO}, \phi, \phi_s)$—which accounts for the statistics $\sigma$ and the normal ordering relation $\le_{NO}$ when inserting $\phi$ into the list—is equal to $1$.

Let $\mathcal{F}$ be a field specification. Suppose $\phi$ is a field operator in $\mathcal{F}$ that is a creation operator, and $\phi_s$ is a list of creation and annihilation operators. The normal ordering sign of the list formed by prepending $\phi$ to $\phi_s$ is equal to the normal ordering sign of $\phi_s$, i.e., $\text{normalOrderSign}(\phi :: \phi_s) = \text{normalOrderSign}(\phi_s)$.

Let $\mathcal{F}$ be a field specification, and let $\mathcal{O}_{\mathcal{F}}$ denote the set of its creation and annihilation field operators. For any field operator $\phi \in \mathcal{O}_{\mathcal{F}}$, the normal ordering sign of the singleton list $[\phi]$ is equal to $1$. That is, $$\text{normalOrderSign}([\phi]) = 1$$

Let $\mathcal{F}$ be a field specification. The normal ordering sign associated with an empty list of creation and annihilation operators is $1$. That is, $$\text{normalOrderSign}([]) = 1$$ where $[]$ denotes the empty list.

Let $\mathcal{F}$ be a field specification, and let $\mathcal{O}_{\mathcal{F}}$ denote the set of its creation and annihilation field operators. Let $\phi, \phi' \in \mathcal{O}_{\mathcal{F}}$ be two field operators and $\Phi_s$ be a list of field operators. If $\phi$ is an annihilation operator, then the Koszul sign incurred by inserting $\phi'$ into the list formed by appending $\phi$ to the end of $\Phi_s$ (with respect to the normal ordering relation $\le_{NO}$) is equal to the Koszul sign of inserting $\phi'$ into $\Phi_s$. That is, $$\text{koszulSignInsert}(\phi', \Phi_s \text{ ++ } [\phi]) = \text{koszulSignInsert}(\phi', \Phi_s)$$

Let $\mathcal{F}$ be a field specification and $\mathcal{O}_{\mathcal{F}}$ be the set of its creation and annihilation field operators. For any list of operators $\Phi_s$ in $\mathcal{O}_{\mathcal{F}}$ and any operator $\phi \in \mathcal{O}_{\mathcal{F}}$, if $\phi$ is an annihilation operator, then the normal ordering sign of the list formed by appending $\phi$ to the end of $\Phi_s$ is equal to the normal ordering sign of $\Phi_s$. That is, $$\text{normalOrderSign}(\Phi_s \text{ ++ } [\phi]) = \text{normalOrderSign}(\Phi_s)$$

Given a field specification $\mathcal{F}$, let $\phi_c$ be a creation operator and $\phi_a$ be an annihilation operator. For any list of field operators $\phi_s$, the Koszul sign associated with inserting $\phi_a$ into the list $\phi_c :: \phi_s$ (the list starting with $\phi_c$ followed by $\phi_s$) according to the normal ordering relation $\le_{NO}$ is equal to the exchange sign $\epsilon(\phi_c, \phi_a)$ multiplied by the Koszul sign associated with inserting $\phi_a$ into the list $\phi_s$.

Let $\mathcal{F}$ be a field specification. Given a creation operator $\phi_c$, an annihilation operator $\phi_a$, and a list of field operators $\Phi_s$, the normal ordering sign of the list formed by prepending $\phi_c$ and $\phi_a$ is related to the list with their positions swapped by the exchange sign: $$\text{normalOrderSign}(\phi_c :: \phi_a :: \Phi_s) = \epsilon(\phi_c, \phi_a) \cdot \text{normalOrderSign}(\phi_a :: \phi_c :: \Phi_s)$$ where $\epsilon(\phi_c, \phi_a)$ is the exchange sign (determined by the statistics of $\phi_c$ and $\phi_a$) and $::$ denotes prepending an element to a list.

For a given field specification $\mathcal{F}$, let $\mathcal{O}_{\mathcal{F}}$ be the set of creation and annihilation operators, $\sigma$ be the field statistics, and $\le_{NO}$ be the normal ordering relation. For any operators $\phi, \phi_c, \phi_a \in \mathcal{O}_{\mathcal{F}}$ and lists of operators $\Phi_s, \Phi_s'$, the Koszul sign associated with inserting $\phi$ into the list $\Phi_s \concat [\phi_a, \phi_c] \concat \Phi_s'$ is equal to the Koszul sign associated with inserting $\phi$ into the list $\Phi_s \concat [\phi_c, \phi_a] \concat \Phi_s'$.

Let $\mathcal{F}$ be a field specification and $\mathcal{O}_{\mathcal{F}}$ denote the set of creation and annihilation operators. For any creation operator $\phi_c \in \mathcal{O}_{\mathcal{F}}$, any annihilation operator $\phi_a \in \mathcal{O}_{\mathcal{F}}$, and any lists of field operators $\Phi_s$ and $\Phi_s'$, the normal ordering sign of the list formed by the concatenation $\Phi_s \concat [\phi_c, \phi_a] \concat \Phi_s'$ is equal to the exchange sign $\epsilon(\phi_c, \phi_a)$ multiplied by the normal ordering sign of the list with $\phi_c$ and $\phi_a$ swapped: $$\text{normalOrderSign}(\Phi_s \concat [\phi_c, \phi_a] \concat \Phi_{s'}) = \epsilon(\phi_c, \phi_a) \cdot \text{normalOrderSign}(\Phi_s \concat [\phi_a, \phi_c] \concat \Phi_{s'})$$ where $\epsilon(\phi_c, \phi_a)$ is the exchange sign determined by the statistics of the operators $\phi_c$ and $\phi_a$.

Let $\mathcal{F}$ be a field specification. For any two field operators $\phi_c, \phi_{c'} \in \mathcal{F}.\text{CrAnFieldOp}$ that are creation operators, and for any list of field operators $\phi_s$, the normal ordering sign of the list beginning with $\phi_c$ and $\phi_{c'}$ is equal to the normal ordering sign of the list where their positions are swapped: $$\text{normalOrderSign}(\phi_c :: \phi_{c'} :: \phi_s) = \text{normalOrderSign}(\phi_{c'} :: \phi_c :: \phi_s)$$

Let $\mathcal{F}$ be a field specification. For any two field operators $\phi_c, \phi_{c'} \in \mathcal{F}.\text{CrAnFieldOp}$ that are both creation operators, and for any lists of field operators $\phi_s$ and $\phi_{s'}$, swapping the adjacent operators $\phi_c$ and $\phi_{c'}$ within the concatenated list preserves the normal ordering sign: $$\text{normalOrderSign}(\phi_s \mathbin{+\mkern-10mu+} [\phi_c, \phi_{c'}] \mathbin{+\mkern-10mu+} \phi_{s'}) = \text{normalOrderSign}(\phi_s \mathbin{+\mkern-10mu+} [\phi_{c'}, \phi_c] \mathbin{+\mkern-10mu+} \phi_{s'})$$