Physlib.QFT.PerturbationTheory.FieldStatistics.ExchangeSign

The exchange sign $\mathcal{S}(a, b)$ is a group homomorphism from the group of field statistics to the group of homomorphisms from field statistics to the complex numbers, $\text{FieldStatistic} \to (\text{FieldStatistic} \to \mathbb{C})$. For two field statistics $a$ and $b$, the value is defined as: $$\mathcal{S}(a, b) = \begin{cases} -1 & \text{if } a = \text{fermionic and } b = \text{fermionic} \\ 1 & \text{otherwise} \end{cases}$$ This sign represents the phase factor acquired when commuting two fields or operators with statistics $a$ and $b$.

The notation $\mathcal{S}(a, b)$ denotes the exchange sign between two field statistics $a$ and $b$. This sign is equal to $-1$ if both field statistics $a$ and $b$ are fermionic, and is $1$ otherwise.

For any two field statistics $a$ and $b$, which categorize a field as either $\text{bosonic}$ or $\text{fermionic}$, the exchange sign $\mathcal{S}(a, b)$ is symmetric, satisfying $\mathcal{S}(a, b) = \mathcal{S}(b, a)$. The exchange sign $\mathcal{S}(a, b)$ is defined to be $-1$ if both $a$ and $b$ are $\text{fermionic}$, and $1$ otherwise.

For any field statistic $a$, the exchange sign $\mathcal{S}(a, \text{bosonic})$ between $a$ and the bosonic statistic is equal to $1$. Here, the exchange sign $\mathcal{S}(a, b)$ is defined to be $-1$ if both $a$ and $b$ are fermionic, and $1$ otherwise.

For any field statistic $a$, the exchange sign $\mathcal{S}(\text{bosonic}, a)$ between the bosonic statistic and $a$ is equal to $1$. Here, the exchange sign $\mathcal{S}(a, b)$ is defined to be $-1$ if both $a$ and $b$ are fermionic, and $1$ otherwise.

The exchange sign $\mathcal{S}(\text{fermionic}, \text{fermionic})$ for two fields with fermionic statistics is $-1$.

For any two field statistics $a, b \in \text{FieldStatistic}$, the exchange sign $\mathcal{S}(a, b)$ is given by: $$\mathcal{S}(a, b) = \begin{cases} -1 & \text{if } a = \text{fermionic and } b = \text{fermionic} \\ 1 & \text{otherwise} \end{cases}$$

For any two field statistics $a$ and $b$, the square of the exchange sign $\mathcal{S}(a, b)$ is equal to $1$, that is, $\mathcal{S}(a, b) \cdot \mathcal{S}(a, b) = 1$.

For any two field statistics $a, b \in \text{FieldStatistic}$, the product of the exchange sign $\mathcal{S}(a, b)$ and the swapped exchange sign $\mathcal{S}(b, a)$ is equal to $1$.

Let $a \in \text{FieldStatistic}$ be a field statistic. Let $s: \mathcal{F} \to \text{FieldStatistic}$ be a mapping that assigns a statistic to each field in a collection $\mathcal{F}$. For any field $\phi \in \mathcal{F}$ and any list of fields $\phi s$, the exchange sign $\mathcal{S}$ between $a$ and the collective statistic of the list formed by prepending $\phi$ to $\phi s$ is given by: $$\mathcal{S}(a, \text{ofList}(s, \phi :: \phi s)) = \mathcal{S}(a, s(\phi)) \cdot \mathcal{S}(a, \text{ofList}(s, \phi s))$$ where $\text{ofList}$ denotes the collective statistic of a list of fields, which is fermionic if the list contains an odd number of fermionic fields and bosonic otherwise.

For any three field statistics $a, b, c \in \text{FieldStatistic}$, the exchange sign $\mathcal{S}$ satisfies the cocycle condition: $$\mathcal{S}(a, b \cdot c) \cdot \mathcal{S}(b, c) = \mathcal{S}(a, b) \cdot \mathcal{S}(a \cdot b, c)$$ Here, $\text{FieldStatistic}$ is the type of quantum statistics (bosonic or fermionic), the operator $\cdot$ denotes the commutative group multiplication of statistics (isomorphic to addition modulo 2, where $\text{bosonic}$ is the identity and $\text{fermionic} \cdot \text{fermionic} = \text{bosonic}$), and $\mathcal{S}(x, y)$ is the phase factor defined as $-1$ if both $x$ and $y$ are fermionic, and $1$ otherwise.