Physlib.QFT.PerturbationTheory.FieldOpFreeAlgebra.NormTimeOrder

For a given field specification $\mathcal{F}$, the normal-time ordering is a $\mathbb{C}$-linear map from the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ to itself. The map is defined by its action on the basis elements, which correspond to products of creation and annihilation operators $\phi_1 \phi_2 \dots \phi_n$ represented by the list $\varphi s = [\phi_1, \dots, \phi_n]$. For any such basis element, the map returns: $$\text{normTimeOrderSign}(\varphi s) \cdot (\phi_{\sigma(1)} \phi_{\sigma(2)} \dots \phi_{\sigma(n)})$$ where the operators are reordered into a new list $\varphi s' = [\phi_{\sigma(1)}, \dots, \phi_{\sigma(n)}]$ according to the relation $\text{normTimeOrderRel}$ (sorting by decreasing time and then by normal-ordering convention), and $\text{normTimeOrderSign}(\varphi s)$ is the sign factor $\pm 1$ arising from the Koszul sign convention for permuting fermionic and bosonic operators.

The notation $\mathcal{TN}^f(a)$ denotes the norm-time ordering of an element $a$ in the free algebra of field operators $\text{FieldOpFreeAlgebra } \mathcal{F}$. It corresponds to the application of the linear map $\text{normTimeOrder}$ to $a$.

For a given field specification $\mathcal{F}$, let $\varphi_s = [\phi_1, \phi_2, \dots, \phi_n]$ be a list of creation and annihilation operators in $\mathcal{F}.\text{CrAnFieldOp}$. Let $\text{ofCrAnListF}(\varphi_s)$ denote the product of these operators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$. The normal-time ordering operator $\mathcal{TN}^f$ acts on this product as: $$\mathcal{TN}^f(\text{ofCrAnListF}(\varphi_s)) = \text{normTimeOrderSign}(\varphi_s) \cdot \text{ofCrAnListF}(\text{normTimeOrderList}(\varphi_s))$$ where $\text{normTimeOrderSign}(\varphi_s)$ is the sign factor (determined by the Koszul sign convention for fermionic/bosonic statistics) and $\text{normTimeOrderList}(\varphi_s)$ is the list of operators reordered chronologically (decreasing time) and according to the normal-ordering convention.