Physlib.QFT.PerturbationTheory.WickAlgebra.Universality

Let $\mathcal{F}$ be a field specification and $A$ be an algebra over $\mathbb{C}$. Given a function $f: \mathcal{F}.\text{CrAnFieldOp} \to A$ such that the unique $\mathbb{C}$-algebra homomorphism $\tilde{f}: \text{FreeAlgebra}(\mathbb{C}, \mathcal{F}.\text{CrAnFieldOp}) \to A$ extending $f$ vanishes on the two-sided ideal generated by the relations in $\mathcal{F}.\text{fieldOpIdealSet}$, this definition provides the induced map from the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ to $A$.

Let $\mathcal{F}$ be a field specification and $A$ be an algebra over the complex numbers $\mathbb{C}$. Let $f: \mathcal{F}.\text{CrAnFieldOp} \to A$ be a map, and let $\tilde{f}: \mathcal{F}.\text{FieldOpFreeAlgebra} \to A$ be the unique $\mathbb{C}$-algebra homomorphism extending $f$ (given by `FreeAlgebra.lift`). Suppose $\tilde{f}$ vanishes on the two-sided ideal generated by the field operator relations $\mathcal{F}.\text{fieldOpIdealSet}$. If $\Phi: \mathcal{F}.\text{WickAlgebra} \to A$ is the universal lift map induced by $f$, then for any element $a$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, it holds that $\Phi(\iota(a)) = \tilde{f}(a)$, where $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is the canonical projection.

Let $\mathcal{F}$ be a field specification and $A$ be an associative algebra over the complex numbers $\mathbb{C}$. Given a function $f: \mathcal{F}.\text{CrAnFieldOp} \to A$, let $\tilde{f} : \text{FreeAlgebra}(\mathbb{C}, \mathcal{F}.\text{CrAnFieldOp}) \to A$ be the unique $\mathbb{C}$-algebra homomorphism extending $f$. If $\tilde{f}$ vanishes on the two-sided ideal generated by the set of relations $\mathcal{F}.\text{fieldOpIdealSet}$ (which define the Wick algebra), then `universalLift` is the resulting $\mathbb{C}$-algebra homomorphism from the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ to $A$.

Let $\mathcal{F}$ be a field specification and $A$ be an associative algebra over the complex numbers $\mathbb{C}$. Given a function $f: \mathcal{F}.\text{CrAnFieldOp} \to A$, let $\tilde{f} : \mathcal{F}.\text{FieldOpFreeAlgebra} \to A$ be the unique $\mathbb{C}$-algebra homomorphism extending $f$. Suppose $\tilde{f}$ vanishes on the two-sided ideal generated by the set of relations $\mathcal{F}.\text{fieldOpIdealSet}$. Let $g: \mathcal{F}.\text{WickAlgebra} \to A$ be the universal lift of $f$ to the Wick algebra (the map `universalLift f h1`). Then for any element $a$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, the identity $g(\iota(a)) = \tilde{f}(a)$ holds, where $\iota : \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is the canonical projection.

Let $\mathcal{F}$ be a field specification and $A$ be an associative algebra over the complex numbers $\mathbb{C}$. Let $f : \mathcal{F}.\text{CrAnFieldOp} \to A$ be a function and $\tilde{f} : \mathcal{F}.\text{FieldOpFreeAlgebra} \to A$ be the unique $\mathbb{C}$-algebra homomorphism extending $f$ (the universal lift of $f$ to the free algebra). If $\tilde{f}$ vanishes on the two-sided ideal generated by the set of relations $\mathcal{F}.\text{fieldOpIdealSet}$, then there exists a unique $\mathbb{C}$-algebra homomorphism $g : \mathcal{F}.\text{WickAlgebra} \to A$ such that $g \circ \iota = \tilde{f}$, where $\iota : \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is the canonical projection.