The Free Fock Space: Creation, Annihilation and Semicircular Operators

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $n\in\mathbb{N}$. A polynomial $p\in\mathcal{P}_{n}$ is a map $W_{n}\to\mathbb{C}$ with finite support $\operatorname{supp}p$ and coefficients $p(w)$, as in The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §polynomials. For $j\in[n]$ and $w\in W_{n}$, $(j)w$ and $w(j)$ are the concatenations of the one-letter word $(j)$ with $w$, on the left and on the right. For $p,q\in\mathcal{P}_{n}$ let

$$h_{n}(p,q)=\sum_{w\in\operatorname{supp}p}\overline{p(w)}\,q(w),$$

a sum over a finite index set, read as $0$ when $p=0$.

1. (Fock space)¶ The map $h_{n}$ satisfies the hypotheses of Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §real-structure: it is additive and complex-homogeneous in $q$ by Properties of a Sum over a Finite Index Set (claims 3 and 4); $h_{n}(q,p)=\overline{h_{n}(p,q)}$, since both sides are $0$ if $\operatorname{supp}p\cap\operatorname{supp}q$ is empty and are otherwise the sum over $\operatorname{supp}p\cap\operatorname{supp}q$ of the conjugate pair of terms, by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing and Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §conjugate; and $h_{n}(p,p)=\sum_{w\in\operatorname{supp}p}|p(w)|^{2}$ is real and nonnegative by Properties of Complex Conjugation and Modulus and Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §nonnegative. The free Fock space $\mathcal{F}_{n}$ is the complex Hilbert completion of $(\mathcal{P}_{n},h_{n})$, a complex Hilbert space by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §hilbert. For $p\in\mathcal{P}_{n}$, $[p]\in\mathcal{F}_{n}$ is the image of $p$ under the canonical map, and $\Omega_{\mathcal{F}}=[1]$ is the vacuum vector.

2. (Creation and annihilation operators)¶ Let $j\in[n]$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension there are linear maps $c^{L}_{j},c^{R}_{j},a^{L}_{j},a^{R}_{j}:\mathcal{P}_{n}\to\mathcal{P}_{n}$ with, for every $w\in W_{n}$,

$$c^{L}_{j}(x_{w})=x_{(j)w},\qquad c^{R}_{j}(x_{w})=x_{w(j)},\qquad a^{L}_{j}(x_{(j)w})=x_{w},\qquad a^{R}_{j}(x_{w(j)})=x_{w},$$

and $a^{L}_{j}(x_{u})=0$ for every word $u$ that is empty or whose first letter is not $j$, $a^{R}_{j}(x_{u})=0$ for every word $u$ that is empty or whose last letter is not $j$; the word $w$ is determined by $(j)w$, and by $w(j)$. Each of these four maps $f$ sends distinct monomials to distinct monomials or to $0$, so every coefficient of $f(p)$ is a coefficient of $p$ at a word determined by it, and $h_{n}(f(p),f(p))\le h_{n}(p,p)$ for every $p$; by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §isometry this is the bound required in The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §extension-linear with $C=1$. Hence for each of $f=c^{L}_{j},c^{R}_{j},a^{L}_{j},a^{R}_{j}$ there is exactly one operator in $\mathcal{L}(\mathcal{F}_{n})$ sending $[p]$ to $[f(p)]$; the left and right creation operators $l_{j},r_{j}\in\mathcal{L}(\mathcal{F}_{n})$ and the left and right annihilation operators $\bar{l}_{j},\bar{r}_{j}\in\mathcal{L}(\mathcal{F}_{n})$ are these operators for $f=c^{L}_{j},c^{R}_{j},a^{L}_{j},a^{R}_{j}$ respectively:

$$l_{j}[p]=[x_{j}p],\qquad r_{j}[p]=[px_{j}],\qquad\bar{l}_{j}[p]=[a^{L}_{j}(p)],\qquad\bar{r}_{j}[p]=[a^{R}_{j}(p)]\qquad(p\in\mathcal{P}_{n}),$$

where $c^{L}_{j}(p)=x_{j}p$ and $c^{R}_{j}(p)=px_{j}$ because both sides are linear in $p$ and agree on monomials by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra.