Q-Gaussian process

q-Gaussian processes are deformations of the usual Gaussian distribution. There are several different versions of this; here we treat a multivariate deformation, also addressed as q-Gaussian process, arising from free probability theory and corresponding to deformations of the canonical commutation relations. For other deformations of Gaussian distributions, see q-Gaussian distribution and Gaussian q-distribution.

History

The q-Gaussian process was formally introduced in a paper by Frisch and Bourret^[1] under the name of parastochastics, and also later by Greenberg^[2] as an example of infinite statistics. It was mathematically established and investigated in papers by Bozejko and Speicher^[3] and by Bozejko, Kümmerer, and Speicher^[4] in the context of non-commutative probability.

It is given as the distribution of sums of creation and annihilation operators in a q-deformed Fock space. The calculation of moments of those operators is given by a q-deformed version of a Wick formula or Isserlis formula. The specification of a special covariance in the underlying Hilbert space leads to the q-Brownian motion,^[4] a special non-commutative version of classical Brownian motion.

q-Fock space

In the following ${\displaystyle q\in [-1,1]}$ is fixed. Consider a Hilbert space ${\displaystyle {\mathcal {H}}}$. On the algebraic full Fock space

${\displaystyle {\mathcal {F}}_{\text{alg}}({\mathcal {H}})=\bigoplus _{n\geq 0}{\mathcal {H}}^{\otimes n},}$

where ${\displaystyle {\mathcal {H}}^{0}=\mathbb {C} \Omega }$ with a norm one vector ${\displaystyle \Omega }$, called vacuum, we define a q-deformed inner product as follows:

${\displaystyle \langle h_{1}\otimes \cdots \otimes h_{n},g_{1}\otimes \cdots \otimes g_{m}\rangle _{q}=\delta _{nm}\sum _{\sigma \in S_{n}}\prod _{r=1}^{n}\langle h_{r},g_{\sigma (r)}\rangle q^{i(\sigma )},}$

where ${\displaystyle i(\sigma )=\#\{(k,\ell )\mid 1\leq k<\ell \leq n;\sigma (k)>\sigma (\ell )\}}$ is the number of inversions of ${\displaystyle \sigma \in S_{n}}$.

The q-Fock space^[5] is then defined as the completion of the algebraic full Fock space with respect to this inner product

${\displaystyle {\mathcal {F}}_{q}({\mathcal {H}})={\overline {\bigoplus _{n\geq 0}{\mathcal {H}}^{\otimes n}}}^{\langle \cdot ,\cdot \rangle _{q}}.}$

For ${\displaystyle -1<q<1}$ the q-inner product is strictly positive.^[3][6] For ${\displaystyle q=1}$ and ${\displaystyle q=-1}$ it is positive, but has a kernel, which leads in these cases to the symmetric and anti-symmetric Fock spaces, respectively.

For ${\displaystyle h\in {\mathcal {H}}}$ we define the q-creation operator ${\displaystyle a^{*}(h)}$, given by

${\displaystyle a^{*}(h)\Omega =h,\qquad a^{*}(h)h_{1}\otimes \cdots \otimes h_{n}=h\otimes h_{1}\otimes \cdots \otimes h_{n}.}$

Its adjoint (with respect to the q-inner product), the q-annihilation operator ${\displaystyle a(h)}$, is given by

${\displaystyle a(h)\Omega =0,\qquad a(h)h_{1}\otimes \cdots \otimes h_{n}=\sum _{r=1}^{n}q^{r-1}\langle h,h_{r}\rangle h_{1}\otimes \cdots \otimes h_{r-1}\otimes h_{r+1}\otimes \cdots \otimes h_{n}.}$

q-commutation relations

Those operators satisfy the q-commutation relations^[7]

${\displaystyle a(f)a^{*}(g)-qa^{*}(g)a(f)=\langle f,g\rangle \cdot 1\qquad (f,g\in {\mathcal {H}}).}$

For ${\displaystyle q=1}$, ${\displaystyle q=0}$, and ${\displaystyle q=-1}$ this reduces to the CCR-relations, the Cuntz relations, and the CAR-relations, respectively. With the exception of the case ${\displaystyle q=1,}$ the operators ${\displaystyle a^{*}(f)}$ are bounded.

q-Gaussian elements and definition of multivariate q-Gaussian distribution (q-Gaussian process)

Operators of the form ${\displaystyle s_{q}(h)={a(h)+a^{*}(h)}}$ for ${\displaystyle h\in {\mathcal {H}}}$ are called q-Gaussian^[5] (or q-semicircular^[8]) elements.

On ${\displaystyle {\mathcal {F}}_{q}({\mathcal {H}})}$ we consider the vacuum expectation state ${\displaystyle \tau (T)=\langle \Omega ,T\Omega \rangle }$, for ${\displaystyle T\in {\mathcal {B}}({\mathcal {F}}({\mathcal {H}}))}$.

The (multivariate) q-Gaussian distribution or q-Gaussian process^[4][9] is defined as the non commutative distribution of a collection of q-Gaussians with respect to the vacuum expectation state. For ${\displaystyle h_{1},\dots ,h_{p}\in {\mathcal {H}}}$ the joint distribution of ${\displaystyle s_{q}(h_{1}),\dots ,s_{q}(h_{p})}$ with respect to ${\displaystyle \tau }$ can be described in the following way,:^[1][3] for any ${\displaystyle i\{1,\dots ,k\}\rightarrow \{1,\dots ,p\}}$ we have

${\displaystyle \tau \left(s_{q}(h_{i(1)})\cdots s_{q}(h_{i(k)})\right)=\sum _{\pi \in {\mathcal {P}}_{2}(k)}q^{cr(\pi )}\prod _{(r,s)\in \pi }\langle h_{i(r)},h_{i(s)}\rangle ,}$

where ${\displaystyle cr(\pi )}$ denotes the number of crossings of the pair-partition ${\displaystyle \pi }$. This is a q-deformed version of the Wick/Isserlis formula.

q-Gaussian distribution in the one-dimensional case

For p = 1, the q-Gaussian distribution is a probability measure on the interval ${\displaystyle [-2/{\sqrt {1-q}},2/{\sqrt {1-q}}]}$, with analytic formulas for its density.^[10] For the special cases ${\displaystyle q=1}$, ${\displaystyle q=0}$, and ${\displaystyle q=-1}$, this reduces to the classical Gaussian distribution, the Wigner semicircle distribution, and the symmetric Bernoulli distribution on ${\displaystyle \pm 1}$. The determination of the density follows from old results^[11] on corresponding orthogonal polynomials.

Operator algebraic questions

The von Neumann algebra generated by ${\displaystyle s_{q}(h_{i})}$, for ${\displaystyle h_{i}}$ running through an orthonormal system ${\displaystyle (h_{i})_{i\in I}}$ of vectors in ${\displaystyle {\mathcal {H}}}$, reduces for ${\displaystyle q=0}$ to the famous free group factors ${\displaystyle L(F_{\vert I\vert })}$. Understanding the structure of those von Neumann algebras for general q has been a source of many investigations.^[12] It is now known, by work of Guionnet and Shlyakhtenko,^[13] that at least for finite I and for small values of q, the von Neumann algebra is isomorphic to the corresponding free group factor.