Q-Gaussian process

The q-Gaussian process was formally introduced in a paper by Frisch and Bourret{{cite journal |last1=Frisch |first1=U. |last2=Bourret |first2=R. |title=Parastochastics |journal=Journal of Mathematical Physics |date=February 1970 |volume=11 |issue=2 |pages=364–390 |doi=10.1063/1.1665149 |bibcode=1970JMP....11..364F }} under the name of parastochastics, and also later by Greenberg{{cite journal |last1=Greenberg |first1=O. W. |title=Example of infinite statistics |journal=Physical Review Letters |date=12 February 1990 |volume=64 |issue=7 |pages=705–708 |doi=10.1103/PhysRevLett.64.705 |pmid=10042057 |bibcode=1990PhRvL..64..705G }} as an example of infinite statistics. It was mathematically established and investigated in

papers by Bozejko and Speicher{{cite journal |last1=Bożejko |first1=Marek |last2=Speicher |first2=Roland |title=An example of a generalized Brownian motion |journal=Communications in Mathematical Physics |date=April 1991 |volume=137 |issue=3 |pages=519–531 |doi=10.1007/BF02100275 |bibcode=1991CMaPh.137..519B |s2cid=123190397 |url=http://projecteuclid.org/euclid.cmp/1104202738 }} and by Bozejko, Kümmerer, and Speicher{{cite journal |last1=Bożejko |first1=M. |last2=Kümmerer |first2=B. |last3=Speicher |first3=R. |title=q-Gaussian Processes: Non-commutative and Classical Aspects |journal=Communications in Mathematical Physics |date=1 April 1997 |volume=185 |issue=1 |pages=129–154 |doi=10.1007/s002200050084 |arxiv=funct-an/9604010 |bibcode=1997CMaPh.185..129B |s2cid=2993071 }} 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, a special non-commutative version of classical Brownian motion.

In the following $q\backslash in[-1,1]$ is fixed.

Consider a Hilbert space $\backslash mathcal\{H\}$. On the algebraic full Fock space

: $$

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

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

:$$

\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^n_{r=1}\langle h_r,g_{\sigma(r)}\rangle q^{i(\sigma)},

where is the number of inversions of $\backslash sigma\backslash in\; S\_n$.

The q-Fock space{{cite journal |last1=Effros |first1=Edward G. |last2=Popa |first2=Mihai |title=Feynman diagrams and Wick products associated with q-Fock space |journal=Proceedings of the National Academy of Sciences |date=22 July 2003 |volume=100 |issue=15 |pages=8629–8633 |doi=10.1073/pnas.1531460100 |pmid=12857947 |pmc=166362 |arxiv=math/0303045 |bibcode=2003PNAS..100.8629E |doi-access=free }} is then defined as the completion of the algebraic full Fock space with respect to this inner product

:$$

\mathcal{F}_q(\mathcal{H})=\overline{\bigoplus_{n\geq 0}\mathcal{H}^{\otimes n}}^{\langle\cdot,\cdot\rangle_q}.

For the q-inner product is strictly positive.{{cite journal |last1=Zagier |first1=Don |title=Realizability of a model in infinite statistics |journal=Communications in Mathematical Physics |date=June 1992 |volume=147 |issue=1 |pages=199–210 |doi=10.1007/BF02099535 |bibcode=1992CMaPh.147..199Z |citeseerx=10.1.1.468.966 |s2cid=53385666 }} For $q=1$ and $q=-1$ it is positive, but has a kernel, which leads in these cases to the symmetric and anti-symmetric Fock spaces, respectively.

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

:$$

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 $a(h)$, is given by

:$$

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.

Those operators satisfy the q-commutation relations{{cite journal |last1=Kennedy |first1=Matthew |last2=Nica |first2=Alexandru |title=Exactness of the Fock Space Representation of the q-Commutation Relations |journal=Communications in Mathematical Physics |date=9 September 2011 |volume=308 |issue=1 |pages=115–132 |doi=10.1007/s00220-011-1323-9 |arxiv=1009.0508 |bibcode=2011CMaPh.308..115K |s2cid=119124507 }}

:$a(f)a^*(g)-q\; a^*(g)a(f)=\backslash langle\; f,g\backslash rangle\; \backslash cdot\; 1\backslash qquad\; (f,g\backslash in\; \backslash mathcal\{H\}).$

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

Operators of the form $s\_q(h)=\{a(h)+a^*(h)\}$

for $h\backslash in\backslash mathcal\{H\}$ are called q-Gaussian (or q-semicircular{{cite journal |last1=Vergès |first1=Matthieu Josuat |title=Cumulants of the q-semicircular Law, Tutte Polynomials, and Heaps |journal=Canadian Journal of Mathematics |date=20 November 2018 |volume=65 |issue=4 |pages=863–878 |doi=10.4153/CJM-2012-042-9 |arxiv=1203.3157 |s2cid=2215028 }}) elements.

On $\backslash mathcal\{F\}\_q(\backslash mathcal\{H\})$ we consider the vacuum expectation state

$\backslash tau(T)=\backslash langle\; \backslash Omega,T\backslash Omega\; \backslash rangle$, for $T\backslash in\backslash mathcal\{B\}(\backslash mathcal\{F\}(\backslash mathcal\{H\}))$.

The (multivariate) q-Gaussian distribution or q-Gaussian process{{cite journal

| last1 = Bryc | first1 = Włodzimierz

| last2 = Wang | first2 = Yizao

| arxiv = 1511.06667

| issue = 2

| journal = Probability and Mathematical Statistics

| mr = 3593028

| pages = 335–352

| title = The local structure of {{mvar|q}}-Gaussian processes

| volume = 36

| year = 2016}} is defined as the non commutative distribution of a collection of q-Gaussians with respect to the vacuum expectation state. For $h\_1,\backslash dots,h\_p\backslash in\backslash mathcal\{H\}$ the joint distribution of $s\_q(h\_1),\backslash dots,s\_q(h\_p)$ with respect to $\backslash tau$ can be described in the following way,: for any $i\backslash \{1,\backslash dots,k\backslash \}\backslash rightarrow\backslash \{1,\backslash dots,p\backslash \}$ we have

:$$

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

For p = 1, the q-Gaussian distribution is a probability measure on the interval $[-2/\backslash sqrt\{1-q\},\; 2/\backslash sqrt\{1-q\}]$, with analytic formulas for its density.{{cite journal |last1=Leeuwen |first1=Hans van |last2=Maassen |first2=Hans |title=A q deformation of the Gauss distribution |journal=Journal of Mathematical Physics |date=September 1995 |volume=36 |issue=9 |pages=4743–4756 |doi=10.1063/1.530917 |bibcode=1995JMP....36.4743V |hdl=2066/141604 |hdl-access=free }} For the special cases $q=1$, $q=0$, and $q=-1$, this reduces to the classical Gaussian distribution, the Wigner semicircle distribution, and the symmetric Bernoulli distribution on $\backslash pm\; 1$. The determination of the density follows from old results{{cite journal |last1=Szegö |first1=G |title=Ein Beitrag zur Theorie der Thetafunktionen |trans-title=A contribution to the theory of theta functions |language=German |journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften, Phys.-Math. Klasse |pages=242–252 |year=1926 }} on corresponding orthogonal polynomials.

The von Neumann algebra generated by $s\_q(h\_i)$, for $h\_i$ running through an orthonormal system $(h\_i)\_\{i\backslash in\; I\}$ of vectors in $\backslash mathcal\{H\}$, reduces for $q=0$ to the famous free group factors $L(F\_\{\backslash vert\; I\backslash vert\})$. Understanding the structure of those von Neumann algebras for general q has been a source of many investigations.{{cite journal

| last = Wasilewski | first = Mateusz

| arxiv = 1907.00730

| doi = 10.4064/cm7968-11-2019

| issue = 1

| journal = Colloquium Mathematicum

| mr = 4162298

| pages = 1–14

| title = A simple proof of the complete metric approximation property for {{mvar|q}}-Gaussian algebras

| volume = 163

| year = 2021}} It is now known, by work of Guionnet and Shlyakhtenko,{{cite journal |last1=Guionnet |first1=A. |last2=Shlyakhtenko |first2=D. |title=Free monotone transport |journal=Inventiones Mathematicae |date=13 November 2013 |volume=197 |issue=3 |pages=613–661 |doi=10.1007/s00222-013-0493-9 |arxiv=1204.2182 |s2cid=16882208 }} that at least for finite I and for small values of q, the von Neumann algebra is isomorphic to the corresponding free group factor.

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Category:Probability distributions