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 is fixed.
Consider a Hilbert space . On the algebraic full Fock space
where with a norm one vector , called vacuum, we define a q-deformed inner product as follows:
where is the number of inversions of .
The q-Fock space[5] is then defined as the completion of the algebraic full Fock space with respect to this inner product
For the q-inner product is strictly positive.[3][6] For and it is positive, but has a kernel, which leads in these cases to the symmetric and anti-symmetric Fock spaces, respectively.
For we define the q-creation operator, given by
Its adjoint (with respect to the q-inner product), the q-annihilation operator, is given by
q-commutation relations
Those operators satisfy the q-commutation relations[7]
For , , and this reduces to the CCR-relations, the Cuntz relations, and the CAR-relations, respectively. With the exception of the case the operators are bounded.
q-Gaussian elements and definition of multivariate q-Gaussian distribution (q-Gaussian process)
Operators of the form
for are called q-Gaussian[5] (or q-semicircular[8]) elements.
On we consider the vacuum expectation state, for .
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 the joint distribution of with respect to can be described in the following way,:[1][3] for any we have
where denotes the number of crossings of the pair-partition . 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 , with analytic formulas for its density.[10] For the special cases , , and , this reduces to the classical Gaussian distribution, the Wigner semicircle distribution, and the symmetric Bernoulli distribution on . The determination of the density follows from old results[11] on corresponding orthogonal polynomials.
Operator algebraic questions
The von Neumann algebra generated by , for running through an orthonormal system of vectors in , reduces for to the famous free group factors . 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.
^Vergès, Matthieu Josuat (20 November 2018). "Cumulants of the q-semicircular Law, Tutte Polynomials, and Heaps". Canadian Journal of Mathematics. 65 (4): 863–878. arXiv:1203.3157. doi:10.4153/CJM-2012-042-9. S2CID2215028.
^Bryc, Włodzimierz; Wang, Yizao (2016). "The local structure of q-Gaussian processes". Probability and Mathematical Statistics. 36 (2): 335–352. arXiv:1511.06667. MR3593028.
^Szegö, G (1926). "Ein Beitrag zur Theorie der Thetafunktionen" [A contribution to the theory of theta functions]. Sitzungsberichte der Preussischen Akademie der Wissenschaften, Phys.-Math. Klasse (in German): 242–252.
^Wasilewski, Mateusz (2021). "A simple proof of the complete metric approximation property for q-Gaussian algebras". Colloquium Mathematicum. 163 (1): 1–14. arXiv:1907.00730. doi:10.4064/cm7968-11-2019. MR4162298.