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 Free probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of independence, and it is connected with free products.
This theory was i ...
and corresponding to deformations of the
canonical commutation relations
In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example,
hat x,\hat ...
. For other deformations of Gaussian distributions, see
q-Gaussian distribution
The ''q''-Gaussian is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints. It is one example of a Tsallis distribution. The ''q''-Gaussian is a generalization of the Gaussian in the sam ...
and
Gaussian q-distribution
In mathematical physics and probability and statistics, the Gaussian ''q''-distribution is a family of probability distributions that includes, as limiting cases, the uniform distribution and the normal (Gaussian) distribution. It was introduc ...
.
History
The q-Gaussian process was formally introduced in a paper by Frisch and Bourret
under the name of ''parastochastics'', and also later by Greenberg as an example of ''infinite statistics''. It was mathematically established and investigated in
papers by Bozejko and Speicher
and by Bozejko, Kümmerer, and Speicher
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.
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''
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.
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
:
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''
(or ''q-semicircular'') elements.
On
we consider the ''vacuum expectation state''
, for
.
The ''(multivariate) q-Gaussian distribution'' or ''q-Gaussian process''
[ 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,:] 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