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 [math]\displaystyle{ q\in[-1,1] }[/math] is fixed. Consider a Hilbert space [math]\displaystyle{ \mathcal{H} }[/math]. On the algebraic full Fock space
- [math]\displaystyle{ \mathcal{F}_\text{alg}(\mathcal{H})=\bigoplus_{n\geq 0}\mathcal{H}^{\otimes n}, }[/math]
where [math]\displaystyle{ \mathcal{H}^0=\mathbb{C}\Omega }[/math] with a norm one vector [math]\displaystyle{ \Omega }[/math], called vacuum, we define a q-deformed inner product as follows:
- [math]\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^n_{r=1}\langle h_r,g_{\sigma(r)}\rangle q^{i(\sigma)}, }[/math]
where [math]\displaystyle{ i(\sigma)=\#\{(k,\ell)\mid 1\leq k\lt \ell\leq n; \sigma(k)\gt \sigma(\ell)\} }[/math] is the number of inversions of [math]\displaystyle{ \sigma\in S_n }[/math].
The q-Fock space[5] is then defined as the completion of the algebraic full Fock space with respect to this inner product
- [math]\displaystyle{ \mathcal{F}_q(\mathcal{H})=\overline{\bigoplus_{n\geq 0}\mathcal{H}^{\otimes n}}^{\langle\cdot,\cdot\rangle_q}. }[/math]
For [math]\displaystyle{ -1 \lt q \lt 1 }[/math] the q-inner product is strictly positive.[3][6] For [math]\displaystyle{ q=1 }[/math] and [math]\displaystyle{ q=-1 }[/math] it is positive, but has a kernel, which leads in these cases to the symmetric and anti-symmetric Fock spaces, respectively.
For [math]\displaystyle{ h\in\mathcal{H} }[/math] we define the q-creation operator [math]\displaystyle{ a^*(h) }[/math], given by
- [math]\displaystyle{ a^*(h)\Omega=h,\qquad a^*(h)h_1\otimes\cdots\otimes h_n=h\otimes h_1\otimes\cdots\otimes h_n. }[/math]
Its adjoint (with respect to the q-inner product), the q-annihilation operator [math]\displaystyle{ a(h) }[/math], is given by
- [math]\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. }[/math]
q-commutation relations
Those operators satisfy the q-commutation relations[7]
- [math]\displaystyle{ a(f)a^*(g)-q a^*(g)a(f)=\langle f,g\rangle \cdot 1\qquad (f,g\in \mathcal{H}). }[/math]
For [math]\displaystyle{ q=1 }[/math], [math]\displaystyle{ q=0 }[/math], and [math]\displaystyle{ q=-1 }[/math] this reduces to the CCR-relations, the Cuntz relations, and the CAR-relations, respectively. With the exception of the case [math]\displaystyle{ q=1, }[/math] the operators [math]\displaystyle{ a^*(f) }[/math] are bounded.
q-Gaussian elements and definition of multivariate q-Gaussian distribution (q-Gaussian process)
Operators of the form [math]\displaystyle{ s_q(h)={a(h)+a^*(h)} }[/math] for [math]\displaystyle{ h\in\mathcal{H} }[/math] are called q-Gaussian[5] (or q-semicircular[8]) elements.
On [math]\displaystyle{ \mathcal{F}_q(\mathcal{H}) }[/math] we consider the vacuum expectation state [math]\displaystyle{ \tau(T)=\langle \Omega,T\Omega \rangle }[/math], for [math]\displaystyle{ T\in\mathcal{B}(\mathcal{F}(\mathcal{H})) }[/math].
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 [math]\displaystyle{ h_1,\dots,h_p\in\mathcal{H} }[/math] the joint distribution of [math]\displaystyle{ s_q(h_1),\dots,s_q(h_p) }[/math] with respect to [math]\displaystyle{ \tau }[/math] can be described in the following way,:[1][3] for any [math]\displaystyle{ i\{1,\dots,k\}\rightarrow\{1,\dots,p\} }[/math] we have
- [math]\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, }[/math]
where [math]\displaystyle{ cr(\pi) }[/math] denotes the number of crossings of the pair-partition [math]\displaystyle{ \pi }[/math]. 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 [math]\displaystyle{ [-2/\sqrt{1-q}, 2/\sqrt{1-q}] }[/math], with analytic formulas for its density.[10] For the special cases [math]\displaystyle{ q=1 }[/math], [math]\displaystyle{ q=0 }[/math], and [math]\displaystyle{ q=-1 }[/math], this reduces to the classical Gaussian distribution, the Wigner semicircle distribution, and the symmetric Bernoulli distribution on [math]\displaystyle{ \pm 1 }[/math]. The determination of the density follows from old results[11] on corresponding orthogonal polynomials.
Operator algebraic questions
The von Neumann algebra generated by [math]\displaystyle{ s_q(h_i) }[/math], for [math]\displaystyle{ h_i }[/math] running through an orthonormal system [math]\displaystyle{ (h_i)_{i\in I} }[/math] of vectors in [math]\displaystyle{ \mathcal{H} }[/math], reduces for [math]\displaystyle{ q=0 }[/math] to the famous free group factors [math]\displaystyle{ L(F_{\vert I\vert}) }[/math]. 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.
References
- ↑ 1.0 1.1 Frisch, U.; Bourret, R. (February 1970). "Parastochastics". Journal of Mathematical Physics 11 (2): 364–390. doi:10.1063/1.1665149. Bibcode: 1970JMP....11..364F.
- ↑ Greenberg, O. W. (12 February 1990). "Example of infinite statistics". Physical Review Letters 64 (7): 705–708. doi:10.1103/PhysRevLett.64.705. PMID 10042057. Bibcode: 1990PhRvL..64..705G.
- ↑ 3.0 3.1 3.2 Bożejko, Marek; Speicher, Roland (April 1991). "An example of a generalized Brownian motion". Communications in Mathematical Physics 137 (3): 519–531. doi:10.1007/BF02100275. Bibcode: 1991CMaPh.137..519B. http://projecteuclid.org/euclid.cmp/1104202738.
- ↑ 4.0 4.1 4.2 Bożejko, M.; Kümmerer, B.; Speicher, R. (1 April 1997). "q-Gaussian Processes: Non-commutative and Classical Aspects". Communications in Mathematical Physics 185 (1): 129–154. doi:10.1007/s002200050084. Bibcode: 1997CMaPh.185..129B.
- ↑ 5.0 5.1 Effros, Edward G.; Popa, Mihai (22 July 2003). "Feynman diagrams and Wick products associated with q-Fock space". Proceedings of the National Academy of Sciences 100 (15): 8629–8633. doi:10.1073/pnas.1531460100. PMID 12857947. Bibcode: 2003PNAS..100.8629E.
- ↑ Zagier, Don (June 1992). "Realizability of a model in infinite statistics". Communications in Mathematical Physics 147 (1): 199–210. doi:10.1007/BF02099535. Bibcode: 1992CMaPh.147..199Z.
- ↑ Kennedy, Matthew; Nica, Alexandru (9 September 2011). "Exactness of the Fock Space Representation of the q-Commutation Relations". Communications in Mathematical Physics 308 (1): 115–132. doi:10.1007/s00220-011-1323-9. Bibcode: 2011CMaPh.308..115K.
- ↑ Vergès, Matthieu Josuat (20 November 2018). "Cumulants of the q-semicircular Law, Tutte Polynomials, and Heaps". Canadian Journal of Mathematics 65 (4): 863–878. doi:10.4153/CJM-2012-042-9.
- ↑ Bryc, Włodzimierz; Wang, Yizao (24 February 2016). The local structure of q-Gaussian processes.
- ↑ Leeuwen, Hans van; Maassen, Hans (September 1995). "A q deformation of the Gauss distribution". Journal of Mathematical Physics 36 (9): 4743–4756. doi:10.1063/1.530917. Bibcode: 1995JMP....36.4743V.
- ↑ Szegö, G (1926). "Ein Beitrag zur Theorie der Thetafunktionen" (in German). Sitzungsberichte der Preussischen Akademie der Wissenschaften, Phys.-Math. Klasse: 242–252.
- ↑ Wasilewski, Mateusz (24 February 2020). A simple proof of the complete metric approximation property for q-Gaussian algebras.
- ↑ Guionnet, A.; Shlyakhtenko, D. (13 November 2013). "Free monotone transport". Inventiones Mathematicae 197 (3): 613–661. doi:10.1007/s00222-013-0493-9.
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