## A q-deformation of the Gauss distribution (2000)

Venue: | J. Math. Phys |

Citations: | 14 - 2 self |

### BibTeX

@ARTICLE{Leeuwen00aq-deformation,

author = {Hans Van Leeuwen and Hans Maassen},

title = {A q-deformation of the Gauss distribution},

journal = {J. Math. Phys},

year = {2000},

volume = {36},

pages = {4743--4756}

}

### OpenURL

### Abstract

The q-deformed commutation relation aa # - qa # a = 11 for the harmonic oscillator is considered with q # [-1, 1]. An explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed. In this representation the distribution of a + a # in the vacuum state is explicitly calculated. This distribution is to be regarded as the natural q- deformation of the Gaussian. 1995 PACS numbers: 02.50.Cw, 05.40.+j, 03.65.Db, 42.50.Lc 1991 MSC numbers: 81S25, 33D90, 81Q10 1 1

### Citations

687 |
Basic Hypergeometric Series
- Gasper, Rahman
- 1990
(Show Context)
Citation Context ... k , a # R + . A good definition for the q-gamma function is now given by: # q (x) := [#] q Z 0 t x-1 exp q qt d q t = (q; q)# (1 - q) n # X k=0 q (n+1)k (q; q) k . 7 The following classical lemma ([=-=GR90-=-], [Jac10], [KS94] and [Koo94]) is given here with proof because it illustrates some techniques needed later on. Lemma 2.3 #n # N : # q (n + 1) = [n] q ! Proof: Note that [#] q Z 0 D q t n exp q (t) ... |

392 | The Askey-scheme of hypergeometric orthogonal polynomials and its qanalogues, Reports of the Faculty
- Koekoek, Swarttouw
- 1998
(Show Context)
Citation Context ...good definition for the q-gamma function is now given by: # q (x) := [#] q Z 0 t x-1 exp q qt d q t = (q; q)# (1 - q) n # X k=0 q (n+1)k (q; q) k . 7 The following classical lemma ([GR90], [Jac10], [=-=KS9-=-4] and [Koo94]) is given here with proof because it illustrates some techniques needed later on. Lemma 2.3 #n # N : # q (n + 1) = [n] q ! Proof: Note that [#] q Z 0 D q t n exp q (t) d q t = 8 > > >... |

307 | Free Random Variables - Voiculescu, Dykema, et al. - 1992 |

168 |
On a Hilbert space of analytic functions and an associated integral transform
- Bargmann
- 1961
(Show Context)
Citation Context ...resented by a qdi#erence operator D q . As q tends to 1,sq will tend to the Gauss measure on C, 4 and D q becomes di#erentiation. Thus one obtains Bargmann's representation of the harmonic oscillator =-=[Bar61]-=-. It should be noted that there is no measuresq for qs0. So far we essentially reproduce the work of Arik and Coon [AC76]. In section 3 we perform the diagonalization of a+a # = D q +Z by constructing... |

86 |
C∗-algebras and W ∗-algebras
- Sakai
- 1971
(Show Context)
Citation Context ... q = 0, we shall simply mean one satisfying i, ii and iii. Proof: Consider a bounded operator a on a Hilbert space H that defines a non trivial irreducible representation of (2). Proposition 1.1.8 in =-=[Sak71]-=- states that #(xy) # { 0 } = #(yx) # { 0 } for all x, y in some unital algebra over C. In the case at hand this implies that #(a # a) # { 0 } = #(aa # ) # { 0 }. Define a linear invertible mapping # :... |

69 | An example of a generalized Brownian motion - Bozejko, Speicher |

62 |
Quantum Probability for Probabilists
- Meyer
- 1995
(Show Context)
Citation Context ...ttention, what distributions are obtained in this limit, if one replaces the classical commutative notion of independence by some other type. Anti-commutative independence, as occuring in Fermi noise =-=[Mey93]-=- and free independence as occurs in large random matrices [Maa92, Spe90, VDN92] have now been studied. In the former case the Gauss distribution is replaced by the measure (# 1 + #-1 )/2 and in the la... |

55 | Recurrence relations, continued fractions and orthogonal polynomials - Askey, Ismail - 1984 |

33 | Addition of freely independent random variables - Maassen - 1992 |

29 |
Compact quantum groups and q-special functions, in Representations of Lie Groups and Quantum Groups
- Koornwinder
- 1993
(Show Context)
Citation Context ...tion for the q-gamma function is now given by: # q (x) := [#] q Z 0 t x-1 exp q qt d q t = (q; q)# (1 - q) n # X k=0 q (n+1)k (q; q) k . 7 The following classical lemma ([GR90], [Jac10], [KS94] and [=-=Koo9-=-4]) is given here with proof because it illustrates some techniques needed later on. Lemma 2.3 #n # N : # q (n + 1) = [n] q ! Proof: Note that [#] q Z 0 D q t n exp q (t) d q t = 8 > > > > > > > > >... |

27 |
Hilbert spaces of analytic functions and generalized coherent states
- Arik, Coon
- 1976
(Show Context)
Citation Context ...iation. Thus one obtains Bargmann's representation of the harmonic oscillator [Bar61]. It should be noted that there is no measuresq for qs0. So far we essentially reproduce the work of Arik and Coon =-=[AC76]-=-. In section 3 we perform the diagonalization of a+a # = D q +Z by constructing a unitary operator W like the one above. The q-Gaussian distribution # q is found naturally for q # [0, 1). We shall mak... |

22 |
Topics in Hardy classes and univalent functions, Birkhäuser
- Rosenblum, Rovnyak
- 1994
(Show Context)
Citation Context ...ft on l 2 (N). In this case a and a # can be quite nicely represented as operators on the Hardy class H 2 of all analytic functions on the unit disc with L 2 limits towards the boundary (for instance =-=[RR94]-=-) via the equivalence l 2 (N) #H 2 given by (#n ) n#N , # X n=0 #nz n , |z|s1. Under this equivalence a # and a change into multiplication by z and the operator (af )(z) := f(z) - f (0) z . The probab... |

19 | Elliptic Functions (Grundlehren der math - Chandrasekharan - 1985 |

6 |
q-canonical commutation relations and stability of the cuntz algebra
- Jrgensen, Schmitt, et al.
- 1994
(Show Context)
Citation Context ...tion operators associated to f satisfying the q-deformed commutation relation a(f)a(g) # - qa(g) # a(f ) = f , g 11, f , g # H. Algebraic aspects of these commutation relations have been studied in [=-=JSW91]-=-. Another interpolation between boson and fermion Brownian motion is described in [Sch91]. There are good reasons to regard the probability distribution # q of these operators a(f ) +a(f ) # in the va... |

3 |
Quantum q-white noise and a q-central limit theorem
- Schurmann
- 1991
(Show Context)
Citation Context ...\Gamma qa(g) a(f) = hf; gi 11; f; g 2 H: Algebraic aspects of these commutation relations have been studied in [JSW91]. Another interpolation between boson and fermion Brownian motion is described in =-=[Sch91]-=-. There are good reasons to regard the probability distributionsq of these operators a(f) + a(f) in the vacuum state as the natural q-deformation of the Gaussian distribution (cf. [Spe90]). By combina... |

2 |
On q-definite integrals, The quarterly journal of pure and applied mathematics
- Jackson
- 1910
(Show Context)
Citation Context ... R + . A good definition for the q-gamma function is now given by: # q (x) := [#] q Z 0 t x-1 exp q qt d q t = (q; q)# (1 - q) n # X k=0 q (n+1)k (q; q) k . 7 The following classical lemma ([GR90], [=-=Jac1-=-0], [KS94] and [Koo94]) is given here with proof because it illustrates some techniques needed later on. Lemma 2.3 #n # N : # q (n + 1) = [n] q ! Proof: Note that [#] q Z 0 D q t n exp q (t) d q t =... |

2 |
A new example of "independence" and "white noise", Probability theory and related fields 84
- Speicher
- 1990
(Show Context)
Citation Context ...ibed in [Sch91]. There are good reasons to regard the probability distribution # q of these operators a(f ) +a(f ) # in the vacuum state as the natural q-deformation of the Gaussian distribution (cf. =-=[Spe90]-=-). By combinatoric means, Bo zejko and Speicher using recursion relations for orthogonal polynomials in a(f ) + a(f ) # , calculated this q-Gaussian distribution # q . For #f# = 1, it is supported by ... |

1 | Some properties of the q-hermite polynomials, Canadian - Allaway - 1980 |

1 | An example of a generalized brownian motion II, Quantum probability and related topics VII - zejko, Speicher - 1992 |

1 |
urmann, Quantum q-white noise and a q-central limit theorem
- Sch
- 1991
(Show Context)
Citation Context ...- qa(g) # a(f ) = f , g 11, f , g # H. Algebraic aspects of these commutation relations have been studied in [JSW91]. Another interpolation between boson and fermion Brownian motion is described in [=-=Sch91]-=-. There are good reasons to regard the probability distribution # q of these operators a(f ) +a(f ) # in the vacuum state as the natural q-deformation of the Gaussian distribution (cf. [Spe90]). By co... |