q -analog

The Q- analogue of a theorem, identity, or expression is a generalization involving a new parameter q , which returns the original theorem, identity, or expression in the limit as q → 1 . Typically, mathematicians are interested in q- analogs that appear naturally, rather than invent arbitrary q- analogs for known results. The earliest q- analog, studied in detail, is the , which were studied in the 19th century ^[1] .

Q- analogs are most often used in combinatorics and in the theory of special functions . Under these conditions, the limit q → 1 is often formal, since q is often discrete (for example, it can represent a prime power ). Q- analogues find application in many fields, including the study of fractals and multifractal measures and for expressing the entropy of chaotic dynamical systems . The connection with fractals and dynamical systems arises from the fact that many fractal objects have symmetries of Fuchsian groups in general (see, for example, the articles and the Apollonius Grid ) and the modular group in particular. The connection passes through hyperbolic geometry and ergodic theory , where elliptic integrals and modular forms play a major role. The themselves are closely related to elliptic integrals.

Q- analogues appear in the study of quantum groups and in q- indignant . The connection here is similar to how string theory is constructed in the language of Riemann surfaces , which leads to a connection with elliptic curves , which, in turn, are associated with .

Content

"Classical" q- theory

The classical q -theory begins with q- analogs for non-negative integers ^[2] . Equality

\lim _{q\rightarrow 1}{\frac {1-q^{n}}{1-q}}=n

{\ displaystyle \ lim _ {q \ rightarrow 1} {\ frac {1-q ^ {n}} {1-q}} = n}

assumes that we define the q- analog of n , known as the q- bracket or q- number of n , equal to

[n]_{q}={\frac {1-q^{n}}{1-q}}=1+q+q^{2}+\ldots +q^{n-1}.

{\ displaystyle [n] _ {q} = {\ frac {1-q ^ {n}} {1-q}} = 1 + q + q ^ {2} + \ ldots + q ^ {n-1} .}

The choice among other possibilities of specifically this q- analogue has no specific reason, however, the analogue arises naturally in several contexts. For example, if we decide to use the notation [ n ] _q for the q- analog of n , we can determine the q- analog of the factorial , which is known as the q- factor , as follows

{\begin{aligned}{\big [}n]_{q}!&=[1]_{q}\cdot [2]_{q}\cdots [n-1]_{q}\cdot [n]_{q}\\[6pt]&={\frac {1-q}{1-q}}\cdot {\frac {1-q^{2}}{1-q}}\cdots {\frac {1-q^{n-1}}{1-q}}\cdot {\frac {1-q^{n}}{1-q}}\\[6pt]&=1\cdot (1+q)\cdots (1+q+\cdots +q^{n-2})\cdot (1+q+\cdots +q^{n-1}).\end{aligned}}

{\ displaystyle {\ begin {aligned} {\ big [} n] _ {q}! & = [1] _ {q} \ cdot [2] _ {q} \ cdots [n-1] _ {q} \ cdot [n] _ {q} \\ [6pt] & = {\ frac {1-q} {1-q}} \ cdot {\ frac {1-q ^ {2}} {1-q}} \ cdots {\ frac {1-q ^ {n-1}} {1-q}} \ cdot {\ frac {1-q ^ {n}} {1-q}} \\ [6pt] & = 1 \ cdot (1 + q) \ cdots (1 + q + \ cdots + q ^ {n-2}) \ cdot (1 + q + \ cdots + q ^ {n-1}). \ end {aligned}}}

This q- analogue appears naturally in several contexts. What is remarkable, while n ! counts the number of permutations of length n , [ n ] _q ! counts permutations taking into account the number of . That is, if inv ( w ) means the number of inversions of the permutation w , and S _{n the} set of permutations of length n , we have

\sum _{w\in S_{n}}q^{{\text{inv}}(w)}=[n]_{q}!

{\ displaystyle \ sum _ {w \ in S_ {n}} q ^ {{\ text {inv}} (w)} = [n] _ {q}!}

In particular, one can obtain the familiar factorial by going to the limit $q\rightarrow 1$ ${\ displaystyle q \ rightarrow 1}$ .

The Q- factorial also has a brief definition in terms of the q- symbol of Pohhammer , the basic building block of all q- theories:

[n]_{q}!={\frac {(q;q)_{n}}{(1-q)^{n}}}.

{\ displaystyle [n] _ {q}! = {\ frac {(q; q) _ {n}} {(1-q) ^ {n}}}.}

From q -factorials, we can pass to q -binomial coefficients , also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients :

{\binom {n}{k}}_{q}={\frac {[n]_{q}!}{[n-k]_{q}![k]_{q}!}}.

{\ displaystyle {\ binom {n} {k}} _ {q} = {\ frac {[n] _ {q}!} {[nk] _ {q}! [k] _ {q}!}} .}

is defined as

e_{q}^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{[n]_{q}!}}.

{\ displaystyle e_ {q} ^ {x} = \ sum _ {n = 0} ^ {\ infty} {\ frac {x ^ {n}} {[n] _ {q}!}}.}

Trigonometric q -functions, together with q -Fourier transform, are defined in the same context.

Q- Analogs in Combinatorics

Gaussian coefficients count the subspaces of a finite vector space . Let q be the number of elements in a finite field . (The number q is then equal to the degree of a prime , q = pe , so using the letter q is advisable.) Then the number of the k- dimensional subspace of the n- dimensional vector space over the field with q elements

{\binom {n}{k}}_{q}.

{\ displaystyle {\ binom {n} {k}} _ {q}.}

As q approaches 1, we obtain a binomial coefficient

{\binom {n}{k}},

{\ displaystyle {\ binom {n} {k}},}

or, in other words, the number of k -element subsets of a set with n elements.

Thus, we can consider a finite vector space as a q- generalization of a set, and subspaces as a q- generalization of subsets of this set. This is a fruitful point of view for finding interesting theorems. For example, there are q- analogues and Ramsey’s theory .

q → 1

Conversely, the permission to change q and consider q- analogs as deviations can be considered the combinatorial case q = 1 as the limit of q- analogs q → 1 (it is often impossible to simply substitute q = 1 in the formula, so you have to take the limit).

This can be formalized in a , where combinatorics is represented as linear algebra over a field with one element. For example, Weil groups are simply algebraic groups over a field with one element.

Physics Applications

Q- analogs are often found in exact solutions to many-body problems. In such cases, the limit as q → 1 corresponds to a relatively simple dynamics, i.e. without nonlinear disturbances, while q <1 makes it possible to look at the complex nonlinear feedback mode.

An example from atomic physics is the model of creating molecular condensate from an ultracold fermion gas under conditions of sweeping an external magnetic field using Feshbach resonance ^[3] . This process is described by a model with a q- perturbed version of the algebra of operators SU (2) and the solution is described by q- perturbed exponential and binomial distributions.

Notes

↑ Exton, 1983 .
↑ Ernst, 2003 , p. 487-525.
↑ Sun, Sinitsyn, 2016 , p. 033808.

Literature

Exton H. q-Hypergeometric Functions and Applications. - New York: Halstead Press, 1983. - ISBN 0853124914 . - ISBN 0470274530 . - ISBN 978-0470274538 .
Thomas Ernst. A Method for q-calculus // Journal of Nonlinear Mathematical Physics. - 2003. - T. 10 , no. 4 . - S. 487–525 .
Sun C., Sinitsyn NA Landau-Zener extension of the Tavis-Cummings model: Structure of the solution // Phys. Rev. A. - 2016 .-- T. 94 , no. 3 . - DOI : 10.1103 / PhysRevA.94.033808 . - .

Links

Hazewinkel, Michiel, ed. (2001), "Umbral calculus" , Encyclopedia of Mathematics , Springer , ISBN 978-1-55608-010-4
q -analog from MathWorld
q -bracket from MathWorld
q -factorial from MathWorld
q -binomial coefficient from MathWorld

[_5c18ebb0ebf1e164-1] Exton, 1983 .

[_8aff156d7ebc4cc8-2] Ernst, 2003 , p. 487-525.

[_5b4c2f2be91daccb-3] Sun, Sinitsyn, 2016 , p. 033808.