Type of mathematical generalization such that the original version is the limit as q approaches 1
In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1. Typically, mathematicians are interested in q-analogs that arise naturally, rather than in arbitrarily contriving q-analogs of known results. The earliest q-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.[1]
Classical q-theory begins with the q-analogs of the nonnegative integers.[2] The equality
suggests that we define the q-analog of n, also known as the q-bracket or q-number of n, to be
By itself, the choice of this particular q-analog among the many possible options is unmotivated. However, it appears naturally in several contexts. For example, having decided to use [n]q as the q-analog of n, one may define the q-analog of the factorial, known as the q-factorial, by
This q-analog appears naturally in several contexts. Notably, while n! counts the number of permutations of length n, [n]q! counts permutations while keeping track of the number of inversions. That is, if inv(w) denotes the number of inversions of the permutation w and Sn denotes the set of permutations of length n, we have
In particular, one recovers the usual factorial by taking the limit as .
The q-factorial also has a concise definition in terms of the q-Pochhammer symbol, a basic building-block of all q-theories:
From the q-factorials, one can move on to define the q-binomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients:
q-trigonometric functions, along with a q-Fourier transform, have been defined in this context.
Combinatorial q-analogs
The Gaussian coefficients count subspaces of a finite vector space. Let q be the number of elements in a finite field. (The number q is then a power of a prime number, q = pe, so using the letter q is especially appropriate.) Then the number of k-dimensional subspaces of the n-dimensional vector space over the q-element field equals
Letting q approach 1, we get the binomial coefficient
or in other words, the number of k-element subsets of an n-element set.
Thus, one can regard a finite vector space as a q-generalization of a set, and the subspaces as the q-generalization of the subsets of the set. This has been a fruitful point of view in finding interesting new theorems. For example, there are q-analogs of Sperner's theorem and Ramsey theory. [citation needed]
Let q = (e2πi/n)d be the d-th power of a primitive n-th root of unity. Let C be a cyclic group of order n generated by an element c. Let X be the set of k-element subsets of the n-element set {1, 2, ..., n}. The group C has a canonical action on X given by sending c to the cyclic permutation (1, 2, ..., n). Then the number of fixed points of cd on X is equal to
Conversely, by letting q vary and seeing q-analogs as deformations, one can consider the combinatorial case of q = 1 as a limit of q-analogs as q → 1 (often one cannot simply let q = 1 in the formulae, hence the need to take a limit).
This can be formalized in the field with one element, which recovers combinatorics as linear algebra over the field with one element: for example, Weyl groups are simple algebraic groups over the field with one element.
Applications in the physical sciences
q-analogs are often found in exact solutions of many-body problems.[citation needed] In such cases, the q → 1 limit usually corresponds to relatively simple dynamics, e.g., without nonlinear interactions, while q < 1 gives insight into the complex nonlinear regime with feedbacks.
An example from atomic physics is the model of molecular condensate creation from an ultra cold fermionic atomic gas during a sweep of an external magnetic field through the Feshbach resonance.[3] This process is described by a model with a q-deformed version of the SU(2) algebra of operators, and its solution is described by q-deformed exponential and binomial distributions.
Ismail, M. E. H. (2005), Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press.
Koekoek, R. & Swarttouw, R. F. (1998), The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, 98-17, Delft University of Technology, Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics.