^{1}

^{*}

^{2}

^{*}

The
relation between noncommutative (or quantum) geometry and themathematics of spacesis in many ways
similar to the relation between quantum physicsand classical physics. One moves from the commutative algebra of
functions on a space (or a commutative algebra of classical observable in
classical physics) to a noncommutative algebra representing a noncommutative
space (or a noncommutative algebra of
quantum observables in quantum physics). The object of this paper is to study
the basic rules governing *q*-calculus
as compared with the classical Newton-Leibnitz calculus.

There exists an intimate relationship between classical geometry and physics, and to appreciate this we will consider the Einstein field equations (EFE) of special relativity written as:

where

tum tensor.

For the moment we are only interested in three basic properties of the above equation.

1) The equation (EFE) is a tensor equation. This is necessarily so, since the principle of invariance under coordinate transformations must hold, in other words the equations of physics must look the same in any frame of reference.

2) We can interpret equation (EFE) more simply as

i.e. it is the presence of matter in space that distorts the neighbouring geometry. Most equations of mathematical physics can be interpreted similarly.

3) The solution to equation (EFE) is a geometrical object, namely a line element given by

where

Every geometry is associated with some kind of space. Quantum (or noncommutative) geometry [

In generalizing classical geometry to the non-commutative level, there are two important conceptual steps:

1) Translation of geometry into a commutative algebra format;

2) Non-commutative generalizations.

It turns out that the geometrical structure on any given topological space X is always completely expressible in the language of some associated *-algebra [

Points

Let X be a compact topological space, and let

be the

-algebra of continuous complex-valued functions on X. Every element

naturally gives rise to a linear functional

defined by

This map is multiplicative in the sense that

Furthermore,

on A. Conversely, consider an arbitrary character

The algebra

is a commutative C^{*}-algebra [^{*}-algebras. This means that for every commutative unital C^{*}-algebra

A there exists (up to homeomerphisms) a unique compact topological space X such that

As we have earlier observed, the points of the space X are recovered as characters of the associated algebra A. In terms of this identification, the topology on X coincides with the weak^{*}-topology, induced from the dual space^{*}-algebras are automatically continuous, in particular characters are continuous linear functionals.

A simple example of quantum geometry is the quantum plane (

In the quantum plane we replace the property

Define

The commutative case is obtained when

It is interesting to know that one can really do geometry in this setting, where coordinates do not commute. This is the remarkable discovery in recent times. For example, we can follow the approach of Newton-Leibnitz defining differentiation by

But when

Thus for example the derivative of the function

We observe here that when

The mathematical study of noncommutative geometry is intimately related to the so-called q-calculus (q- numbers, q-factorials, q-differentials and integrals, basic q-hypergeometric functions, and q-orthogonal polynomials). Here we give a brief introduction to q-numbers and q-factorials which will be required in the subsequent sections.

For any nonzero complex number q, the q-number

The quantum plane (xy ≠ yx)

We observe that,

Thus given

The q-numbers satisfy the following relations derived from the property of the exponential function

1)

2)

3)

4)

5)

The proof of 1) is easy to see from the fact that;

Let

Hence;

The rest of the identities can be proved similarly. It is important to note that the relations 1)-5) remain valid

when q is considered an indeterminate. Consequently any q-number

Suppose

The following expression is quite useful in the theory of hypergeometric functions as well as in combinatorics.

It is now possible for us to relate the above with the q-factorials. We observe that;

From equation (2.3) we note that

which converges

Remark: We can also define and show that;

The q-binomial coefficients are defined by the formula;

Remark: There exist a close analogy between the classical binomial coefficients

their q-analogues. Many of the identities satisfied by the former have their counterparts for the q-binomial coefficients. For example the classical identity;

simply translates to;

Let

In case q is a primitive p^{th} root of unity, and p is odd, then

Equation (2.9) can be established by induction on n, and employing the first identity in (2.8). The second assertion follows directly from (2.9). Observe that:

From (2.7)

and

The following are important basic notions derived from their analogue in classical calculus, and will be employed subsequently.

For

Note that

Provided the expression on the right hand side exists.

Let

Setting

The formula for the product of two functions is given by

We can now define the q-analogue of the Newton-Leibnitz rule:

Thus,

By induction on n, it follows from (3.3) that:

As a special case when

The q-integral operator will be defined as the inverse of the q-differential operator.

Given

It then follows that;

Hence, summing these relations over

Assuming

We can now formally define the q-integral of a function

On the semi-infinite interval

Over any closed interval

We now define the integral over the interval

The integration by parts formula of Newton-Leibnitz calculus is interpreted in the present noncommutative context as:

There are a number of applications of the foregoing, we mention here just two, namely:

1) q-binomial formulae in two variables satisfying a quadratic relation, this has recently been published in [

2) A recent trend in modern physics is the study of the quantum anti-de Sitter space [

This work began at the African Institute for mathematical sciences (AIMS) in Muizenberg South Africa, when the first author visited in 2010. I wish to thank the director Prof Fritz Hahne for giving me a postdoctoral fellowship, and for providing a conducive environment which made this research possible.