How is the Second Equation in Quantum Geometry Derived?

[MathematicalPhysicist]

in the next link there is an article about quantum geometry:http://www.maths.qmw.ac.uk/~majid/qgeom.html
i would like to know how was the second equation derived?:
f'(x)=f(x)-f(qx)/(1-q)x.

 

[marcus]

Originally posted by loop quantum gravity
in the next link there is an article about quantum geometry:http://www.maths.qmw.ac.uk/~majid/qgeom.html
i would like to know how was the second equation derived?:
f'(x)=f(x)-f(qx)/(1-q)x.

Hello Loop,

What Majid means by "quantum geometry" is very different from what Abhay Ashtekar means by it----that is, what it means in the context of LQG.

Majid's website is largely an advertisement for Majid's personality and Majid's book. He does not explain enough so that one can derive anything.

I think you left off a couple of parentheses when you copied
f'(x) = (f(x)-f(qx))/(1-q)x.

This is a finite difference approximation to the ordinary derivative f'(x)
which is the limit of that expression as q --> 1
If you make q very close to 1 then that formula will tend to approach the ordinary derivative. Like, try 0.999 for q.
So what's the big deal?

It strikes me as teasing instead of teaching----he tells just enough to get you interested and then tries to sell you his book.

 

[tacman]

Quantum geometry is a branch of mathematics that combines concepts from quantum mechanics and differential geometry to study the geometric structure of space at the quantum level. It is a relatively new field that has emerged from the need to reconcile the principles of quantum mechanics, which govern the behavior of particles at the subatomic level, with the principles of general relativity, which describe the behavior of space and time at large scales.

The article in the provided link discusses the basic principles of quantum geometry and its applications in physics and mathematics. The second equation, f'(x)=f(x)-f(qx)/(1-q)x, is derived from the concept of a q-derivative, which is a generalization of the ordinary derivative used in calculus. In this case, the q-derivative is defined as a function that satisfies the following properties:

1. Linearity: f'(x+y) = f'(x) + f'(y)
2. Leibniz rule: f'(xy) = xf'(y) + qyf'(x)
3. q-Chain rule: f'(qx) = (1-q)x f'(x)

Using these properties, the author of the article derived the second equation by applying the q-Chain rule to the function f(qx). This allows us to express the q-derivative of f(qx) in terms of the q-derivative of f(x), which is given by f'(x). The resulting equation is then rearranged to obtain the desired form, f'(x)=f(x)-f(qx)/(1-q)x.

Overall, the second equation is derived from the fundamental principles of q-derivatives and is an important tool in studying the geometry of space at the quantum level. It allows us to understand how the geometry of space changes when we consider quantum effects, and has applications in various fields such as quantum gravity, quantum field theory, and quantum information theory.