Deformed Numbers and Calculus

When I first heard of fractional calculus, I thought that it sounds contrived. But then again, so are many abstractions and generalizations of established ideas in mathematics. There is one good motivation for this idea, which strikes at the heart of fundamental physics i.e. the Dirac operator \hat{D} which is understood as the square root of the Laplacian \Delta i.e. \hat{D}^2=\Delta. If it is one dimensional with \Delta=\partial_x^2, this would have been straightforward. For more than one dimension, this is nontrivial – what is the square root of say \partial_x^2 + \partial_y^2?

The reason my attention got to this is through a visitor of mine, Prof. Won Sang Chung from Gyeongsang National University who visited the institute for the period of February 2-14, 2017. His research has been mainly on deformation of physical and mathematical structures. His interest in noncommutative quantum mechanics was the initial reason of our contact. With respect to the above topic, Prof. Won’s many works on deformation has led him to study the deformation of units for instance in the published work “On the q-deformed circular imaginary unit and hyperbolic imaginary unit: q-deformed rotation in two dimension and q-deformed special relativity in 1+1 dimension“. It is interesting that this work was carried out with an undergraduate, of which I’m told, work together with Prof. Won a few times a week at late evening hours. There are other works by them that touches on arithmetic operations of numbers, for which they call it \alpha-deformed numbers. In a way, it shifts the idea of deformation of a complex operation of say the derivative to arithmetic ones.

They start with the operation of addition for which in general it is deformed as

x\oplus_f y=f^{-1}(f(x)+f(y))

where f are some specified function and x,y are integers. By making f(x)=x^\alpha, one then has the alpha-deformed addition

x\oplus_\alpha y= (x^\alpha + y^\alpha)^{1/\alpha}

for x,y>0. For more cases of the integer, it is

x\oplus_\alpha y = \vert \vert x\vert^{\alpha-1}x +\vert y\vert^{\alpha-1}y\vert^{\frac{1}{\alpha}-1} (\vert x\vert^{\alpha-1}x + \vert y\vert^{\alpha-1}y)\ .

Suffice for our discussion to take the simpler case. It is easy to show that the additive identity is preserved

x\oplus_\alpha 0=0\oplus_\alpha x= x

The additive inverse requires the definition of \alpha-deformed subtraction:

x\ominus_\alpha y = \begin{cases} (x^\alpha - y^\alpha)^{1/\alpha}\quad &(x>y)\\ -(y^\alpha - x^\alpha)^{1/\alpha}\quad &(x<y) \end{cases}\ .

Consecutive addition can be done easily. Now we can define the \alpha-deformed numbers by noting the following:

0_\alpha=0,\quad 1_\alpha=1,\quad (-1)_\alpha=-1


(n)_\alpha = 1\oplus_\alpha 1\oplus_\alpha \cdots \oplus_\alpha 1=n^{1/\alpha}\ .

The multiplication and division operations are taken as the normal ones (undeformed) and

(mn)_\alpha = (m)_\alpha (n)_\alpha\quad;\quad \left(\cfrac{m}{n}\right)_\alpha=\cfrac{(m)_\alpha}{(n)_\alpha}\ .

Thus, the \alpha-deformed numbers form a commutative ring and essentially there is a 1-1 correspondence between the \alpha-deformed numbers and the integers. By the division operation defined above, I guess one could easily extend these deformed integers to \alpha-deformed rational numbers.

The topic stopped there will not be too interesting. The authors went on to define the \alpha-deformed derivation to build up calculus. It is here, there is a leap of faith to say that the \alpha-deformed rational numbers can be completed to form \alpha-deformed real numbers. Suppose that this can be done, one defines the \alpha-derivative by

D^\alpha_x F(x) =\lim_{y\rightarrow x}\ \cfrac{F(y)\ominus_\alpha F(x)}{y\ominus_\alpha x}\ .

One can show indeed that this definition gives a derivation (obeying Leibniz rule) – see paper. Hence, one finds

D^\alpha_x x\ominus_\alpha xD^\alpha_x = 1\quad\Rightarrow D^\alpha_x x= 1\oplus_\alpha xD^\alpha_x\ .

So one can easily show that for example, D^\alpha_x x^n= n_\alpha x^{n-1} i.e. it works like the ordinary derivative with numbers replace by the deformed numbers. One could generalise this further to power series functions. For instance the \alpha-exponential function defined by

D^\alpha_x e_\alpha (x) = e_\alpha (x)\ .

One can extend the deformed numbers to that of \alpha-deformed complex numbers z_\alpha=x_\alpha\oplus_\alpha iy_\alpha and form the \alpha-trigonometric function via the complex \alpha-exponential function. In fact one can show that these functions obey the usual Euler relation.

Once we build up these function and their calculus, one can start to think about solving standard physics problems via \alpha-deformed differential equations. We will state only the case of the quantum harmonic oscillator which was worked out in the paper. The differential equation to solve is

\left(-\cfrac{\hbar^2}{2m}(D^\alpha_x)^2\oplus_\alpha\cfrac{1}{2}m\omega^2 x^2\right)\ u= Eu\quad .

Suffice for our discussions, to state the results for its spectra, namely


Note that the energy levels almost retain its functional form but note that the energy level spacings

E_{n+1} \ominus_\alpha E=2^{\frac{1}{\alpha}-1}\hbar\omega

are only equidistant in the \alpha-deformed sense! So, there are nontrivialities associated with the earlier deformations of the arithmetic operations. In one of his talks, he had mentioned that the case of \alpha-deformed hydrogen atom has also been worked out with similar functional form of the spectra but of course again deformed. The applications that Prof. Won had mentioned are the case of quantum systems whose primary potential is known but there are missing information about them. He envisaged that by doing some fitting of the spectra, better physical understanding of previously unsolved systems (but spectrally known) can be achieved using these \alpha-deformed theories.


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