Fractional Klein–Gordon equation composed of Jumarie fractional derivative and its interpretation by a smoothness parameter

Abstract

Klein–Gordon equation is one of the basic steps towards relativistic quantum mechanics. In this paper, we have formulated fractional Klein–Gordon equation via Jumarie fractional derivative and found two types of solutions. Zero-mass solution satisfies photon criteria and non-zero mass satisfies general theory of relativity. Further, we have developed rest mass condition which leads us to the concept of hidden wave. Classical Klein–Gordon equation fails to explain a chargeless system as well as a single-particle system. Using the fractional Klein–Gordon equation, we can overcome the problem. The fractional Klein–Gordon equation also leads to the smoothness parameter which is the measurement of the bumpiness of space. Here, by using this smoothness parameter, we have defined and interpreted the various cases.

Appendix: Description of fractional quantities used in the paper

1.1 Fractional mass and momentum

Fractional mass $m_\alpha$ is defined as $m_\alpha =\int \rho \mathrm {d}x^\alpha$, where $\rho$ is the fractional linear mass density in one dimension; for constant $\rho$, it is $m_\alpha =\rho \int \mathrm {d}x^\alpha$. This integration is with respect to coarse-grained space, i.e. $\mathrm {d}x^\alpha$. We have considered that the density is the same as in the case $\alpha =1$ [18]. The fractional change of displacement, i.e. $\mathrm {d}^\alpha x(=\Gamma (1+\alpha )\mathrm {d}x)$ per unit change in fractional time differential $(\mathrm {d}t)^{\alpha }$ is the fractional velocity, i.e. for $0<\alpha <1$: $v_\alpha =\frac{\mathrm {d}^\alpha x}{\mathrm {d}t^\alpha }=\alpha !\frac{\mathrm {d} x}{(\mathrm {d}t)^\alpha }$.

Then we can write [18]

$$\begin{aligned} \frac{\mathrm {d}^{1-\alpha }}{\mathrm {d}t^{1-\alpha }}(v_\alpha )=\frac{\mathrm {d}x}{\mathrm {d}t}=v;\ \quad 0<\alpha <1. \end{aligned}$$

Now we shall define fractional momentum. When a body of fractional mass $m_\alpha$ is moving with fractional velocity $v_\alpha$, we can define fractional momentum in the form $p_\alpha =m_\alpha v_\alpha$. It coincides with classical momentum at $\alpha =1$. Now, in terms of fractional momentum, we write the following expressions for fractional mass:

$$\begin{aligned} m_\alpha =\frac{p_\alpha }{v_\alpha }=\frac{p_\alpha /c}{\mu (\alpha )}, \quad \hbox { where }\mu (\alpha )=\frac{v_\alpha }{c}. \end{aligned}$$

Thus, smoothness parameter is directly related to the fractional mass of the system.

Fig. 3

Plot showing fractional wavelength.

1.2 Fractional wavelength

Fractional wavelength is demonstrated by a plot (figure 3) of a fractional wave of the order $\alpha =0.8$.

The wavelength is the distance, which is covered by a fractional wave in a full fractional cycle. Fractional wavelength is not a fixed quantity. It changes with the evolution of fractional time, like a damped oscillating wave.

1.3 Fractional time period

Consider a fractional wave propagating in a medium. It must repeat its initial phase after a certain time. This time of repetition for a fractional wave is defined as the fractional time period [18]. The time taken $N_\alpha$ for a wave to cover the distance AB (figure 3) is the fractional time period. We considered this as the first-order time period. As wavelength changes, the time period also changes with the wave propagation. We assume that $\lambda _\alpha =\upsilon _\alpha N_\alpha$.

1.4 Fractional angular frequency

Figure 3 is the polar plot of the fractional wave of the order $\alpha =0.8$. In this polar plot, we can easily see that the wave returns to the same point after completing a fractional cycle, i.e. to its origin. By polar plot, we can say that the angle traversed in a full fractional cycle is $2\pi$. Thus, the fractional angular frequency can be assigned as $\omega _\alpha ={2\pi }/{N\alpha }$.

Fig. 4

Plot showing the concept of fractional angular frequency.

In figure 4, the initial line is along the horizontal line. From this, we can see that the product of fractional angular momentum and fractional time period $T_\alpha$ is always $2\pi$ though both are varying. In limiting condition, we have

$$\begin{aligned} \lim _{\alpha \rightarrow 1}\omega _\alpha =\lim _{\alpha \rightarrow 1}\frac{\omega _\alpha }{\omega }=1. \end{aligned}$$

1.5 Fractional reduced Planck constant

In this section, we shall show that the fractional Planck constant and Planck constant are the same. Similarly, fractional reduced Planck constant and reduced Planck constant are also the same. Fractional reduced Planck constant $\hbar _\alpha$ is a basic constant for $\alpha$ ordered fractional system. At $\alpha =1$, this constant is the same as the reduced Planck constant $\hbar$. This fractional reduced Planck constant $\hbar _\alpha$ is used for quantisation of fractional space phenomena at microscopic level. In this paper, we have introduced fractional Planck constant $\hbar _\alpha$ as a basic constant. For the limiting condition of $\alpha$, this constant is of the form of reduced Planck constant $\hbar$. Consider integer (i.e. classical) and fractional energies $E=\hbar \omega =h\upsilon$ and $\epsilon _\alpha =\hbar _\alpha \omega \alpha =h_\alpha \upsilon _\alpha$. Then, we can write for the limiting condition, i.e. with $\alpha \rightarrow 1$, $E=h\upsilon =h_\alpha \upsilon _\alpha$. Here, $\upsilon$ is the integer-order frequency and $\upsilon _\alpha$ is the fractional-order frequency. We write the following expression with above description $h=\lim _{\alpha \rightarrow 1}({\upsilon _\alpha }{/}{\upsilon })h_\alpha$ and

$$\begin{aligned} \hbar =\lim _{\alpha \rightarrow 1}\frac{\omega _\alpha }{\omega }\hbar _{\alpha }. \end{aligned}$$

Using Appendix A.5, we have

$$\begin{aligned} \hbar =\lim _{\alpha \rightarrow 1}\hbar _{\alpha }. \end{aligned}$$

We have energy proportional to angular frequency and this assumption does not depend upon the value of $\alpha$. Therefore, we have $E\propto \omega$ and $\epsilon _{\alpha }\propto \omega _\alpha$. If we remove the proportionality, we can have a constant such that the following condition is satisfied:

$$\begin{aligned} \frac{\epsilon _\alpha }{\omega _\alpha }=\frac{\epsilon _\beta }{\omega _\beta }=\cdots =\hbar \quad (0<\alpha<\beta<\cdots <1) .\end{aligned}$$

Therefore, we have $\hbar _\alpha = \hbar$ and hence $h_\alpha =h$. We can conclude that the fractional Planck constant is nothing but Planck constant. The reduced fractional Planck constant is also the reduced Planck constant.

1.6 Fractional energy

Fractional energy is defined as the sum of fractional kinetic energy and fractional potential energy. We previously defined [18] fractional kinetic energy as

$$\begin{aligned} T_\alpha =\frac{1}{2^{\alpha } m_\alpha } p_\alpha ^2 \end{aligned}$$

and fractional potential energy as in the general form of $V(x^\alpha )$. Thus, the total fractional energy is

$$\begin{aligned} \epsilon _ \alpha =\frac{1}{2^{\alpha }m_\alpha }p_\alpha ^2+V(x^\alpha ). \end{aligned}$$

At $\alpha =1$, total fractional energy is the same as the classical total energy.