Comment on "On the modification of the Hamiltonians' spectrum in gravitational quantum mechanics" by Pedram P.

We will show the results in paper [1] were incorrect. Quantum gravity theories predict the existence of a minimum measurable length which needs in turn a modification in the commutation relation between position and momentum [2], the commutators that are consistent with string theory, black hole physics and double special relativity and ensure $\left [x_i,x_j\right ]=\left [p_i,p_j\right ]=0$ are

$$\begin{equation} \left[x_i,p_j\right]=i\hbar \left[ \delta_{ij}-\alpha \left( p\alpha_{ij}+\frac{p_ip_j}{p} \right)+\alpha^2(p^2\delta_{ij} + 3p_ip_j ) \right]\!; \end{equation} \tag{ 1 }$$

position and momentum operators take the form

$$\begin{equation} x_i=x_{i0}, \qquad p_i=p_{i0}(1-\alpha p_0+2\alpha^2p_0^2). \end{equation} \tag{ 2 }$$

The Hamiltonian will be

$$\begin{equation} H=H_0-\frac{\alpha}{m}p_0^3+\frac{5\alpha^2}{2m}p_0^4,\qquad H_0=\frac{p_0^2}{2m}+v(r); \end{equation} \tag{ 3 }$$

let

$$\begin{equation} H_1=-\frac{\alpha}{m}p_0^3 \ \ {\rm and} \ \ H_2=\frac{5\alpha^2}{2m}p_0^4, \end{equation} \tag{ 4 }$$

there is a factor $\frac {1}{2}$ in H₂, eq. (4), that does not exist in ref. [1], eq. (7).

First, for a particle in a box, using the perturbation theory, the corrections in eigenvalues are

$$\begin{equation} E_n^1=\alpha \langle n\mid H_1\mid n\rangle, \end{equation} \tag{ 5 }$$

$$\begin{equation} E_n^{2,1}=\alpha^2 \sum_{k\neq n} \frac{{\mid \langle k\mid H_1\mid n\rangle \mid}^2}{E_n^0-E_k^0}, \end{equation} \tag{ 6 }$$

$$\begin{equation} E_n^{2,2}=\alpha^2\langle n\mid H_2\mid n\rangle=-\frac{1}{2}\frac{\alpha^2n^2\pi^2\hbar^4}{ml^4}. \end{equation} \tag{ 7 }$$

The total change in the energy

$$\begin{equation} \Delta E_n=E_n^{2,1}+E_n^{2,2}=\frac{2\alpha^2n^2\pi^2\hbar^4}{ml^4}. \end{equation} \tag{ 8 }$$

So the energy change in eq. (8) in terms of unperturbed energy is $\Delta E_n=8m\alpha^2{E_n^0}^2$. This result does not coincide with [1], eq. (12).

Second, for the harmonic-oscillator case, the Hamiltonian

$$\begin{equation} H_0=\frac{p_0^2}{2m}+\frac{1}{2}m \omega^2x^2. \end{equation} \noindent \tag{ 9 }$$

Define $p^3=\frac {i(m\hbar \omega )^{3/2}}{2\surd 2} (a-a^*)^3$ and $p^4=\frac {(m\hbar \omega )^2}{4} (a-a^*)^4$, using the perturbation theory

$$\begin{equation} E_n^1=0, \end{equation} \noindent \tag{ 10 }$$

$$\begin{equation} E_n^{2,1}=-\frac{1}{8}\alpha^2m\hbar^2\omega^2(30n^2+30n+11), \end{equation} \noindent \tag{ 11 }$$

$$\begin{equation} E_n^{2,1}=\frac{1}{8}\alpha^2m\hbar^2\omega^2(30n^2+30n+15). \end{equation} \noindent \tag{ 12 }$$

The total change in the energy is

$$\begin{equation} \Delta E_n=E_n^{2,1}+E_n^{2,2}=\frac{1}{2}\alpha^2m\hbar^2\omega^2. \end{equation} \noindent \tag{ 13 }$$

This result coincides with [3], in which only the ground state is calculated, and is differing from the corresponding one in [1]. The effect due to GUP, which was proposed by Ali et al. [2], does not depend on the state quantum number so it cannot appear in the spectrum of the coherent states of the harmonic oscillator.