##
**Lorentz-covariant deformed algebra with minimal length and application to the \((1 + 1)\)-dimensional Dirac oscillator.**
*(English)*
Zbl 1168.81014

Lorentz-covariant generalization of Kempf’s \(D\)-dimensional \((\beta,\beta')\)-two-parmeter deformed algebra [cf. A. Kempf, G. Mangano and R. B. Mann, Phys. Rev. D (3) 52, No. 2, 1108–1118 (1995)], which reproduces B. Snyder algebra [Phys. Rev., II. Ser. 71, 38–41 (1947; Zbl 0035.13101)] in the case \(D= 3\) and \(\beta= 0\), is presented. It describes a \((D+ 1)\)-dimensional quantized spacetime and exists a nonzero minimal uncertainty in position (minimal length). In the case \(D= 1\) and \(\beta= 0\), the Dirac oscillator described by such an algebra is studied and exact bound-state energies and wavefunctions are obtained. It is shown physically acceptable states exist for \(\beta< 1/(m^2c^2)\) and in contrast with the conventional Dirac oscillator, the energy spectrum is bounded.

To describe the deformed algebra, in the conventional \((D+1)\)-dimensional continuous spacetime, authors take \(g^{\mu\nu}= \text{diag}(1,-1,\dots,-1)= g^{\mu\nu}\) and set \(p^\nu= (E/c,{\mathbf p})\), \(p_\mu= g_{\mu\nu} p^\nu\). In the coordinate representation of quantum mechanics, the momentum operator and position opertor become \[ p^\mu= i\hslash{\partial\over\partial x_\mu}= i\hslash g^{\mu\nu}{\partial\over\partial x^\nu},\quad x^\mu= -i\hslash{\partial\over\partial p_\mu}. \] By using these notions, the deformed position and momentum operators \(X^i\), \(P^i\) are represented by \[ X^\mu= (1-\beta p_\nu p^\mu) x^\mu- \beta' p^\mu p_\nu x^\nu+ i\hslash\gamma p^\mu,\quad P^\mu= p^\mu. \] By definition, \([P^\mu, P^\nu]= 0\). Other commutators are computed to be \[ [X^\mu, P^\nu]= -i\hslash[(1-\beta P_\rho P^\rho) g^{\mu\nu}- \beta'P^\mu P^\nu], \]

\[ [X^\mu, X^\nu]= i\hslash{2\beta- \beta'- (2\beta+ \beta')\beta P_\rho P^\rho\over 1-\beta P_\rho P^\rho} (P^\mu X^\nu- P^\nu X^\mu). \] \(X^\mu\) and \(P^\mu\) are Hermitian with respect to the modified scalar product \[ \langle\psi|\phi\rangle= \int {d^D{\mathbf p}\over [1-(\beta+ \beta') p_\nu p^\nu]^\alpha} \psi^*(p^\mu) \phi(p^m u), \] in moment \(m\)-space. Here \(\alpha= {2\beta* \beta'(D+ 2)- 2\gamma\over 2(\beta+ \beta')}\). By this inner product, the energy of physically acceptable states satisfies the condition \((\beta+ \beta')(p^0)^2< 1\) (§2.1).

The invariance of this deformed algebra under Lorentz transformation is shown in §2.2. Invariance of the above commutation relations under the infinitesimal translations of the Poincaré algebra is also shown. Then in §2.3, by using invariance under the rotation, \(\Delta X'\) is evaluated and show the isotropic absolutely smallest uncertainty in position is given by \[ (\Delta X)_0= (\Delta X^i)_0= \hslash\sqrt{(D\beta+ \beta')[1- \beta\langle({\mathbf P}^0)^2\rangle]}. \] In §3, assuming \(\beta'= 0\) and \(\beta> 0\), \((1+1)\)-dimensional Dirac oscillator with above deformed algebra is treated. In momentum space, the equation is \[ (\sigma_x P- m\omega\sigma_x X+ mc\sigma_x) \psi(p, p^0)= P^0\psi(p, p^0). \] Then introducing dimensionless position and momentum operators \(\widetilde X^\mu= X^\mu/a\), \(\widetilde P^\mu= a/\hslash P^\mu\), \(\mu= 0, 1\), and separating the wavefunction \(\psi\) into large \(\psi_1\) and small \(\psi_2\) compoents, the Dirac oscillator equation is written as two coupled equations \[ B^+\psi_2(\widetilde p,\widetilde p^0)= (\widetilde p^0- 1)\psi_1(\widetilde p,\widetilde p^0),\quad B^-\psi_1(\widetilde p,\widetilde p^0)= (\widetilde p^0+ 1)\psi_2(\widetilde p,\widetilde p^0), \] where \(B^\pm=\widetilde P\pm i\widetilde\omega\widetilde X=\widetilde p\mp\widetilde f(\widetilde p,\widetilde p^0) \partial/\partial\widetilde p\). Then introducing a hierarchy of Hamiltonians \(H_i= B+ (g_i)B^-(g_i)+ \sum^i_{j=0} \varepsilon_j\), the Dirac oscillator energy spectrum is computed to be \[ E_{n,\tau}= {\tau c\over\sqrt{\beta}} \Biggl(1+{\beta m^2 c^2- 1\over (1+\beta m\hslash\omega m)- 2}\Biggr)^{1/2}. \] Therefore the energy spectrum is bounded: \(mc^2\leq|E_{n,\tau}|\leq c/\sqrt{\beta}\) It is remarked if the deformation vanishes, this result recovers the unbounded energy spectrum of the conventional spectrum. Relations to the results obtained by using Kempf algebra [C. Quesne and V. M. Tkachuk, J. Phys. A, Math. Gen. 38, No. 8, 1747–1765 (2005; Zbl 1061.81023)] is also noted (§3.1). Then in §3.2, normalized wavefunctions are explicitly computed by using Gegenbauer polynomials.

To describe the deformed algebra, in the conventional \((D+1)\)-dimensional continuous spacetime, authors take \(g^{\mu\nu}= \text{diag}(1,-1,\dots,-1)= g^{\mu\nu}\) and set \(p^\nu= (E/c,{\mathbf p})\), \(p_\mu= g_{\mu\nu} p^\nu\). In the coordinate representation of quantum mechanics, the momentum operator and position opertor become \[ p^\mu= i\hslash{\partial\over\partial x_\mu}= i\hslash g^{\mu\nu}{\partial\over\partial x^\nu},\quad x^\mu= -i\hslash{\partial\over\partial p_\mu}. \] By using these notions, the deformed position and momentum operators \(X^i\), \(P^i\) are represented by \[ X^\mu= (1-\beta p_\nu p^\mu) x^\mu- \beta' p^\mu p_\nu x^\nu+ i\hslash\gamma p^\mu,\quad P^\mu= p^\mu. \] By definition, \([P^\mu, P^\nu]= 0\). Other commutators are computed to be \[ [X^\mu, P^\nu]= -i\hslash[(1-\beta P_\rho P^\rho) g^{\mu\nu}- \beta'P^\mu P^\nu], \]

\[ [X^\mu, X^\nu]= i\hslash{2\beta- \beta'- (2\beta+ \beta')\beta P_\rho P^\rho\over 1-\beta P_\rho P^\rho} (P^\mu X^\nu- P^\nu X^\mu). \] \(X^\mu\) and \(P^\mu\) are Hermitian with respect to the modified scalar product \[ \langle\psi|\phi\rangle= \int {d^D{\mathbf p}\over [1-(\beta+ \beta') p_\nu p^\nu]^\alpha} \psi^*(p^\mu) \phi(p^m u), \] in moment \(m\)-space. Here \(\alpha= {2\beta* \beta'(D+ 2)- 2\gamma\over 2(\beta+ \beta')}\). By this inner product, the energy of physically acceptable states satisfies the condition \((\beta+ \beta')(p^0)^2< 1\) (§2.1).

The invariance of this deformed algebra under Lorentz transformation is shown in §2.2. Invariance of the above commutation relations under the infinitesimal translations of the Poincaré algebra is also shown. Then in §2.3, by using invariance under the rotation, \(\Delta X'\) is evaluated and show the isotropic absolutely smallest uncertainty in position is given by \[ (\Delta X)_0= (\Delta X^i)_0= \hslash\sqrt{(D\beta+ \beta')[1- \beta\langle({\mathbf P}^0)^2\rangle]}. \] In §3, assuming \(\beta'= 0\) and \(\beta> 0\), \((1+1)\)-dimensional Dirac oscillator with above deformed algebra is treated. In momentum space, the equation is \[ (\sigma_x P- m\omega\sigma_x X+ mc\sigma_x) \psi(p, p^0)= P^0\psi(p, p^0). \] Then introducing dimensionless position and momentum operators \(\widetilde X^\mu= X^\mu/a\), \(\widetilde P^\mu= a/\hslash P^\mu\), \(\mu= 0, 1\), and separating the wavefunction \(\psi\) into large \(\psi_1\) and small \(\psi_2\) compoents, the Dirac oscillator equation is written as two coupled equations \[ B^+\psi_2(\widetilde p,\widetilde p^0)= (\widetilde p^0- 1)\psi_1(\widetilde p,\widetilde p^0),\quad B^-\psi_1(\widetilde p,\widetilde p^0)= (\widetilde p^0+ 1)\psi_2(\widetilde p,\widetilde p^0), \] where \(B^\pm=\widetilde P\pm i\widetilde\omega\widetilde X=\widetilde p\mp\widetilde f(\widetilde p,\widetilde p^0) \partial/\partial\widetilde p\). Then introducing a hierarchy of Hamiltonians \(H_i= B+ (g_i)B^-(g_i)+ \sum^i_{j=0} \varepsilon_j\), the Dirac oscillator energy spectrum is computed to be \[ E_{n,\tau}= {\tau c\over\sqrt{\beta}} \Biggl(1+{\beta m^2 c^2- 1\over (1+\beta m\hslash\omega m)- 2}\Biggr)^{1/2}. \] Therefore the energy spectrum is bounded: \(mc^2\leq|E_{n,\tau}|\leq c/\sqrt{\beta}\) It is remarked if the deformation vanishes, this result recovers the unbounded energy spectrum of the conventional spectrum. Relations to the results obtained by using Kempf algebra [C. Quesne and V. M. Tkachuk, J. Phys. A, Math. Gen. 38, No. 8, 1747–1765 (2005; Zbl 1061.81023)] is also noted (§3.1). Then in §3.2, normalized wavefunctions are explicitly computed by using Gegenbauer polynomials.

Reviewer: Akira Asada (Takarazuka)

### MSC:

81R60 | Noncommutative geometry in quantum theory |

53D55 | Deformation quantization, star products |

35Q40 | PDEs in connection with quantum mechanics |