Hermitian versus non-Hermitian representations for minimal length uncertainty relations (original) (raw)

Abstract

: We investigate four different types of representations of deformed canonical variables leading to generalized versions of Heisenberg's uncertainty relations resulting from noncommutative spacetime structures. We demonstrate explicitly how the representations are related to each other and study three characteristically different solvable models on these spaces, the harmonic oscillator, the manifestly non-Hermitian Swanson model and an intrinsically noncommutative model with Pöschl-Teller type potential. We provide an analytical expression for the metric in terms of quantities specific to the generic solution procedure and show that when it is appropriately implemented expectation values are independent of the particular representation. A recently proposed inequivalent representation resulting from Jordan twists is shown to lead to unphysical models. We suggest an anti-PT -symmetric modification to overcome this shortcoming.

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References (25)

  1. A. Kempf, Uncertainty relation in quantum mechanics with quantum group symmetry, J. Math. Phys. 35, 4483-4496 (1994).
  2. A. Kempf, G. Mangano, and R. B. Mann, Hilbert space representation of the minimal length uncertainty relation, Phys. Rev. D52, 1108-1118 (1995).
  3. B. Bagchi and A. Fring, Minimal length in Quantum Mechanics and non-Hermitian Hamiltonian systems, Phys. Lett. A373, 4307-4310 (2009).
  4. A. Fring, L. Gouba, and F. G. Scholtz, Strings from dynamical noncommutative space-time, J. Phys. A43, 345401(10) (2010).
  5. A. Fring, L. Gouba, and B. Bagchi, Minimal areas from q-deformed oscillator algebras, J. Phys. A43, 425202 (2010).
  6. S. Dey, A. Fring, and L. Gouba, PT-symmetric noncommutative spaces with minimal volume uncertainty relations, J. Phys. A45, 385302 (2012).
  7. P. Aschieri, M. Dimitrijevic, F. Meyer, and J. Wess, Noncommutative geometry and gravity, Class. Quant. Grav. 23, 1883-1912 (2006).
  8. M. Gomes and V. Kupriyanov, Position-dependent noncommutativity in quantum mechanics, Phys. Rev. D79, 125011 (2009).
  9. M. Arik and D. D. Coon, Hilbert Spaces of Analytic Functions and Generalized Coherent States, J. Math. Phys. 17, 524-527 (1976).
  10. P. Kulish and E. Damaskinsky, On the q oscillator and the quantum algebra SU(q) (1,1), J. Phys. A23, L415-L419 (1990).
  11. P. Castro, R. Kullock, and F. Toppan, Snyder Noncommutativity and Pseudo-Hermitian Hamiltonians from a Jordanian Twist, J. Math. Phys. 52, 062105 (2011).
  12. E. Wigner, Normal form of antiunitary operators, J. Math. Phys. 1, 409-413 (1960).
  13. C. M. Bender and S. Boettcher, Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry, Phys. Rev. Lett. 80, 5243-5246 (1998).
  14. A. Bhattacharjie and G. Sudarshan, A class of solvable potentials, Nuovo Cim. 25, 864-879 (1962).
  15. T. Jana and P. Roy, Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty, SIGMA 5, 083 (2009).
  16. G. Levai, A search for shape invariant sovable potentials, J. Phys. A22, 689-702 (1989).
  17. F. G. Scholtz, H. B. Geyer, and F. Hahne, Quasi-Hermitian Operators in Quantum Mechanics and the Variational Principle, Ann. Phys. 213, 74-101 (1992).
  18. A. Mostafazadeh, Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. Maths. Phys. 43, 202-212 (2002).
  19. C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rept. Prog. Phys. 70, 947-1018 (2007).
  20. A. Mostafazadeh, Pseudo-Hermitian Representation of Quantum Mechanics, Int. J. Geom. Meth. Mod. Phys. 7, 1191-1306 (2010).
  21. C. Bender, A. Fring, U. Günther, and H. Jones, Special Issue: Quantum physics with non-Hermitian operators, J. Phys. A45, 440301 (2012).
  22. I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, London) (2007).
  23. D. P. Musumbu, H. B. Geyer, and W. D. Heiss, Choice of a metric for the non-Hermitian oscillator, J. Phys. A40, F75-F80 (2007).
  24. M. S. Swanson, Transition elements for a non-Hermitian quadratic Hamiltonian, J. Math. Phys. 45, 585-601 (2004).
  25. G. Pöschl and E. Teller, Bemerkungen zur Quantenmechanik des anharmonischen Oszillators, Z. Phys. 83, 143-151 (1933).