Relating pseudospin and spin symmetries through charge conjugation and chiral transformations: The Case of the relativistic harmonic oscillator

We solve the generalized relativistic harmonic oscillator in 1+1 dimensions, i.e., including a linear pseudoscalar potential and quadratic scalar and vector potentials which have equal or opposite signs. We consider positive and negative quadratic potentials and discuss in detail their bound-state solutions for fermions and antifermions. The main features of these bound states are the same as the ones of the generalized three-dimensional relativistic harmonic oscillator bound states. The solutions found for zero pseudoscalar potential are related to the spin and pseudospin symmetry of the Dirac equation in 3+1 dimensions. We show how the charge conjugation and {gamma}{sup 5} chiral transformations relate the several spectra obtained and find that for massless particles the spin and pseudospin symmetry-related problems have the same spectrum but different spinor solutions. Finally, we establish a relation of the solutions found with single-particle states of nuclei described by relativistic mean-field theories with scalar, vector, and isoscalar tensor interactions and discuss the conditions in which one may have both nucleon and antinucleon bound states.

[1]  Radosław Szmytkowski,et al.  Completeness of the Dirac oscillator eigenfunctions , 2001 .

[2]  P. Ring,et al.  Pseudospin symmetry in relativistic mean field theory , 1998 .

[3]  J. Bell,et al.  Dirac equations with an exact higher symmetry , 1975 .

[4]  P. Ring,et al.  Pseudo spin-orbit potential in relativistic self-consistent models , 2000 .

[5]  Jian-You Guo,et al.  Pseudospin symmetry in the relativistic harmonic oscillator , 2005 .

[6]  J. Meng,et al.  Pseudospin symmetry in Zr and Sn isotopes from the proton drip line to the neutron drip line , 1999 .

[7]  Bound states of the Dirac equation for a class of effective quadratic plus inversely quadratic potentials , 2003, hep-th/0311087.

[8]  J. Ginocchio.  Pseudospin as a relativistic symmetry , 1996, nucl-th/9611044.

[9]  Cohen,et al.  From QCD sum rules to relativistic nuclear physics. , 1991, Physical review letters.

[10]  L. Ryder,et al.  Quantum Field Theory , 2001, Foundations of Modern Physics.

[11]  P. Alberto,et al.  Role of the Coulomb and the vector-isovector $ρ$ potentials in the isospin asymmetry of nuclear pseudospin , 2003, nucl-th/0301063.

[12]  M. Moshinsky,et al.  A Dirac equation with an oscillator potential and spin-orbit coupling , 1991 .

[13]  Bernd Thaller,et al.  The Dirac Equation , 1992 .

[14]  A. D. Castro,et al.  Tensor coupling and pseudospin symmetry in nuclei , 2004, nucl-th/0411120.

[15]  Bounded solutions of neutral fermions with a screened Coulomb potential , 2005, hep-th/0505025.

[16]  P. Strange.  Relativistic Quantum Mechanics: With Applications in Condensed Matter and Atomic Physics , 1998 .

[17]  Bounded solutions of fermions in the background of mixed vector–scalar inversely linear potentials , 2004, hep-th/0409294.

[18]  F. M. Toyama,et al.  Coherent state of the Dirac oscillator , 1996 .

[19]  Zhang Shuang-quan,et al.  Pseudospin Symmetry in Relativistic Framework with Harmonic Oscillator Potential and Woods-Saxon Potential , 2003 .

[20]  K. A. Semendyayev,et al.  Handbook of mathematics , 1985 .

[21]  F. M. Toyama,et al.  Harmonic oscillators in relativistic quantum mechanics , 1999 .

[22]  D. Itô,et al.  An example of dynamical systems with linear trajectory , 1967 .

[23]  F. M. Toyama,et al.  Behaviour of wavepackets of the 'Dirac oscillator': Dirac representation versus Foldy - Wouthuysen representation , 1997 .

[24]  M. Fiolhais,et al.  Pseudospin symmetry and the relativistic harmonic oscillator , 2003, nucl-th/0310071.

[25]  J. Ginocchio.  Relativistic symmetries in nuclei and hadrons , 2005 .

[26]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[27]  P. Ring,et al.  Spin symmetry in the antinucleon spectrum. , 2003, Physical review letters.

[28]  M. Fiolhais,et al.  PERTURBATIVE BREAKING OF THE PSEUDOSPIN SYMMETRY IN THE RELATIVISTIC HARMONIC OSCILLATOR , 2004, nucl-th/0410096.

[29]  M. Moshinsky,et al.  The Dirac oscillator , 1989 .