Light-front dynamics of the covariant harmonic oscillator.

Abstract

Following Leutwyler and Stern, the two-particle covariant harmonic-oscillator model of Feynman, Kislinger, and Ravndal is reformulated within the framework of light-front dynamics. The light-front dynamics of the oscillator is obtained by constraining the manifestly covariant formulation to the null plane, ct+z=0. An inner product for the null-plane wave functions is developed, and it is shown that the light-front mass operator M and spin operator scrJ are Hermitian with respect to this inner product. Interaction dependent, nonlocal relative position and momentum operators Q and P, are introduced, and the mass and spin operators are expressed in terms of them. It is found that the part of ${\mathit{M}}^{2}$ that contains the dynamics is proportional to a nonrelativistic harmonic-oscillator Hamiltonian in P and Q, and that the spin operator scrJ=Q\ifmmode\times\else\texttimes\fi{}P. The eigenstates of Q and P are determined, and are used to construct position and momentum representations for the null-plane oscillator. In these representations the light-front dynamics takes on the appearance of nonrelativistic quantum mechanics. In particular, the nonlocal inner product for the null-plane wave functions goes over to the local nonrelativistic form. The obvious relevance of these results to the quark model of the hadrons is briefly discussed.