The double spin asymmetry for exclusive J/ψ pair production in the polarized p − p collisions at RHIC is investigated. Our study shows that the asymmetry in this process is measurable at RHIC-SPIN experiments in the near future, and, hence it can be used to determine the gluon distribution function ∆G(x) in the polarized nucleon.PACS numbers: 13.88.+e, 21.10.Hw, 13.60.Le Since the first measurement of the polarized structure function by the European Muon Collaboration [1], an enormous amount of researches have been done on the nucleon spin structure both experimentally and theoretically. The unpolarized deep inelastic scattering (DIS) experiments show that the gluon shares a large portion of the parent proton's momentum. However, how the gluons share the spin of proton is still an open question. In the conventional DIS process, the gluon contributions are the Quantum Chromodynamics(QCD) higher order effects and we can not avoid some ambiguities coming from e.g. the factorization scheme dependence, when determining the polarized gluon distributions. Although there are some efforts [2,3] to parameterize them, it seems difficult to obtain clear understanding of the role of gluons.Therefore, it is desirable to study other processes than DIS in which a direct measurement of the polarized gluon is possible. It is now expected that the polarized protonproton collisions (RHIC-Spin) at BNL relativistic heavyion collider RHIC will provide copious experimental data to unveil the polarized parton distributions. Since the major emphasis and strength of RHIC-Spin is to measure the gluon polarization, it is important and interesting to investigate various processes which are attainable experimentally to this aim.Up to now, three processes are considered to be promising for measuring the polarized gluons, which are thought to be feasible at RHIC technically. Those are • High-p T Prompt Photon Production • Jet production • Heavy Flavor ProductionThere are some advantages and disadvantages in each of these processes. The prompt photon production with po-larized beams at RHIC is a useful process, but we must know precisely the quark distributions from the polarized DIS experiments. For jet and heavy flavor production, the effects of hadronization and higher order corrections are not yet well controlled theoretically. For detailed discussions, see recent review paper in Ref. [4]. Therefore, to develop more practical ways for measuring the gluon polarization at RHIC experiments is one of the urgent theoretical tasks today.The quarkonium production and decays have long been taken as ideal processes to investigate the nature of QCD. Due to the approximately non-relativistic nature, the heavy quark and antiquark bound state is considered as one of the simplest system in QCD. The very clean signals of quarkonium leptonic decays lead to the experimental observations with a high precision, and therefore the quarkonium may play a crucial role in investigating many quantities and phenomena, e.g. the parton distribution, the QGP and even n...