There are Standard Model processes in which hadrons do not dissociate in the interaction. Without hadron dissociation there is no underlying event in the interaction - only a very clean, "exclusive", central system is observerd in the detectror. I am searching for two such processes; exclusive production of two photons via QCD (gluon exchange), and electron pair production via QED (through photon exchange). Theoretically, the QCD process can produce a Higgs Boson. If suitable detectors are installed at the LHC, the outgoing protons can be used like a spectrometer to measure the Higgs mass. The QED process cross section can be accurately calculated, meaning that it could be a useful tool to measure the luminosity at the LHC. Before either of these processes can be used at the LHC, they must be understood here at the Tevatron - the goal of my thesis research is to observe and understand these unique events at CDF.