Greene received his bachelor's degree from Swarthmore College in 1971 and his doctorate from Harvard University in 1974 with Norman Ramsey, a Nobel laureate in physics.
As a post-doctoral fellow he was at the Rutherford Appleton Laboratory and the Institut Laue-Langevin.
He was an assistant professor at Yale University and then at the National Institute of Standards and Technology (NIST).
He had a leading role in determining fundamental neutron properties such as lifetime, mass, magnetic moment and various experiments investigating possible symmetry breaking in nuclear reactions.
In 2021, he received the Tom W. Bonner Prize in Nuclear Physics “for foundational work establishing the field of fundamental neutron physics in the U.S., for developing experimental techniques for in-beam measurements of the neutron lifetime and other experiments and for realizing a facility for the next generation of fundamental neutron physics measurements.”[4]