Kroll received in 1942 his bachelor's degree from
Columbia University after 2 years of study, having studied from 1938 to 1940 at
Rice University in
Houston. During WW II he did theoretical radar research (
magnetron theory), during 1943–1945, at Columbia under the supervision of
Willis Lamb and
I. I. Rabi. In 1943 Kroll received his master's degree and in 1948 his PhD from Columbia University with Lamb as thesis advisor. In the academic year 1948–1949 he was a visiting scholar at the
Institute for Advanced Study, where he, with
Robert Karplus, calculated the QED two-loop contributions for the
anomalous magnetic moment of the electron. Kroll was, with Lamb, one of the first (including
Victor Weisskopf and his student
Bruce French) to calculate the relativistic Lamb shift (after
Hans Bethe made a rough, non-relativistic estimate for it). This work was part of the pioneering efforts that led to the QED formalism developed by
Richard Feynman,
Julian Schwinger, and
Sin-Itiro Tomonaga. Kroll became at Columbia an assistant professor in 1949 and was promoted to associate professor and then full professor before leaving for
UCSD. In 1960–1981 he was a member of the
JASON Defense Advisory Group. Upon his death, Kroll was survived by his wife, four children, and nine grandchildren. == Honors ==