Lance Nielsen

Dr. Lance Nielsen

Dr. Lance Nielsen
Professor of Mathematics

Dr. Lance NielsenThe research I pursue is in the area of Feynman’s Operational Calculus. I started working in this area during my graduate work at the University of Nebraska - Lincoln under the guidance of the late G. W. Johnson.

Feynman’s Operational Calculus was originated by the Nobel laureate Richard Feynman in the late 1940’s to the very early 1950’s. In particular, the paper that established the operational calculus was “An operator calculus having applications in quantum electrodynamics” (Phys. Rev. 84 (1951), 108—128). In this paper, Feynman remarks that:

“The mathematics is not completely satisfactory. No attempt has been made to maintain mathematical rigor. The excuse is not that it is expected that rigorous demonstrations can be easily supplied. Quite the contrary, it is believed that to put present methods on a rigorous basis may be quite a difficult task, beyond the abilities of the author.”

In the 65 years that have followed the appearance of this paper, mathematicians of various stripes have tried to put the operational calculus into a rigorous mathematical framework. However, even though the operational calculus is widely and very successfully used in quantum mechanics and quantum field theory, it is not well understood mathematically. My work since receiving my Ph.D. has been to work towards making the operational calculus rigorous (or more rigorous, anyway) and I’ve published approximately 20 journal articles concerning the operational calculus during my time at Creighton. I have also, in 2015, along with M. L. Lapidus (University of California, Riverside) and G. W. Johnson (University of Nebraska—Lincoln) published the research monograph

“Feynman’s Operational Calculus and Beyond: Noncommutativity and Time-Ordering”

(Oxford Mathematical Monographs, Oxford Science Publications, Oxford University Press, Oxford, UK.

An earlier, and closely related, monograph is

“The Feynman Integral and Feynman’s Operational Calculus”

by M. L. Lapidus and G. W. Johnson (Oxford Mathematical Monographs, Oxford Science Publications, Oxford University Press, Oxford, UK, 2002.

Education

Ph.D. (Mathematics), University of Nebraska, 1999

Interests

  • Analysis
  • Operator Theory
  • Feynman Integrals
  • Feynman's Operational Calculus