## Lance W. Nielsen, PhD

Professor

College of Arts and Sciences

### Expertise/Specializations

- Feynman's operational calculus
- Fractal Strings and Complex Dimensions

### Academic Appointments

### Department

- Mathematics

### Position

- Professor

### Biography

I am originally from rural western Iowa, outside of Onawa, IA. I received a B.S. double major in mathematics and physics from the University of South Dakota, a M.S. in mathematics from the University of New Hampshire and a Ph.D. in mathematics from the University of Nebraska-Lincoln. My specialization is in the area referred to as

*Feynman's operational calculus*, an area of mathematics closely related to quantum mechanics and quantum electrodynamics which was originated by R. P. Feynman in approximately 1950.### Publications and Presentations

### Books

- Blending Instantaneous and Continuous Phenomena in Feynman’s Operational Calculus with G. W. Johnson, in “Stochastic Analysis and Mathematical Physics (SAMP/ANESTOC 2002) Proceedings of the Mathematical Legacy of R. P. Feynman”, World Scientific, Sept. 2004.
- Noncommutativity and Time-Ordering: Feynman’s Operational Calculus and Beyond, with Michel Lapidus and G. W. Johnson, published by Oxford University Press, in print September 2015., Oxford University Press

### Articles

- Nielsen, Lance A distributional approach to Feynman's operational calculus, New York Journal of Mathematics, 20, 377-398, 2014
- A Distributional Approach to Feynman’s Operational Calculus, New York J. of Math. 20 (2014), 1-22., New York Journal of Mathematics, 2014, 1-22, 2014
- Towards a Comprehensive Stability Theory for Feynman’s Operational Calculus: The Time Independent Setting, Acta Appl. Math., 2014, in press. DOI 10.1007/s10440-014-9957-1., Journal of Doctoral Nursing Practice, 2014, 2014
- Fuzzy Cores in Spatial Models, J. of Fuzzy Mathematics, Vol. 21, No. 4, 2013., Journal of Fuzzy Mathematics, 21, 4, 2013
- Nielsen, Lance Weak Convergence and Banach Space-Valued Functions: Improving the Stability Theory of Feynman's Operational Calculi, Mathematical Physics, Analysis and Geometry, 14, 279-294, 2011
- Weak Convergence and Banach Space-Valued Functions: Improving the Stability Theory of Feynman’s Operational Calculi, Math. Phys. Anal. Geom. 14 (2011), 279-294., Mathematical Physics, Analysis and Geometry, 14, 2014, 2011
- Nielsen, Lance Feynman's Operational Calculi: Disentangling Away from the Origin, Acta Applicandae Mathematicae, 110, 409-429, 2010
- Clark, T. D., Mordeson, J. N., Nielsen, L., Wierman, M. J. Fuzzy geometry: applied to comparative politics, Critical Review, 2, 1-12, 2008
- Nielsen, L. Weak convergence and vector-valued functions: Improving the stability theory of feynman's operational calculi, Mathematical Physics, Analysis and Geometry, 10, 271-295, 2007
- Jefferies, B., Johnson, G. W., Nielsen, L. Feynman's operational calculi: Spectral theory for noncommuting self-adjoint operators, Mathematical Physics, Analysis and Geometry, 10, 65-80, 2007
- Nielsen, L. Stability properties for feynman's operational calculus in the combined continuous/discrete setting, Acta Applicandae Mathematicae, 88, 47-79, 2005
- Nielsen, L. Time dependent stability for Feynman's operational calculus, Rocky Mountain Journal of Mathematics, 35, 1347-1368, 2005
- Feynman’s Operational Calculi for Time Dependent Noncommuting Operators, with Brian Jefferies and G. W. Johnson, J. Korean Math. Soc. 38 (2001), 193 – 226, Journal of the Korean Mathematical Society, 38, 2001, 193-226
- A Stability Theorem for Feynman’s Operational Calculus, with G. W. Johnson, Conference Proc., Canadian Mathematical Society, 29 (2000), 351 – 365, Conference Proceedings, Canadian Mathematical Society, 29, 2000, 351-365
- Effects of Absolute Continuity in Feynman’s Operational Calculus, Proc. Amer. Math. Soc. 131 (2003), 781-791, Proceedings of the American Mathematical Society, 131, 2003, 781-791
- Stability Properties of Feynman’s Operational Calculus for Exponential Functions of Noncommuting Operators, Acta Applicandae Mathematicae 74, 2002, 265 – 292, Acta Applicandae Mathematicae, 74, 2002, 265-292
- Time Dependent Stability for Feynman’s Operational Calculus, Rocky Mountain Journal of Mathematics 35 no. 4 (2005), 1347 - 1368, Rocky Mountain Journal of Mathematics, 4, 2005, 1347-1368
- Stability Properties for Feynman’s Operational Calculus in the Combined Continuous/Discrete Setting, Acta Applicandae Mathematicae 88 (2005), 47 – 79., Acta Applicandae Mathematicae, 88, 2005, 47-79
- Feynman’s Operational Calculi: Spectral Theory for Noncommuting Self – Adjoint Operators, with B. Jefferies and G. W. Johnson, Mathematical Physics, Analysis, and Geometry, 10 (2007), 65 - 80, Mathematical Physics, Analysis and Geometry, 10, 2007, 65-80
- Weak Convergence and Vector-Valued Functions: Improving the Stability Theory of Feynman’s Operational Calculi, Mathematical Physics, Analysis, and Geometry, 10 (2007), 271 - 295., Mathematical Physics, Analysis and Geometry, 10, 2007, 271-295
- An Integral Equation for Feynman’s Operational Calculus, Integration: Mathematical Theory and Applications, Volume 1, Number 1, 2008, pp. 49 - 66., Integration: Mathematical Theory and Applications, 1, 2008, 49-66
- Feynman’s Operational Calculi: Disentangling Away from the Origin, Acta Applicandae Mathematicae ,110 (2010), 409-429., Acta Applicandae Mathematicae, 110, 2010, 409-429
- á Feynman’s Operational Calculus : Using Cauchy’s Integral Formula, New York J. of Math. (16) 2010, 1 - 26., New York Journal of Mathematics, 16, 2010, 1-26
- Weak Convergence and Banach Space-Valued Functions: Improving the Stability Theory of Feynman’s Operational Calculi, Math. Phys. Anal. Geom. 14 (2011), 279-294., Mathematical Physics, Analysis and Geometry, 14, 2011, 279-294
- A Distributional Approach to Feynman’s Operational Calculus, New York J. of Math. 20 (2014), 1-22., New York Journal of Mathematics, 20, 2014, 1-22
- Towards a Comprehensive Stability Theory for Feynman’s Operational Calculus: The Time Independent Setting, Acta Appl. Math., 138 (2015), 59-79., Acta Applicandae Mathematicae, 138, 2015, 59-79
- Towards a Comprehensive Stability Theory for Feynman’s Operational Calculus: The Time-Dependent Setting, submitted for publication, July 30, 2015., Acta Applicandae Mathematicae, 152 (2017), 1-31
, Integral Equations and Operator Theory, Nielsen, L. Integr. Equ. Oper. Theory (2018) 90: 12. https://doi.org/10.1007/s00020-018-2428-8*Combining Continuous and Discrete Phenomena for Feynman’s Operational Calculus in the Presence of a (C0) Semigroup and Feynman-Kac Formulas with Lebesgue-Stieltjes Measures*- "Two Approaches to the Use of Unbounded Operators in Feynman's Operational Calculus", New York Journal of Mathematics

### Editing and Reviews

- Analytic Tools for Feynman Integrals, by Vladimir Smirnov, Springer Tracts in Modern Physics 250, Springer, 2012. (Monograph), 2012

### Presentations

- Invited lecture at the South Dakota State University department of mathematics seminar. Title: "Essential Ideas and Properties of Feynman's Operational Calculus." , 2018
- Presented the invited talk "Feynman's Operational Calculus: Background and Esential Properties" at the spring 2017 Nebraska-Iowa Functional Analysis Seminar at Creighton University., 2017
- Presented the talk: "Combining Continuous and Discrete Phenomena in Feynman's Operational Calcuus in the Presence of a (C_0) Semigroup: Feynman-Kac Formulas with Lebesgue-Stieltjes Measures" at the Joint Meetings of the AMS in Atlanta, GA. , 2017
- Presented the talk "An Evolution Equation for Feynman's Operational Calculus in the Combined Continuous/Discrete Setting" at the Joint Meetings of the AMS in Seattle, WA, Jan. 2016., 2016
- Presented the invited talk “Towards a Comprehensive Stability Theory for Feynman’s Operational Calculus: The Time Independent Setting” during the special session “AMS Special Session on Advances in Analysis and PDEs” at the Joint Meetings of the American Mathematical Society in Baltimore, Jan. 2014., 2014
- Presented the contributed talk “Feynman’s Operational Calculus: Using Cauchy’s Integral Formula” at the Joint Meetings of the AMS, Boston, January 2012., 2012
- Presented "Stability Properties for Feynman's Operational Calculus" at the International Conference on Feynman Integrals and Related Topics., 1999
- Eight lectures on on-going research at the Functional Integration Seminar at UNL in the fall of 2004
- Presented 6 lectures on current research at the Functional Integration Seminar at UNL during the Fall of 2000.
- Presented "Effects of Absolute Continuity in Feynman's Operational Calculus" at the Joint Meetings of the AMS, January 2001.
- Presented 6 lectures on current research at the Functional Integration Seminar at UNL in the Spring of 2001
- Presented 8 lectures on current research at the Functional Integration Seminar during the Fall of 2002.
- Presented 5 lectures on current research at the Functional Integration Seminar at UNL in Spring 2003.
- Presented "Blending Instantaneous and Continuous Phenomena in Feynman's Operational Calclus" at the Joint Meetings of the AMS, Jan. 2003
- Invited Lecture - "A Survey of Feynman's Operational Calculus" at the Nebraska Iowa Functional Analysis Seminar, April 2003.
- Presented 8 lectures on current research at the Functional Integration Seminar at UNL in Fall 2004
- Presented "An Integral Equation for Feynman's Operational Calculus" at the Joint Meetings of the AMS, Jan. 2005.
- Presented 5 lectures on current research at the Functional Integration Seminar at UNL, spring 2005.
- Presented 5 lectures on current research at the Functional Integration Seminar at UNL in Spring 2006.
- Presented "An Integral Equation for Feynman's Operational Calculus" at "The Feynman Integral and Related Topics in Mathematics and Physics" at UNL in 2006.
- Presented 6 lectures at the Functional Integration Seminar, Spring 2007.
- Presented 5 lectures at the Functional Integration Seminar at UNL in Fall 2007.
- Presented "Weak Convergence and Vector-Valued Functions: Improving the Stability Theory of Feynman's Operational Calculus" at the Joint Meetings of the AMS, Jan. 2008
- Presented 4 lectures on current research at the Functional Integration Seminar at UNL, Fall 2008.
- Presented "Feynman's Operational Calculus Beyond an Introduction" at the University of California, Riverside, March 2009.
- Presented the invited lecture “Feynman’s Operational Calculi : Background and a Survey of Current Research” at the Nebraska - Iowa Functional Analysis Seminar in Des Moines, IA
- Presented the invited talk “An Integral Equation for Feynman’s Operational Calculi” at “The 10th International Conference on Path Integrals” held at Howard University, Washington, D.C.
- Presented three lectures at the functional integration seminar at the University of Nebraska, Lincoln, spring 2010.
- Presented two lectures at the functional integration seminar at the University of Nebraska, Lincoln, fall 2010.
- Presented two lectures at the functional integration seminar at the University of Nebrask, Lincoln, spring 2011.
- Presented the invited talk “Feynman’s Operational Calculus: Background and a Survey of Current Research” at the University of Nebraska, Omaha, April 22, 2011
- Presented three lectures at the functional integration seminar at the University of Nebraska, Lincoln, fall 2011.
- Presented three lectures at the functional integration seminar at the University of Nebraska, Lincoln, spring 2012.
- Presented three lectures at the functional integration seminar at the University of Nebraska, Lincoln, fall 2012.
- Presented the contributed talk “Feynman’s Operational Calculus: Using Cauchy’s Integral Formula” at the Joint Meetings of the AMS, Boston, January 2012.
- Presented the invited talk “Towards a Comprehensive Stability Theory for Feynman’s Operational Calculus: The Time Independent Setting” during the special session “AMS Special Session on Advances in Analysis and PDEs” at the Joint Meetings of the American Mathematical Society in Baltimore, Jan. 2014.
- Presented the talk: “Towards a Comprehensive Stability Theory for Feynman’s Operational Calculus: The Time Independent and the Time-Dependent Settings” at the Joint Mathematics Meetings, Jan. 2015, in San Antonio.
- Presented the talk: “An Evolution Equation for Feynman’s Operational Calculus in the Combined Continuous/Discrete Setting” at the Joint Mathematics Meetings, Jan. 2016, in Seattle, WA

### Research and Scholarship

### Research and Scholarship Interests

- Feynman's operational calculus, complex dimensions and fractal strings.

### Current Research Projects

- Feynman's operational calculus:

The 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 two dozen 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.