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Dr. Neil Christensen

Associate Professor
Moulton Hall - MLT 312D
  • About
  • Education
  • Awards & Honors
  • Research

Current Courses

390.016Computational Research In Physics

417.001Mathematical Methods

387.001Methods Of Mathematical Physics

112.001Physics For Science And Engineering III

112.002Physics For Science And Engineering III

Research Interests & Areas

My research is in the area of theoretical particle physics. Currently, I am interested in constructive amplitude methods. This is an alternative to and complementary to Feynman diagrams.

Feynman diagrams are derived from traditional field theory, which is constructed to be manifestly Lorentz invariant, symmetry group invariant, Hermitian and local. In order to achieve this, we must use fields that are product representations under the Lorentz group. This includes spinors (usually Dirac spinors), four vectors and combinations of these. The only fields that contain the right transformation properties for photons, gluons, W and Z bosons are vector fields. However, vector fields have four degrees of freedom, while photons and gluons only have two (positive and negative helicity) and W and Z bosons only have the three of a spin-1 particle. Therefore, in order to use a vector field for these particles, we must add extra unphysical degrees of freedom. These extra unphysical degrees of freedom would ruin the calculations of physical observables if not removed or cancelled. However, long ago it has been found that, if we "gauge" the symmetry groups for electroweak and strong transformations, the bad consequences of these extra unphysical degrees of freedom will exactly cancel.

Feynman diagrams result from gauged field theory. They have the nice properties of those field theories. They are manifestly Lorentz invariant, symmetry group invariant, Hermitian and local. However, as a consequence of the unphysical degrees of freedom, the contribution from the diagrams expands beyond what is physical. On the other hand, due to the gauged symmetries, these extra diagram contributions exactly cancel among themselves in gauge-invariant subsets of diagrams, leaving the physical result at the end. Although this has allowed the calculation of any scattering amplitude at any order in principle, it appears that it may be unnecessarily uneconomical and inefficient. It might be better to never add extra unphysical degrees of freedom and avoid the expansion of diagram contributions and avoid the required cancellation.

Recently, a new method for calculating scattering amplitudes has been emerging, where we never add extra unphysical degrees of freedom, and therefore, never need to have cancellations between diagrams in order to have physically meaningful results, and therefore, never need to "gauge" any symmetries. Indeed, each diagram in such a method is trivially gauge invariant. This new method is determined from the ground up, based on the symmetries. For this reason, it is called the "constructive amplitude" method. It turns out that the symmetries, including the Lorentz symmetry as well as the global symmetries of electromagnetism and the strong force, are sufficient to determine most of the fundamental vertices in the constructive amplitude method. We now have all the vertices for the Constructive Standard Model (CSM).

We are still developing the constructive amplitude method. We have calculated all the four-point amplitudes in the CSM and we have shown that it is perturbatively unitary at tree-level four-point. We have showed that it only includes four four-point vertices, as opposed to the seven four-point vertices of Feynman diagrams. We have seen that many of its amplitudes at this level are simpler, more economical and more efficient than their Feynman counterparts. This theory clearly is also Lorentz invariant, satisfies all the symmetries and is Hermitian. However, one very interesting place where constructive theory differs at a fundamental level from Feynman diagrams is that the vertices are non-local. They also involve on-shell methods for intermediate steps of the calculation, although they are correct both on-shell and off-shell at the end of the calculation. A current deficiency that we would like to understand better is why all the couplings are related to each other. This has an elegant explanation in gauge field theory, but not currently in constructive theory.

A listing of my publications can be found on the inSpire website.

Ph D Theoretical Physics

Stony Brook University
Stony Brook, NY

BA Mathematics

University of Utah
Salt Lake City, UT


Snow College
Ephraim, UT

Pre-tenure Faculty Initiative Grant

Illinois State University

New Faculty Initiative Grant

Illinois State University

Honorable Teaching Award

Society of Physics Students, Illinois State University

Summer Undergraduate Research Award for Independent Research

The Dietrich School of Arts and Sciences

Research Grant

Department of Energy

The Pennsylvania Space Grant Consortium Research Scholarship


Samuel P. Langley PITT-PACC Fellowship

Pittsburgh Particle physics, Astrophysics and Cosmology Center

LHC-TI Fellowship

National Science Foundation

Max Dresden Theoretical Physics Thesis Prize

Dept. of Physics, Stony Brook University


National Science Foundation

Journal Article

  • N. Christensen, ``Perturbative unitarity and the four-point vertices in the constructive standard model,'' Phys. Rev. D \textbf{109}, no.11, 116014 (2024).
  • N. Christensen, ``SPINAS: spinor amplitude subroutines for constructive diagram evaluations,'' Eur. Phys. J. C 84, no.5, 520 (2024).
  • ``A Complete Set of 4-Point Amplitudes in the Constructive Standard Model,'' Eur. Phys. J. C. 84, 620 (2024).
  • N. Christensen, H. Diaz-Quiroz, B. Field, J. Hayward and J. Miles,``Challenges with internal photons in constructive QED,'' Nucl. Phys. B 993, 116278 (2023).
  • N. Christensen and B. Field, “The Constructive Standard Model,” Physics Review D98 (2018) 016014.
  • Presentations

  • "4-Point Vertices, Amplitudes and Phase-Space Calculations in the Constructive Standard Model," DPF-Pheno 2024, May 14, 2024.
  • “Challenges with Internal Photons in Constructive QED,” Phenomenology 2023 Symposium, May 5, 2023.
  • “Challenges with Internal Photons in Constructive QED," UC Davis QMAP Particles/Cosmology seminar, December 5, 2022.
  • "The Constructive Standard Model," Phenomenology Group Seminar, Department of Physics, University of Illinois at Urbana-Champaign, September 7, 2018.
  • “The Constructive SM,” N. Christensen, Phenomenology 2018 Symposium, University of Pittsburgh, Pittsburgh, PA, May 7, 2018.
  • “A New Approach to Scattering Amplitudes”, N. Christensen, Phenomenology 2017 Symposium, University of Pittsburgh, Pittsburgh, PA, May 9, 2017.
  • “The S-Matrix without Fields,” N. Christensen, High Energy Physics Seminar, Department of Physics and Astronomy, Michigan State University, East Lansing, MI, February 15, 2017.
  • “Relativistic Quantum Mechanics and the Higgs Boson”, N. Christensen, Physics Seminar, Wichita State University, Wichita, KS, December 2, 2015.
  • Electroweak Symmetry Breaking”, Physics Department Colloquium, Utah State University, Logan, UT, January 7, 2014.
  • “Model Building and Compact Lie Algebras in Galileo”, Geometry, Mathematical Physics and Computer Algebra group seminar, Utah State University, Logan, UT, January 8, 2014.
  • Grants & Contracts

    Samuel P. Langley PITT-PACC Fellowship. PITTsburgh Particle-physics Astro-physics and Cosmology Center (PITT-PACC). Other.