Mathematical construction of quantum field theory 2018-2023

I am writing a almost purely mathematical paper to clarify and elaborate the formulation of the work I did on the mathematical foundation for quantum fields during 2004-2017, which is recorded in the book manuscript described below. The rigorous treatment of quantum fields is illustrated with the explicit construction of the φ_d⁴ scalar field in any spacetime dimension d, including 4-dimensional spacetime. The traditional processes of regularization and renormalization are discussed. The reason the existing methods lead to triviality (that interaction theories have a scattering matrix that is the identity) is exhibited. The paper presents a coherent and rigorous theory of quantum fields, formulates a full derivation of the covariant perturbation theory used in the Standard Model and the Feynman diagram techniques for computing the scattering matrix, using the scalar field with a polynomial potential as an explicit example.

Timeline

• Jan 30, 2023: The detailed steps to carry out the covariant perturbation theory for an interacting field theory from a free field reference have been analyzed and proven. I finished writing a short paper with high-level story of QFT to keep the big picture clear to the reader. The rigorous presentation is long and the mathematical details, while essential, are distracting from the big picture. That story will then guide how to finish writing the full mathematical paper, which was left in an almost complete state in Nov 2019.
• April 3, 2021: The covariant perturbation theory used to define the Standard Model can be derived from the framework developed in the paper. I am using Gerald Folland’s excellent book “Quantum Field Theory: A Tourist Guide for Mathematicians” AMS 2008 ISBN 978-0-8218-4705-3, as the reference formulation to present these results.
• Jul 11, 2020: The writing of the full paper is progressing; 150 pages and counting… The construction of general quantum fields, scalar and spinor is complete. The formulation of gauge theory for quantum mechanics, emphasizing the (local) gauge symmetry over imposing gauge conditions as constraints, has also been written up. Now the derivation of the well-known Feynman diagrams in the constructive quantum field theory framework is being tackled. Finishing up the writing of the construction of a non-Abelian Yang-Mills gauge theory with confinement remains.
• Sep 30, 2018: Further clarifications and derivations, going beyond the content of the book manuscript, strengthen and clarify the construction of quantum fields. Scalar fields are done, now working on fields with a Yukawa velocity coupling.
• Jan 27, 2018: I continue to expand, elaborate, and refine the presentation with the material developed in the book. The paper has the more focused title “On the construction of quantum fields“.
• Dec 6, 2017: The paper is not accepted for publication or even for revision, with little meaningful commentary from the reviewer. Maybe because it presents insufficient context and does not clearly state the full argument.
• Nov 7, 2017: A significantly revised and expanded manuscript, presenting the material in the forms of theorems at the suggestion of the referee, was submitted to Phys. Rev. D1.
• Sep 4, 2017: The first version of the paper presented a high-level overview of the construction. The manuscript “Quantum Mechanics of Interacting Fields” is submitted to Phys Rev. D1.

Architecture and design of quantum computers 2020

From teaching a course on quantum information science in the spring of 2020, I have a new project in the works: To apply the work on the measurement problem in quantum mechanics to the exploding field of quantum computers. The things I have learned shed some new light on problems and possible solutions facing the scientists and engineers trying to build quantum computers.

Foundations of quantum mechanics 2017-2019

The work for the book described below (2004-2017), has led to a refinement of the concepts and the interpretation of the wave function and probability in quantum mechanics. The changes that need to be made to quantum mechanics consists of removing concepts from classical mechanics that have been retained in the Copenhagen interpretation. The mathematics of Heisenberg, Schrödinger, and Dirac is retained in full and remains unchanged.

Timeline

• Mar 5, 2023: I have written a series of notes to tell the story of this fresh view on quantum mechanics. These notes were highly inspired by the workshop held in Arlington, VA on Jan 30-31, 2023 on “Quantum education for quantum workforce development.” It showed the success people have had recently in teaching quantum concepts to a broader audience, including high-school students.
• Dec 7, 2019: The paper is published, cite as: “On Classical Systems and Measurements in Quantum Mechanics”, Erik Deumens, Quantum Studies: Mathematics and Foundation, (2019) 6(4), 481-517 (37 pages)
• Mar 27, 2019: The paper is published online with doi: 10.1007/s40509-019-00189-3 (published online) (PDF)
• Mar 16, 2019: The paper has been accepted for publication.
• Feb 1, 2019: The significantly revised manuscript “On classical systems and measurements in quantum mechanics” has been re-submitted to Quantum Studies: Mathematics and Foundations. All comments from the reviewers have been addressed. The paper is significantly improved because of these changes.

ABSTRACT

The recent rigorous derivation of the Born rule from the dynamical law of quantum mechanics [Phys. Rep. 525(2013) 1-166] is taken as incentive to reexamine whether quantum mechanics has to be an inherently probabilistic theory.
It is shown, as an existence proof, that an alternative perspective on quantum mechanics is possible where the fundamental ontological element, the ket, is not probabilistic in nature and in which the Born rule can also be derived from the dynamics.
The probabilistic phenomenology of quantum mechanics follows from a new definition of statistical state in the form of a probability measure on the Hilbert space of kets is a replacement for the von Neumann statistical operator to address the lack of uniqueness in recovering the pure states included in mixed states, as was pointed out by Schrödinger.
From the statistical state of a quantum system, classical variables are defined as collective variables with negligible dispersion. In this framework, classical variables can be chosen to define a derived classical system that obeys, by Ehrenfest’s theorem, the laws of classical mechanics and that describes the macroscopic behavior of the quantum system.
The Born rule is derived from the dynamics of the statistical state of the quantum system composed of the observed system interacting with the measurement system and the role of the derived classical system in the process is exhibited.
The approach suggests to formulate physical systems in second quantization in terms of local quantum fields to ensure conceptually equivalent treatment of space and time.
A real double-slit experiment, as opposed to a thought experiment, is studied in detail to illustrate the measurement process.

• Jan 11, 2019: Two reviewers gave extensive comments and ask for a major revision of the paper by Feb 11, 2019
• Dec 2, 2018: The manuscript “On classical systems and measurements in quantum mechanics” was transferred to Quantum Studies: Mathematics and Foundations, with the assistance of Springer Submission Services. In the twp-column format for that journal it is 27 pages.
• July 31, 2018: The manuscript “On classical systems and measurements in quantum mechanics” has been submitted to Foundation of Physics. It has 42 pages single spaced.
• Jan 27, 2018: From the feedback received, I am reworking the material from the book into a new paper. The first paper was written as a summary and cannot be fully understood by itself.
• Oct 26, 2017: Paper rejected as being too condensed. The referee feels that the subject is extensive and needs to be discussed in a book(!).
• Apr 30, 2017: Manuscript “Quantum Mechanics of Macroscopic Systems” with a summary from the book is submitted to Foundations of Physics.

Research for a book on quantum mechanics 2004-2017

Starting from an idea that I had in graduate school in the summer of 1980, I have been working on a book on quantum mechanics since 2004. From 2013 through March 2017, I wrote the results in a manuscript for a book “The principles of Quantum Mechanics“. It was written to present the new ideas in a coherent way to a wide audience ranging from experimental and theoretical physicists, chemists, and engineers, as well as mathematical physicists and mathematicians.

The manuscript of a book “The Principles of Quantum Mechanics” was submitted to Oxford University Press on April 18, 2017. Publication is expected in early 2018. The book takes a fresh look at the foundations of quantum mechanics with the approach to the some foundation thoroughly revised in some places. With this new point of view, some problems are no longer insurmountable.

• The table of contents is available in PDF format in the file Principles-TOC.pdf.
• Apr 18, 2017: Manuscript submitted.
• Jan 18, 2018: Oxford University Press notified me that they will not publish the manuscript.

The reviewers unanimously found the layout of the book too complex and too hard to follow. In retrospect, I agree with them. My plan to present the material to the very wide audience caused me to arrange the material in a complex way, so that it was hard to follow. There are two main new ideas in the book putting a quantum mechanics in a new perspective. The organization I chose hides the new results and completely fails to bring out a this new perspective. I will work on presenting the ideas separately and publish them first as papers. The ideas are:

1. Mathematically rigorous formulation of quantum fields with a clear identification of the problem that is traditionally addressed by renormalization. This is the content of chapters 10, 11, 12, and 3. Part III of the manuscript together with chapter 3 properly expanded form the content of a (long) paper on the construction of quantum fields.
2. A new perspective of the interpretation of the wave function and the role of probability in quantum mechanics with a new way to look at classical systems and measurements in quantum mechanics. This is the content of chapters 2, 4, and 13 of the manuscript. I will work on publishing this material in a paper.

Earlier Research Projects

• 2010 Together with Frank Harris, I organized an exploratory workshop with an NSF grant into the organization of a Scientific Software Innovation Institute for the Atomistic Modeling and simulation community. The report for the one we organized is at S2I2Report.pdf. In July 2016, a Scientific Software Institute MolSSI was funded by NSF with Dan Crawford as PI.
• 2003 – 2010 With Beverly Sanders of the UF CISE department and Victor Lotrich, Mark Ponton, Ajith Perera, and Rod Bartlet we are developing a domain-specific-language (DSL) based approach to massively parallel programming of complex software systems for molecular and materials processes. Funding comes from DOE and NSF, as well as DOD Army and Air Force. I was the software architect and project manager with Rod Bartlett and ACES Q.C.in the design of the super instruction architecture (SIA) for scalable parallel software development. This environment has been used to write ACES III, a parallel implementation of Coupled Cluster methods. During the Summer of 2009, we managed to run a CCSD(T) calculation in the molecule RDX with ACES III on 60,000 cores of the Cray XT5 Jaguarpf of the DOE facility at Oak Ridge, Tennessee. During the Spring of 2010, we got up to 86,000 cores with a new SIAL program for building the Fock matrix.
• 2004 – 2005 John Klauder and I worked a project to investigate self interacting field theories using lattice Monte-Carlo computations.
• 1986 – 2008 With Yngve Öhrn and several postdocs and graduate students and I have developed the theory of the Electron Nuclear Dynamics (END). I am the principal author of the software package ENDyne implementing this theory. The effort from 2005 until 2008 has been to extend END to multi-configuration wave function. We call this dynamic MC wave function method vector Hartree-Fock (VHF). The code was completed and debugged just before Christmas 2008. We are now starting to run the first VHF calculations on atoms and some simple molecules to test the theory and the algorithms. I use the development of a high performance, portable, parallel software library for quantum chemical integrals, called QTIP as testing ground for research and teaching of high quality software engineering.

Updated 5 Mar 2023