Eliminating Electron Self-Repulsion

in Foundations of Physics (2023) [journal version] [arXiv version]

Problems of self-interaction arise in both classical and quantum field theories.  To understand how such problems are to be addressed in a quantum theory of the Dirac and electromagnetic fields (quantum electrodynamics), we can start by analyzing a classical theory of these fields.  In such a classical field theory, the electron has a spread-out distribution of charge that avoids some of the problems of self-interaction facing point charge models.  However, there remains the problem that the electron will experience self-repulsion.  This self-repulsion cannot be eliminated within classical field theory without also losing Coulomb interactions between distinct particles.  But, electron self-repulsion can be eliminated from quantum electrodynamics in the Coulomb gauge by fully normal-ordering the Coulomb term in the Hamiltonian.  After normal-ordering, the Coulomb term contains pieces describing attraction and repulsion between distinct particles and also pieces describing particle creation and annihilation, but no pieces describing self-repulsion.

Absorbing the Arrow of Electromagnetic Radiation

with Mario Hubert, in Studies in History and Philosophy of Science (2023) [journal version] [arXiv version]

We argue that the asymmetry between diverging and converging electromagnetic waves is just one of many asymmetries in observed phenomena that can be explained by a past hypothesis and statistical postulate (together assigning probabilities to different states of matter and field in the early universe).  The arrow of electromagnetic radiation is thus absorbed into a broader account of temporal asymmetries in nature.  We give an accessible introduction to the problem of explaining the arrow of radiation and compare our preferred strategy for explaining the arrow to three alternatives: (i) modifying the laws of electromagnetism by adding a radiation condition requiring that electromagnetic fields always be attributable to past sources, (ii) removing electromagnetic fields and having particles interact directly with one another through retarded action-at-a-distance, (iii) adopting the Wheeler-Feynman approach and having particles interact directly through half-retarded half-advanced action-at-a-distance.  In addition to the asymmetry between diverging and converging waves, we also consider the related asymmetry of radiation reaction.

The Disappearance and Reappearance of Potential Energy in Classical and Quantum Electrodynamics

in Foundations of Physics (2022) [journal version] [arXiv version] [talk and slides]

In electrostatics, we can use either potential energy or field energy to ensure conservation of energy.  In electrodynamics, the former option is unavailable.  To ensure conservation of energy, we must attribute energy to the electromagnetic field and, in particular, to electromagnetic radiation.  If we adopt the standard energy density for the electromagnetic field, then potential energy seems to disappear.  However, a closer look at electrodynamics shows that this conclusion actually depends on the kind of matter being considered.  Although we cannot get by without attributing energy to the electromagnetic field, matter may still have electromagnetic potential energy.  Indeed, if we take the matter to be represented by the Dirac field (in a classical precursor to quantum electrodynamics), then it will possess potential energy (as can be seen by examining the symmetric energy-momentum tensor of the Dirac field).  Thus, potential energy reappears.  Upon field quantization, the potential energy of the Dirac field becomes an interaction term in the Hamiltonian operator of quantum electrodynamics.

The Fundamentality of Fields

in Synthese (2022) [journal version] [arXiv version]

There is debate as to whether quantum field theory is, at bottom, a quantum theory of fields or particles.  One can take a field approach to the theory, using wave functionals over field configurations, or a particle approach, using wave functions over particle configurations.  This article argues for a field approach, presenting three advantages over a particle approach: (1) particle wave functions are not available for photons, (2) a classical field model of the electron gives a superior account of both spin and self-interaction as compared to a classical particle model, and (3) the space of field wave functionals appears to be larger than the space of particle wave functions.  The article also describes two important tasks facing proponents of a field approach: (1) legitimize or excise the use of Grassmann numbers for fermionic field values and in wave functional amplitudes, and (2) describe how quantum fields give rise to particle-like behavior.

The Mass of the Gravitational Field

in The British Journal for the Philosophy of Science (2022) [journal version] [arXiv version] [notes on gravity, electromagnetism, and gravitoelectromagnetism]

By mass-energy equivalence, the gravitational field has a relativistic mass density proportional to its energy density.  I seek to better understand this mass of the gravitational field by asking whether it plays three traditional roles of mass: the role in conservation of mass, the inertial role, and the role as source for gravitation.  The difficult case of general relativity is compared to the more straightforward cases of Newtonian gravity and electromagnetism by way of gravitoelectromagnetism, an intermediate theory of gravity that resembles electromagnetism.

Particles, Fields, and the Measurement of Electron Spin

in Synthese (2021) [journal version] [arXiv version] 

This article compares treatments of the Stern-Gerlach experiment across different physical theories, building up to a novel analysis of electron spin measurement in the context of classical Dirac field theory.  Modeling the electron as a classical rigid body or point particle, we can explain why the entire electron is always found at just one location on the detector (uniqueness) but we cannot explain why there are only two locations where the electron is ever found (discreteness).  Using non-relativistic or relativistic quantum mechanics, we can explain both uniqueness and discreteness. Moving to more fundamental physics, both features can be explained within a quantum theory of the Dirac field.  In a classical theory of the Dirac field, the rotating charge of the electron can split into two pieces that each hit the detector at a different location.  In this classical context, we can explain a feature of electron spin that is often described as distinctively quantum (discreteness) but we cannot explain another feature that could be explained within any of the other theories (uniqueness).

Electron Charge Density: A Clue from Quantum Chemistry for Quantum Foundations

in Foundations of Physics (2021) [journal version] [arXiv version] 

Within quantum chemistry, the electron clouds that surround nuclei in atoms and molecules are sometimes treated as clouds of probability and sometimes as clouds of charge.  These two roles, tracing back to Schrödinger and Born, are in tension with one another but are not incompatible.  Schrödinger's idea that the nucleus of an atom is surrounded by a spread-out electron charge density is supported by a variety of evidence from quantum chemistry, including two methods that are used to determine atomic and molecular structure: the Hartree-Fock method and density functional theory.  Taking this evidence as a clue to the foundations of quantum physics, Schrödinger's electron charge density can be incorporated into many different interpretations of quantum mechanics (and extensions of such interpretations to quantum field theory).

Possibility of Small Electron States

in Physical Review A (2020) [journal version] [arXiv version] 

Some authors have claimed that there exists a minimum size (on the order of the Compton radius) for electron states composed entirely of positive-frequency solutions to the free Dirac equation.  Other authors have put forward counterexamples to such claims.  This article asks how the counterexamples of A. J. Bracken and G. F. Melloy [J. Phys. A. 32, 6127 (1999)] bypass two arguments against their possibility.  The first is an old argument that, because of the prohibition on faster-than-light motion, the electron must be larger than a certain minimum size if it is to have the correct angular momentum and magnetic moment.  This challenge can be addressed by analyzing the flow of energy and charge for the counterexample states.  The second argument is an explicit proof (presented in C.-P. Chuu et al., [Solid State Commun. 150, 533 (2010)]) that there is a minimum size for purely positive-frequency electron states.  This proof hinges on the assumption of a small spread in momentum space, which is violated by the counterexamples that have been put forward.

[Note: Bialynicki-Birula and Bialynicka-Birula have written a response, "Comment on 'Possibility of Small Electron States'" (2022).]

Putting Positrons into Classical Dirac Field Theory

in Studies in History and Philosophy of Modern Physics (2020) [journal version] [arXiv version]

One way of arriving at a quantum field theory of electrons and positrons is to take a classical theory of the Dirac field and then quantize.  Starting with the standard classical field theory and quantizing in the most straightforward way yields an inadequate quantum field theory.  It is possible to fix this theory by making some modifications (such as redefining the operators for energy and charge).  Here I argue that we ought to make these modifications earlier, revising the classical Dirac field theory that serves as the starting point for quantization (putting positrons into that theory and removing negative energies).  Then, quantization becomes straightforward.  Also, the physics of the Dirac field is made more similar to the physics of the electromagnetic field and we are able to better understand electron spin.

How Electrons Spin

in Studies in History and Philosophy of Modern Physics (2019) [journal version] [arXiv version]

There are a number of reasons to think that the electron cannot truly be spinning.  Given how small the electron is generally taken to be, it would have to rotate superluminally to have the right angular momentum and magnetic moment.  Also, the electron's gyromagnetic ratio is twice the value one would expect for an ordinary classical rotating charged body.  These obstacles can be overcome by examining the flow of mass and charge in the Dirac field (interpreted as giving the classical state of the electron).  Superluminal velocities are avoided because the electron's mass and charge are spread over sufficiently large distances that neither the velocity of mass flow nor the velocity of charge flow need to exceed the speed of light.  The electron's gyromagnetic ratio is twice the expected value because its charge rotates twice as fast as its mass.

[Note: There is an important correction to this account of electron spin in "Possibility of Small Electron States" (above).]

[Note: The 4/3 in equation (2) of "How Electrons Spin" should be 3/4.]

Electromagnetism as Quantum Physics

in Foundations of Physics (2019) [journal version] [arXiv version]

One can interpret the Dirac equation either as giving the dynamics for a classical field or a quantum wave function.  Here I examine whether Maxwell's equations, which are standardly interpreted as giving the dynamics for the classical electromagnetic field, can alternatively be interpreted as giving the dynamics for the photon's quantum wave function.  I explain why this quantum interpretation would only be viable if the electromagnetic field were sufficiently weak, then motivate a particular approach to introducing a wave function for the photon (following Good, 1957).  This wave function ultimately turns out to be unsatisfactory because the probabilities derived from it do not always transform properly under Lorentz transformations.  The fact that such a quantum interpretation of Maxwell's equations is unsatisfactory suggests that the electromagnetic field is more fundamental than the photon.

Forces on Fields

in Studies in History and Philosophy of Modern Physics (2018) [journal version] [arXiv version]

In electromagnetism, as in Newton's mechanics, action is always equal to reaction.  The force from the electromagnetic field on matter is balanced by an equal and opposite force from matter on the field.  We generally speak only of forces exerted by the field, not forces exerted upon the field.  But, we should not be hesitant to speak of forces acting on the field.  The electromagnetic field closely resembles a relativistic fluid and responds to forces in the same way.  Analyzing this analogy sheds light on the inertial role played by the field's mass, the status of Maxwell's stress tensor, and the nature of the electromagnetic field.

Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics

with Sean Carroll, in The British Journal for the Philosophy of Science (2018) [journal version] [arXiv version] [blog I, II, III] [quanta]

A longstanding issue in attempts to understand the Everett (Many-Worlds) approach to quantum mechanics is the origin of the Born rule: why is the probability given by the square of the amplitude?  Following Vaidman, we note that observers are in a position of self-locating uncertainty during the period between the branches of the wave function splitting via decoherence and the observer registering the outcome of the measurement.  In this period it is tempting to regard each branch as equiprobable, but we argue that the temptation should be resisted.  Applying lessons from this analysis, we demonstrate (using methods similar to those of Zurek's envariance-based derivation) that the Born rule is the uniquely rational way of apportioning credence in Everettian quantum mechanics.  In doing so, we rely on a single key principle: changes purely to the environment do not affect the probabilities one ought to assign to measurement outcomes in a local subsystem.  We arrive at a method for assigning probabilities in cases that involve both classical and quantum self-locating uncertainty.  This method provides unique answers to quantum Sleeping Beauty problems, as well as a well-defined procedure for calculating probabilities in quantum cosmological multiverses with multiple similar observers.

Constructing and Constraining Wave Functions for Identical Quantum Particles

in Studies in History and Philosophy of Modern Physics (2016) [journal version] [arXiv version]

I address the problem of explaining why wave functions for identical particles must be either symmetric or antisymmetric (the symmetry dichotomy) within two interpretations of quantum mechanics which include particles following definite trajectories in addition to, or in lieu of, the wave function: Bohmian mechanics and Newtonian quantum mechanics (a.k.a. many interacting worlds).  In both cases I argue that, if the interpretation is formulated properly, the symmetry dichotomy can be derived and need not be postulated.

Killer Collapse: Empirically Probing the Philosophically Unsatisfactory Region of GRW

in Synthese (2015) [published version] [PhilSci archive version]

GRW theory offers precise laws for the collapse of the wave function.  These collapses are characterized by two new constants, λ and σ.  Recent work has put experimental upper bounds on the collapse rate, λ.  Lower bounds on λ have been more controversial since GRW begins to take on a many-worlds character for small values of λ.  Here I examine GRW in this odd region of parameter space where collapse events act as natural disasters that destroy branches of the wave function along with their occupants.  Our continued survival provides evidence that we don't live in a universe like that.  I offer a quantitative analysis of how such evidence can be used to assess versions of GRW with small collapse rates in an effort to move towards more principled and experimentally-informed lower bounds for λ.

Quantum Mechanics as Classical Physics

in Philosophy of Science (2015) [published version] [arXiv version] [blog] [featured in Nature News]

Here I explore a novel no-collapse interpretation of quantum mechanics which combines aspects of two familiar and well-developed alternatives, Bohmian mechanics and the many-worlds interpretation.  Despite reproducing the empirical predictions of quantum mechanics, the theory looks surprisingly classical.  All there is at the fundamental level are particles interacting via Newtonian forces.  There is no wave function.  However, there are many worlds.

Many Worlds, the Born Rule, and Self-Locating Uncertainty

with Sean Carroll, in Quantum Theory: A Two-Time Success Story, Yakir Aharonov Festschrift (2014) [published version (original)] [arXiv version (updated)]

We provide a derivation of the Born Rule in the context of the Everett (Many-Worlds) approach to quantum mechanics.  Our argument is based on the idea of self-locating uncertainty: in the period between the wave function branching via decoherence and an observer registering the outcome of the measurement, that observer can know the state of the universe precisely without knowing which branch they are on.  We show that there is a uniquely rational way to apportion credence in such cases, which leads directly to the Born Rule.