2206.09472.pdf

Eliminating Electron Self-Repulsion

Charles T. Sebens

Division of the Humanities and Social Sciences

California Institute of Technology

arXiv v2 - May 4, 2023

Abstract

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.

Contents

1 Introduction

2 Self-Interaction and Self-Repulsion

3 Classical Field Theory

4 Quantum Field Theory

5 Conclusion

arXiv:2206.09472v2 [quant-ph] 4 May 2023

1 Introduction

There are pervasive problems of self-interaction in both classical and quantum

electrodynamics. In classical electrodynamics, the severity of these problems depends

on whether matter is modeled as point charges or as continuous charge distributions.

The latter option dodges the most serious problems of self-interaction, avoiding infinite

self-energy and yielding well-defined dynamics that includes radiation reaction. However,

the continuous charge distributions will experience self-repulsion.

If the electron is modeled classically as a cloud of negative charge, the different parts

of that cloud would be expected to repel one another. This self-repulsion would increase

the energy of such an electron cloud and would cause the electron to rapidly explode (if

there is no force countering the self-repulsion

). We do not observe such self-repulsion.

As is apparent in quantum chemistry, the ground state energy of an atom or molecule

does not include a contribution from electron self-repulsion.

Quantum electrodynamics can be arrived at by quantizing a classical theory of

interacting Dirac and electromagnetic fields, where the Dirac field describes charged

matter (electrons and positrons) as having a continuous distribution of charge. Although

electron self-repulsion is present in this classical starting point, it should be absent in the

quantum field theory that we get upon field quantization: quantum electrodynamics.

The question to be tackled here is how self-repulsion is eliminated in quantum field

theory when it is present in classical field theory. Answering this question is important

to better understanding the transition from classical to quantum theories of the Dirac

and electromagnetic fields, and thus to better understanding quantum field theory. The

classical theory of interacting Dirac and electromagnetic fields is rarely studied, but it

is worthy of our attention because of its relation to quantum electrodynamics [15].

In section 2, I briefly discuss the problems of self-interaction that arise for point

charges and charge distributions in classical electromagnetism. In section 3, I review

1

Early models of the electron posited forces holding the electron together, now known as “Poincar ́e

stresses” [1, ch. 28]; [2–5]; [6, ch. 16]; [7, sec. 5].

However, no such forces appear in quantum

electrodynamics or the standard model. Thus, for our purposes here in understanding the relation

between classical and quantum field theories, we need not consider such forces.

2

The Hartree-Fock method for approximating the ground state energy of an atom or molecule includes

contributions from Coulomb repulsion between distinct electrons but explicitly excludes self-repulsion,

and a more recent method for calculating ground state energies, called “density functional theory,”

requires a self-interaction correction that corrects for the initial inclusion of electron self-repulsion [8, pg.

436]; [9]; [10, sec. 8.3]; [11, pg. 559]; [12].

3

Barut and his collaborators have studied such a theory of interacting classical interacting Dirac and

electromagnetic fields, considering it to be an alternative to standard quantum electrodynamics. They

have shown that a number of important phenomena can be explained without quantum physics [13, 14].

These authors retain self-interaction effects, including self-repulsion. In the context of a discussion of

the hydrogen atom, Barut [13, pg. 39] explains why we do not observe electron self-repulsion (even

though he thinks that it does in fact occur) as follows: “in a static situation, the interaction potential

between electron and the proton is just

Ze/r

[where

= 1 and

is the distance from the proton], hence

we expect that the static self-field of the electron should have no effect - it is already taken into account

by the physical charge and mass of the electron.” I do not see how one could explain the absence of

any observed self-repulsion by simply redefining the mass and/or charge of the electron. The classical

Coulomb energy associated with electron self-repulsion (8) would vary depending on how compactly the

electron’s charge is spread.

the classical theory of the Dirac field interacting with the electromagnetic field,

explaining how self-repulsion arises in that context. Then, in section 4, I move to

quantum field theory via field quantization in the Coulomb gauge and argue that

it is the full normal-ordering of creation and annihilation operators in the Coulomb

interaction term of the Hamiltonian that eliminates electron self-repulsion. The full

normal-ordering of the Coulomb term (not merely normal-ordering each instance of the

charge density operator) is rarely mentioned and has not been recognized as removing

electron self-repulsion. To understand the role of the normal-ordered Coulomb term

in the Hamiltonian, in section 4 we first imagine that it is the only interaction term

in a simplified quantum theory of the Dirac field (that could be called “quantum

electrostatics”) and then move to full quantum electrodynamics.

Before diving in, let me clarify the kind of explanation of the absence of electron

self-repulsion that is being sought in this paper. Ultimately, the presence or absence

of self-repulsion will be a consequence of the dynamical laws of a given theory. The

dynamical laws of classical electrodynamics include self-repulsion. The dynamical laws

of quantum electrodynamics, properly formulated, avoid self-repulsion. I take the laws

posited by a theory to be brute facts about the world that do not themselves require

explanation

(at least not if that theory is being treated as fundamental physics). Thus,

finding that the laws of quantum electrodynamics avoid self-repulsion is sufficient for

explaining the absence of electron self-repulsion in nature.

The goal of this paper is to understand how electron self-repulsion is removed in the

shift from classical to quantum field theories, not to resolve all puzzles of self-interaction

in quantum electrodynamics. In textbook discussions of Feynman diagram methods for

calculating scattering amplitudes in quantum electrodynamics, the electron is standardly

described as having an infinite self-energy that must be rendered finite via some

procedure of mass renormalization [17, sec. 15a]; [18, ch. 8]; [19, sec. 5.3]; [20, sec. 18.2].

This is somewhat surprising if you take the starting point for quantum electrodynamics

to be a classical theory of interacting Dirac and electromagnetic fields where electron

charge is spread out and the electrostatic self-energy is finite. However, the relation

between this infinite quantum self-energy and the classical electromagnetic self-energy

of a point charge or charge distribution is not entirely clear. Schweber [17, pg. 514], for

example, writes that “the self-energy problem is of a rather different nature in quantum

theory from that found in classical theory” and remarks that it “seems unlikely that

a solution of the classical self-energy problem is either necessary or sufficient for the

solution of the quantum mechanical one.” I believe that more work needs to be done to

better understand the connections (or lack thereof) between the classical and quantum

self-energy problems. This project is intended to be just one step towards that deeper

understanding.

4

See [16, ch. 4].

2 Self-Interaction and Self-Repulsion

In classical electromagnetism, one can take the charged matter interacting with the

electromagnetic field to be either point charges or charge distributions [21]. Point charges

come with a variety of problems of self-interaction. The electric field of a point charge

increases in strength as one approaches the charge, becoming infinite in strength and

indeterminate in direction at the location of the charge (figure 1). The density of energy

in the charge’s electromagnetic field is

2

2

, which yields an infinite amount of

“self-energy” when integrated over any finite volume around the charge. A consequence

of the electric field not being well-defined at the location of the point charge is that the

standard Lorentz force,

, does not yield a well-defined force on the point

charge from its own field. One could attempt to alleviate this problem by positing that a

point charge does not react to its own field, but then one would miss an important effect:

radiation reaction. To accelerate a charged body from rest, you must put in additional

energy, beyond the energy acquired by the body, to produce the electromagnetic wave

that is emitted when the body accelerates (carrying energy away). If charges only react

to one another’s fields through the Lorentz force law, energy-bearing electromagnetic

waves will be produced without that additional energy having to be put in. Because

deleting self-interaction leads to violations of energy conservation, Frisch [22–25] has

described classical electromagnetism as an inconsistent theory (see also [26]). There are

a variety of strategies that might be pursued to fix-up classical electromagnetism with

point charges [27, sec. 3], but let us not wade into those waters here. Our focus will be

on charge distributions because we will be viewing quantum field theory as built from

a classical field theory where charge is spread-out (not concentrated at points). In that

context, there is only one problem of self-interaction: self-repulsion.

Figure 1: The left image depicts the electric field of a point negative charge at rest. The

image on the right depicts the electric field of a static Gaussian distribution of negative

charge (shown as a gray cloud).

Moving away from point charges, we can model the electron classically as a spread-out

cloud-like charge distribution (figure 1). The electric field of the electron grows in

strength as one approaches from far away, eventually reaching a maximum strength and

then becoming weaker as one moves towards the center of the electron. Integrating the

energy density

2

2

yields a finite amount of energy in the electron’s electromagnetic

field. The Lorentz force density,

, is always well-defined.

If such an

electron charge distribution is accelerated, the electromagnetic wave that is emitted will

exert a force on the electron as it exits. Thus, the electron will experience a radiation

reaction force and energy will be conserved. The problems for point charges presented

in the previous paragraph are not problems for such a charge distribution.

Still, there is a problem of self-interaction for a spread-out electron charge

distribution. Looking at figure 1, the electric field at any location within the electron

points inward and thus the

electric force density points outward. These outward

forces are forces of self-repulsion. One can perfectly well include forces of self-repulsion

without having any ill-defined or infinite forces. Indeed, it makes sense to do so when

modeling macroscopic bodies. However, we know that electrons do not experience

self-repulsion and thus the presence of such forces here is a cause for concern. We

must show that this self-repulsion, though present in classical field theory, is absent

in quantum field theory. In the next section, we will fill in the above classical picture

of electrons with spread-out distributions of charge, using the classical Dirac field to

represent electrons. Then, in section 4 we will move to quantum field theory and see

how self-repulsion can be eliminated.

3 Classical Field Theory

Textbooks on quantum field theory generally begin their treatment of electrons and

positrons with a discussion of the Dirac equation, viewed as giving the time-evolution

of a four-component complex-valued

entity

. Dirac originally thought of

as a

single-electron wave function, in which case the Dirac equation would be part of a

relativistic single-particle quantum theory. Alternatively,

can be thought of as a

classical field (like the electromagnetic field) and the Dirac equation as part of a classical

field theory (playing a similar role to Maxwell’s equations).

Here we will adopt the latter

interpretation and start with a classical theory of the Dirac field. Allowing this field to

interact with the electromagnetic field, we have a classical field theory that can be used

to arrive at quantum electrodynamics through field quantization. This classical field

5

Given that the electron will be modeled classically in the next section as a lump of energy and

charge in the classical Dirac field, one might wonder whether it makes sense to think of the above

density of force as acting on a field (the Dirac field). For defense of the idea that forces act on both the

electromagnetic and Dirac fields, see [28, 29].

6

Although I will not do so here, there are reasons (coming from quantum field theory) to treat the

components of the Dirac field

at a given location as anticommuting Grassmann numbers instead of

complex numbers (see [15] and references therein).

7

These two interpretations of the Dirac equation and their relation to quantum field theory are

discussed in [15].

theory is sometimes called “Maxwell-Dirac theory.”

Before proceeding, let me emphasize that classical Maxwell-Dirac field theory is of

interest not because it itself accurately describes the behavior of electrons, but because

studying it can shed light on quantum field theory. The classical and quantum field

theories are deeply connected. It is classical Maxwell-Dirac field theory that yields

quantum electrodynamics upon field quantization. That being said, it is important to

note that one cannot “go the other way” and derive Maxwell-Dirac field theory in the

classical limit as an approximation to quantum electrodynamics [30, pg. 221].

In classical Maxwell-Dirac field theory, the Dirac field

evolves in accordance with

the Dirac equation (including interactions with the electromagnetic field):

~

∂ψ

∂t

(

−

~

cγ

~γ

∇

)

eγ

~γψ

−

eψφ .

(1)

The electromagnetic field evolves by Maxwell’s equations, with the charge and current

densities of the Dirac field,

−

eψ

†

(2)

−

ecψ

†

~γψ ,

(3)

acting as source terms.

In (1), electromagnetic interactions are written in terms of the

vector and scalar potentials, which specify the electric and magnetic fields via

−

∇

−

∂

∂t

(4)

∇×

A .

(5)

All equations in this paper appear in Gaussian cgs units.

The Hamiltonian, giving the total energy for the interacting Dirac and

electromagnetic fields, is

(

†

(

−

~

cγ

~γ

∇

)

eψ

†

~γψ

)

x .

(6)

Adopting the Coulomb gauge

(

∇·

= 0) and using Gauss’s law (

∇·

= 4

πρ

) as well

8

In [31] I propose changing the charge and current densities so that the (free) classical Dirac field

describes both negatively charged electrons and positively charged positrons. For our purposes here,

where we are concerned first and foremost with electrons, we can stick with the standard charge and

current densities. However, ultimately it would be better to extend the treatment in [31] to include

interactions with the electromagnetic field.

9

In [32] I discuss the interpretation of different terms in this Hamiltonian, arguing that the first two

terms give the energy of the electromagnetic field and the remainder gives the energy of the Dirac field

(with the final term being a potential energy of the Dirac field).

10

For presentations of quantum electrodynamics in the Coulomb gauge (including discussion of the

Hamiltonian for interacting Dirac and electromagnetic fields), see [33, sec. 15.2 and 17.9]; [34, sec. 5.2

and 8.1]; [35, sec. 8.3]; [36, sec. 6.4]. To limit the scope of this paper, I will work entirely within the

Coulomb gauge and not ask how the points made here might be extended to other gauges.

as (2), (4), and (5), it is possible to rewrite this Hamiltonian exclusively in terms of

and

(

∣

∂

∂t

∣

πc

(

∇×

)

†

(

−

~

cγ

~γ

∇

)

eψ

†

~γψ

†

(

)

(

)

†

(

)

(

)

−

)

x .

(7)

The last term gives the standard classical energy of Coulomb repulsion for the (always

negative) charge density (2) of the Dirac field,

(

)

(

)

−

y .

(8)

It is this term that captures the energy of Coulomb repulsion between electrons and

also, inconveniently, the energy of electron self-repulsion.

To see the problem, let us first decompose the Dirac field via the standard plane

wave expansion,

(

) =

(

)

︷

︸︸

︷

~

)

√

(

)

∑

(

)

(

)

i

~

)

~

)

√

(

)

∑

(

†

(

)

(

)

−

i

~

)

︸

︷︷

︸

−

(

)

(9)

where

is a spin index,

(

) =

√

(

) and

(

) are basis spinors, and

(

) and

†

(

) are complex coefficients specifying the particular state of the Dirac field.

In the absence of electromagnetic interactions,

(

) is a sum of positive-frequency

(electron) modes with different momenta

, each evolving by

−

i

~

(

)

independently

of one another and independently of the negative-frequency (positron) modes,

−

(

Including interactions, the evolution of

(

) and

−

(

) becomes more complicated.

Let us take a state of the classical Dirac field representing a single electron and

nothing else to be a state where the positron part of the Dirac field,

−

(

), is zero

everywhere and the integral of the charge density (2),

−

eψ

†

(

)

(

), over all space is

−

(the charge of the electron) [29, 37, 38]. For such a state, the Coulomb term in the

Hamiltonian (8) gives the energy of self-repulsion for the electron’s cloud of negative

charge,

(

)

(

)

−

†

(

)

(

)

†

(

)

(

)

−

y .

(10)

For a state of the Dirac field representing two electrons, where

−

eψ

†

(

)

(

)

−

, the Coulomb term (8) includes three kinds of energy: the energy of self-repulsion

for the first electron, the energy of self-repulsion for the second electron, and the energy

of repulsion between the two electrons. Ideally, we would like to only include the third

kind of energy.

Within a classical theory of the electromagnetic and Dirac fields, there is no way

to include the energy of Coulomb repulsion between electrons while excluding the

self-repulsion within each electron’s charge distribution. The problem stems from the

lack of separation between electrons in the classical Dirac field. If we have a state of the

Dirac field with total charge

−

representing two electrons, what is the contribution of

each electron to the total charge distribution? There would be multiple ways to divide

the total charge density into a density of charge for one electron and a density of charge

for the other (figure 2). This is a problem with classical field theory that we will shortly

see resolved in quantum field theory, where it is possible to eliminate self-repulsion from

the Hamiltonian.

Figure 2: If the classical Dirac field has a spherically symmetric charge density with

total charge

−

, there are many ways that one could imagine decomposing that charge

density into separate contributions from two distinct electrons. There might be two

electrons with the same charge distribution, or, one electron that is responsible for the

top half of the total charge distribution and another that is responsible for the bottom

half. The second alternative would have a larger energy of self-repulsion and a smaller

energy of repulsion between the electrons (with the total energy of electrostatic repulsion

the same as in the first alternative).

It is worth noting that self-repulsion can be removed in classical field theory if one

shifts from a theory where there is a single Dirac field to a theory where there are many

Dirac fields, one for each electron or positron. One could then modify the Hamiltonian

so that the only contributions to the Coulomb energy (8) come from multiplying the

charge densities of distinct Dirac fields. Let us set this option for removing self-repulsion

aside because it is a significant departure from the classical field theory being considered

and it is a departure that seems to fit poorly with quantum electrodynamics, where there

appears to be only a single (quantum) Dirac field.

4 Quantum Field Theory

Moving to quantum field theory allows us to do something that could not be done within

classical field theory: including Coulomb attraction and repulsion between distinct

electrons while excluding self-repulsion (and retaining radiation reaction). We will

see that self-repulsion can be eliminated by normal-ordering the Coulomb term in

the Hamiltonian. Although I have seen the Coulomb term written with the requisite

normal-ordering in one place [34, eq. 8.28], I have not seen any author explain how this

maneuver removes electron self-repulsion.

In quantum field theory, the time evolution of a quantum state

is given by a

Schr ̈odinger equation of the general form

~

Ψ(

)

〉

Ψ(

)

〉

(11)

For quantum electrodynamics in the Coulomb gauge, the Hamiltonian is an operator

version of the classical Hamiltonian discussed in the previous section (7). The Coulomb

term becomes

†

(

)

(

)

†

(

)

(

) :

−

y ,

(12)

where the colons indicate normal-ordering (rearranging so that creation operators

appear to the left of annihilation operators, including a minus sign whenever a creation

operator for an electron or positron is moved past an annihilation operator). Often, the

normal-ordering of the Coulomb term is not explicitly mentioned when the Hamiltonian

for quantum electrodynamics is presented.

Bjorken and Drell [33, eq. 15.28 and

17.60] give what I believe to be an incorrect

version of the Coulomb term, separately

normal-ordering each appearance of the charge density operator:

†

(

)

(

) : :

†

(

)

(

) :

−

(

)

(

)

−

y ,

(13)

11

Greiner and Reinhardt [39, sec. 8.6] do not explicitly write down the normal-ordered Coulomb

term, but they do seem to think that the entire Hamiltonian (in both spinor and scalar quantum

electrodynamics) should be fully normal-ordered. Greiner and Reinhardt [39, pg. 238] explain that

normal-ordering removes an undesirable kind of self-interaction, writing that “the prescription of

normal-ordering of the interaction operator eliminates the interaction of a particle with itself at the

same point

.” However, they do not directly discuss self-repulsion.

12

There are a number of competing proposals as to the nature of these quantum states. Elsewhere, I

have argued that states in quantum field theory should be viewed as wave functionals assigning quantum

amplitudes to classical field configurations [15].

13

See [35, eq. 8.3.12]; [36, sec. 6.4]; [40].

14

When Bjorken and Drell [33, sec. 17.9] use the Coulomb term in deriving the Feynman rules for

quantum electrodynamics, they only consider its contribution to diagrams involving interactions between

distinct particles. Thus, although they do not explicitly normal order the entire term, they treat it as

if it was normal-ordered.

where the charge density operator is

(

) =

−

†

(

)

(

) :

(14)

This partially normal-ordered Coulomb term (13) is the operator version of the classical

energy of electric repulsion in (8). Like that equation, it includes a form of electron

self-interaction. We will see shortly that it is the full normal-ordering in (12) that allows

us to avoid this self-interaction.

Before analyzing the role that the Coulomb term (12) plays in quantum

electrodynamics, let us first imagine that this term describes the only way in which

electrons and positrons interact and ask how the free dynamics are altered. Put another

way, we can start with a free quantum theory of the Dirac field and then see how

the dynamics change if the Coulomb term is added. When this term is added, we

get a quantum theory of a single self-interacting field that might be called “quantum

electrostatics,” because the only interaction that is included is the Coulomb interaction.

(Although the title “quantum electrostatics” is not in wide use, Kay [40] has used it

to refer to a different theory: a quantum theory of the electromagnetic field interacting

with a fixed classical distribution of charge.) Alternatively, the theory could simply be

called “quantum Dirac field theory with Coulomb interactions.” This theory is a natural

object of study, but I have not seen it examined elsewhere. One reason this theory might

be avoided is that it is not a relativistic quantum field theory (because we only include

instantaneous Coulomb interactions).

Let us briefly review the quantum theory of the Dirac field without any interactions.

In this theory, the dynamics (11) are given by the Hamiltonian operator

†

(

−

~

cγ

~γ

∇

)

x .

(15)

The time-independent (Schr ̈odinger picture) field operator can be written in its standard

plane wave expansion by putting the appropriate hats on (9),

(

) =

(

)

︷

︸︸

︷

~

)

√

(

)

∑

(

)

(

)

i

~

)

~

)

√

(

)

∑

(

†

(

)

(

)

−

i

~

)

︸

︷︷

︸

−

(

)

(16)

where

(

) is an electron annihilation operator,

†

(

) is a positron creation operator,

(

) is a field operator for the electron modes, and

−

(

) is a field operator for the

15

See, e.g., [17, sec. 8b]; [33, pg. 90]; [41, pg. 58]; [36, ch. 5].

positron modes. A single-electron quantum state can be written as a superposition of

momentum eigenstates

~p,s

〉

√

(

)

†

(

)

〉

, taking the general form

∑

(

~p,s

)

√

(

)

†

(

)

〉

(17)

where

〉

is the zero-particle vacuum state and

(

~p,s

) is an appropriately normalized

function of the momentum

and spin index

, determining the particular single-electron

state.

We can now consider how the dynamics change if we modify the Hamiltonian (15)

by adding the Coulomb interaction term in (12),

†

(

−

~

cγ

~γ

∇

)

†

(

)

(

)

†

(

)

(

) :

−

y .

(18)

Expanding the field operators in terms of

(

) and

−

(

) from (16), the Coulomb term

can be written as a sum of terms involving 0, 1, 2, 3, or 4 creation operators. There

are terms describing the creation of four particles (two electrons and two positrons), the

annihilation of four particles, an electron creating an electron-positron pair, etc. Here

are the terms that leave the total numbers of electrons and positrons unchanged:

−

∑

i,j

†

(

)

†

(

)

(

)

(

)

−

∑

i,j

−

(

)

†

(

)

†

−

(

)

(

)

−

∑

i,j

−

(

)

−

(

)

†

−

(

)

†

−

(

)

−

y .

(19)

The sums over the Dirac field indices

and

, that are implicit elsewhere, need to be

written out explicitly here because the normal ordering does not always place daggered

operators immediately to the left of their undaggered partners. The first term describes

Coulomb repulsion between electrons, the second describes Coulomb attraction between

electrons and positrons, and the last describes Coulomb repulsion between positrons.

As we are asking about the fate of electron self-repulsion and not about the

interactions between electrons and positrons or among positrons, let us now focus on the

one term in the expansion of the original Coulomb term (12) that includes only

electron

creation and annihilation operators,

−

∑

i,j

†

(

)

†

(

)

(

)

(

)

−

y ,

(20)

the first term in (19). Looking back to (16), we can recognize this term as having two

electron creation operators to the left of two electron annihilation operators. This term

resembles the energy of self-repulsion for a classical cloud of electron charge in the Dirac

field (10), but we will see shortly that it describes only Coulomb repulsion between

electrons and not electron self-repulsion.

If we time evolve a two-electron state by the Schr ̈odinger equation (11), (20) will alter

its free dynamics to include electron-electron repulsion. In the Schr ̈odinger evolution of

a single-electron state (17), (20) will have no effect because the sequence of two electron

annihilation operators will return zero when acting on this single-particle state. Thus,

we have included electron-electron repulsion and excluded electron self-repulsion. There

is no self-repulsion in the dynamics and no contribution to the energy from self-repulsion.

If we had used the incorrect normal-ordering in (13), then the piece of the Coulomb

term that contains only electron creation and annihilation operators would be

†

(

)

(

)

†

(

)

(

)

−

y ,

(21)

instead of the similar expression in (20).

Because (21) alternates creation and

annihilation operators, it would alter the evolution of a single-electron state. Thus,

(21) does not merely describe electron-electron repulsion. This term also includes a

kind of electron self-interaction that could be called “electron self-repulsion” because

(21) is the operator version of the classical energy of self-repulsion (10). By positing

that (12) is the correct Coulomb term to include in the Hamiltonian, not (13), we can

avoid this kind of electron self-interaction.

To fully understand the kind of electron self-interaction that has been removed,

and whether it indeed acts as a form of self-repulsion, would require further study.

One question that could be asked is whether a lone electron would rapidly expand as

its parts repel one another. That question can be put more precisely. One way of

understanding the quantum state that evolves by the Schr ̈odinger equation (11) is as

a wave functional that describes a quantum superposition of different classical field

configurations [15]. Each classical configuration of the Dirac field gives a precise charge

distribution and thus a superposition of field configurations is also a superposition of

different charge distributions. One could begin by using a superposition of compact

charge distributions to represent a lone electron in empty space and then ask whether

this superposition would evolve by the Schr ̈odinger equation (11) into a superposition

of more widely-spread wave packets, seeing if the above electron self-interaction (21)

would yield more rapid expansion than might occur under the free dynamics (15). To

better understand the above self-interaction, one could also ask how it would change

the shapes of atoms. For example, one could study the ground state of the hydrogen

atom and see if the superposed electron charge distributions are spread wider when the

partially normal-ordered Coulomb term (13) is used instead of the fully normal-ordered

Coulomb term (12). The electron cloud should be larger (and the ground state energy

higher) because, with self-repulsion included, the inner part of the electron’s charge

distribution would partially shield the outer part from the charge of the nucleus.

Let us set that self-interaction aside and return to the theory of quantum

electrostatics under investigation here (using the correct normal-ordering).

With

the full Coulomb interaction term (12) included in the Hamiltonian (18), not just

the electron-only term in (20) or the particle-number-preserving terms in (19), the

zero-particle vacuum state

〉

from the free theory is no longer the ground state (a

common feature of quantum field theories with interactions

). The Coulomb term will

generate particles from the vacuum. There should be a minimum energy ground state

Ω

〉

that evolves trivially under the dynamics (11) (only changing its global phase),

with

Ω

〉

Ω

〉

, where

is the ground state energy. Depending on what we

deem worthy of the title “particle,” one might either say that this ground state of

the interacting theory is the zero-particle vacuum state for that theory (even though

this state is distinct from the zero-particle state of the free theory) or that the ground

state of the interacting theory has particle content that can be treated as a background

upon which we can look for deviations. For our purposes here, let us adopt the latter

interpretation and view the ground state as containing particles. On this interpretation,

one can ask about the evolution of what might be called “single-extra-electron states”

that are arrived at by acting on the minimum energy ground state

Ω

〉

with electron

creation operators as in (17). These are states where an electron has been added to

the particle content of the ground state. The evolution for these states will be more

complicated than for the simple single-electron states discussed above. The electron-only

term in (20) will continue to omit self-repulsion, but it will include Coulomb repulsion

between the extra electron and background electrons.

Moving from the quantum theory of the Dirac field with Coulomb interactions

that we have been discussing to full quantum electrodynamics, quantum states become

more complicated (as they describe both the Dirac and electromagnetic fields) and the

evolution of states becomes more complicated (as the Hamiltonian includes more terms).

The Hamiltonian of quantum electrodynamics in the Coulomb gauge can be written as





∣

∂

∂t

∣

πc

(

∇×

)

+ :

†

(

−

~

cγ

~γ

∇

)

+ :

†

~γ

: +

†

(

)

(

)

†

(

)

(

) :

−

)

x ,

(22)

putting hats on the expression for the energy of the interacting Dirac and Maxwell

fields in (7) and normal-ordering every term. This Hamiltonian includes the Coulomb

interaction term (12) that we have been analyzing. In this theory, the evolution for a

single-electron state (17) will be altered from the free evolution by both the Coulomb

term—though not the parts in (19)—and the :

†

~γ

: term (which should account

for radiation reaction).

16

See the discussion of Haag’s theorem in [42, sec. 3]; [43, sec. 3]; [15, sec. 4.3].

5 Conclusion

If the electron is modeled classically as a cloud of energy and charge in the Dirac field,

then it will experience self-repulsion (section 3). This self-repulsion can be eliminated

in quantum field theory by fully normal-ordering the Coulomb term in the Hamiltonian

operator (section 4). The Hamiltonian of quantum field theory includes Coulomb

attraction and repulsion between distinct particles and excludes self-repulsion.

Acknowledgments

Thank you to Jacob Barandes, Maaneli Derakhshani, Michael

Miller, Logan McCarty, Simon Streib, and anonymous reviewers for helpful feedback

and discussion.

References

[1] Feynman, R. P., Leighton, R. B., and Sands, M. (1964) The Feynman Lectures on

Physics, Vol. II, Addison-Wesley Publishing Company.

[2] Rohrlich, F. (1973) The Electron: Development of the first elementary particle

theory. In Mehra, J., (ed.),

The Physicist’s Conception of Nature

, pp. 331–369 D.

Reidel Publishing Company.

[3] Rohrlich, F. (2007) Classical Charged Particles, World Scientific, 3rd edition.

[4] Pearle, P. (1982) Classical Electron Models. In Teplitz, D., (ed.),

Electromagnetism:

Paths to Research

, pp. 211–295 Plenum Press.

[5] Schwinger, J. (1983) Electromagnetic Mass Revisited.

Foundations of Physics,

(3), 373–383.

[6] Jackson, J. D. (1999) Classical Electrodynamics, Wiley, 3rd edition.

[7] Griffiths, D. J. (2012) Resource Letter EM-1:

Electromagnetic Momentum.

American Journal of Physics,

, 7–18.

[8] Blinder, S. M. (1965) Basic Concepts of Self-Consistent-Field Theory.

American

Journal of Physics,

(6), 431–443.

[9] Perdew,

J. P. and Zunger,

A. (1981) Self-Interaction Correction to

Density-Functional Approximations for Many-Electron Systems.

Physical Review

(10), 5048–5079.

[10] Parr, R. G. and Yang, W. (1989) Density-Functional Theory of Atoms and

Molecules, Oxford University Press.

[11] Levine, I. N. (2014) Quantum Chemistry, Pearson, 7th edition.

[12] Sebens, C. T. (2021) Electron Charge Density: A clue from quantum chemistry for

quantum foundations.

Foundations of Physics,

, 75.

[13] Barut, A. O. (1989) The Schr ̈odinger and the Dirac Equation - Linear Nonlinear

and Integrodifferential. In De Filippo, S., Marinaro, M., Marmo, G., and Vilasi,

G., (eds.),

Geometrical and Algebraic Aspects of Nonlinear Field Theory

, pp. 37–51

Elsevier.

[14] Barut, A. O. (1991) Foundations of Self-Field Quantumelectrodynamics. In Barut,

A. O., (ed.),

New Frontiers in Quantum Electrodynamics and Quantum Optics

, pp.

345–365 Plenum Press.

[15] Sebens, C. T. (2022) The Fundamentality of Fields.

Synthese,

200

(5), 380.

[16] Lange, M. (2002) An Introduction to the Philosophy of Physics: Locality, Energy,

Fields, and Mass, Blackwell.

[17] Schweber, S. S. (1961) Introduction to Relativistic Quantum Field Theory, Harper

& Row.

[18] Bjorken, J. D. and Drell, S. D. (1964) Relativistic Quantum Mechanics,

McGraw-Hill.

[19] Greiner, W. and Reinhardt, J. (2003) Quantum Electrodynamics, Springer-Verlag,

3rd edition.

[20] Schwartz, M. D. (2014) Quantum Field Theory and the Standard Model, Cambridge

University Press.

[21] Maudlin,

(2018)

Ontological

Clarity

via

Canonical

Presentation:

Electromagnetism and the Aharonov-Bohm effect.

Entropy,

(6), 465.

[22] Frisch, M. (2004) Inconsistency in Classical Electrodynamics.

Philosophy of Science,

(4), 525–549.

[23] Frisch, M. (2005) Inconsistency, Asymmetry, and Non-Locality: A philosophical

investigation of classical electrodynamics, Oxford University Press.

[24] Frisch, M. (2008) Conceptual Problems in Classical Electrodynamics.

Philosophy

of Science,

(1), 93–105.

[25] Frisch, M. (2009) Philosophical Issues in Electromagnetism.

Philosophy Compass,

(1), 255–270.

[26] Belot, G. (2007) Is Classical Electrodynamics an Inconsistent Theory?.

Canadian

Journal of Philosophy,

(2), 263–282.

[27] Lazarovici, D. (2018) Against Fields.

European Journal for Philosophy of Science,

(2), 145–170.

[28] Sebens, C. T. (2018) Forces on Fields.

Studies in History and Philosophy of Modern

Physics,

, 1–11.

[29] Sebens, C. T. (2021) Particles, Fields, and the Measurement of Electron Spin.

Synthese,

198

(12), 11943–11975.

[30] Duncan, A. (2012) The Conceptual Framework of Quantum Field Theory, Oxford

University Press.

[31] Sebens, C. T. (2020) Putting Positrons into Classical Dirac Field Theory.

Studies

in History and Philosophy of Modern Physics,

, 8–18.

[32] Sebens, C. T. (2021) The Disappearance and Reappearance of Potential Energy in

Classical and Quantum Electrodynamics.

arXiv preprint arXiv:2112.14643,

[33] Bjorken, J. D. and Drell, S. D. (1965) Relativistic Quantum Fields, McGraw-Hill.

[34] Hatfield, B. (1992) Quantum Field Theory of Point Particles and Strings,

Addison-Wesley, Frontiers in Physics, Volume 75.

[35] Weinberg, S. (1995) The Quantum Theory of Fields, Volume 1: Foundations,

Cambridge University Press.

[36] Tong, D. Lectures on Quantum Field Theory. (2007)

http://www.damtp.cam.ac.

uk/user/tong/qft.html

[37] Sebens, C. T. (2019) How Electrons Spin.

Studies in History and Philosophy of

Modern Physics,

, 40–50.

[38] Sebens, C. T. (2020) Possibility of Small Electron States.

Physical Review A,

102

052225.

[39] Greiner, W. and Reinhardt, J. (1996) Field Quantization, Springer-Verlag.

[40] Kay, B. S. (2020) Quantum Electrostatics, Gauss’s Law, and a Product Picture

for Quantum Electrodynamics; or, the Temporal Gauge Revised.

arXiv preprint

arXiv:2003.07473,

[41] Peskin, M. E. and Schroeder, D. V. (1995) An Introduction to Quantum Field

Theory, Westview Press.

[42] Earman, J. and Fraser, D. (2006) Haag’s Theorem and its Implications for the

Foundations of Quantum Field Theory.

Erkenntnis,

, 305–344.

[43] Fraser, D. (2008) The Fate of ‘Particles’ in Quantum Field Theories with

Interactions.

Studies in History and Philosophy of Modern Physics,

(4), 841–859.