# Gauge-underdetermination and shades of locality in the Aharonov-Bohm
effect111Preprint—accepted for Foundations of Physics (Springer). Please cite
published version, if available.

Ruward A. Mulder222Freudenthal Institute, History and Philosophy of Science
Department, Utrecht University, Princetonplein 5, 3584CC Utrecht, The
Netherlands. ✉<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

I address the view that the classical electromagnetic potentials are shown by
the Aharonov-Bohm effect to be physically real (which I dub: ‘the potentials
view’). I give a historico-philosophical presentation of this view and assess
its prospects, more precisely than has so far been done in the literature.
Taking the potential as physically real runs prima facie into ‘gauge-
underdetermination’: different gauge choices represent different physical
states of affairs and hence different theories. This fact is usually not
acknowledged in the literature (or in classrooms), neither by proponents nor
by opponents of the potentials view. I then illustrate this theme by what I
take to be the basic insight of the AB effect for the potentials view, namely
that the gauge equivalence class that directly corresponds to the electric and
magnetic fields (which I call the Wide Equivalence Class) is too wide, i.e.,
the Narrow Equivalence Class encodes additional physical degrees of freedom:
these only play a distinct role in a multiply-connected space. There is a
trade-off between explanatory power and gauge symmetries. On the one hand,
this narrower equivalence class gives a local explanation of the AB effect in
the sense that the phase is incrementally picked up along the path of the
electron. On the other hand, locality is not satisfied in the sense of signal
locality, viz. the finite speed of propagation exhibited by electric and
magnetic fields. It is therefore intellectually mandatory to seek desiderata
that will distinguish even within these narrower equivalence classes, i.e.
will prefer some elements of such an equivalence class over others. I consider
various formulations of locality, such as Bell locality, local interaction
Hamiltonians, and signal locality. I show that Bell locality can only be
evaluated if one fixes the gauge freedom completely. Yet, an explanation in
terms of signal locality can be accommodated by the Lorenz gauge: the
potentials propagate in waves at finite speed. I therefore suggest the Lorenz
gauge potentials theory—an even narrower gauge equivalence relation—as the
ontology of electrodynamics.

###### Contents

  1. 1 Preamble: What there is in the lecture halls
  2. 2 Outline of the problem and the argument
  3. 3 Aharonov-Bohm: non-locality and gauge potentials
  4. 4 Electrodynamics in terms of potentials
  5. 5 ‘Gauge-underdetermination’ and equivalence classes
  6. 6 Gauge-underdetermination in the scientific literature
  7. 7 Desiderata: Bell locality and signal locality
  8. 8 Local interaction and narrowing equivalence classes
  9. 9 A suggestion: the Lorenz Equivalence Class
  10. 10 Conclusion

##  1 Preamble: What there is in the lecture halls

How do our theories of physics relate to what actually exists in the world? In
the everyday life of physicists, both in the lab of the experimenter and on
the paper of the theorist, such a question is rarely helpful for directly
solving specific problems. Nevertheless, questions like ‘what is the quantum
state exactly?’, ‘is there really energy in the world?’ or ‘does light
actually consist of photons or of electromagnetic waves?’ are commonplace in
the minds of students. Often, such questions are evaded, as if asking for
things that are not obscure by accident and into which one should not delve
too deeply. Unmistakeably, part of this reluctance about dealing with such
questions lies in the difficulty of conceptual reasoning in mathematical
theories that are designed to solve practical problems.

In lectures on electrodynamics during undergraduate years in physics, the
introduction of the electromagnetic potentials as follows is commonplace:

> The fact that Maxwell’s equations show that the electric and magnetic fields
> can propagate on their own and carry energy over long distances, shows that
> they are physically real. Let us now introduce the electromagnetic
> potentials through their relation with these fields,
> $\textbf{E}=-\nabla\phi-\partial\textbf{A}/\partial t$ and
> $\textbf{B}=\nabla\times\textbf{A}$. These are introduced for mathematical
> convenience, mind you, and have more degrees of freedom than the actual
> physical degrees of freedom.

This appears unproblematic: the ‘actual physical’ degrees of freedom are
encoded in the E and B fields and the role of the ‘extra’ structure can be
understood as convenient for the practical purposes of the calculations that
await the student. After all, the task of finding the value of the potential
between two conducting plates that is high enough to ionize the hydrogen atom
might not leave much time to question the physical nature of that voltage.

But what is to stop an eccentric from adhering to a ‘potentials view’, where
$\phi$ and A correspond to real things in the world? Empirical data does not
rule out such a move, since the potential theory is empirically equivalent (at
least in the classical domain). Of course, the term ‘Ockham’s razor’ (which is
often bandied about) applies in the sense that we presumably should not add
extra structure when it is unnecessary. Yet opinions over what is necessary
diverge. Particularly for these two electromagnetic theories—apparently making
the same predictions—additional criteria come into play.

Near the end of the semester, the professor admits that his initial claim,
that the potentials merely serve an auxiliary role, has a caveat:

> Remember the claim that $\phi$ and A are merely convenient constructs that
> only help us solve problems? It turns out that in quantum theory this is no
> longer really true, because of the Aharonov-Bohm effect. I will not go into
> it now, but you will see that gauge degrees of freedom have a direct effect
> on quantum observables.

Although it makes for an exciting cliff-hanger, the student is often left
hanging there. When the AB effect is discussed in the subsequent advanced
quantum mechanics courses, the reasons for taking $\phi$ and A to be
physically real, and the nature of gauge degrees of freedom, are often
omitted. Indeed, although some sense of locality is the sole reason for taking
$\phi$ and A as real, the word ‘locality’ is rarely mentioned.

##  2 Outline of the problem and the argument

The Aharonov-Bohm effect shows that there are situations in which the phase of
the wavefunction of a charged quantum probe is influenced at locations where
the magnetic (and electric) field vanishes, whereas the vector potential A
does not. Thus, in brief, one must conclude that either the magnetic field B
acts in a non-local way or the A-field plays a physical role.333This dichotomy
of course assumes that there is no third candidate that provides a mechanism
to explain the effect but that is hidden somewhere in the formalism and
therefore overlooked; cf. footnote 4. But if the former: in what way? And if
the latter: in what way? The Aharonov-Bohm effect (AB effect) thereby
highlights the difficulty of understanding theories that admit local gauge
transformations; it puts a question mark on our understanding of what gauge
transformations are in the first place.

This paper is a historico-philosophical evaluation of the ‘potentials view’ as
opposed to the ‘fields view’. On the potentials view—adhered to by most
physicists—the potentials, rather than the E and B fields, are taken as the
physical players of the theory, so that the potentials are not just auxiliary
mathematical conveniences.

In experimental physics, aspects of the AB effect have flourished in recent
years (for example the occurrence of a magnetic edge in graphene rings [1]),
but progress has been slower in finding an explanation of the AB effect, as
pointed out by Batelaan and Tonomura in 2009 [2, p.38] “the investigation and
exploitation of the AB effect remain far from finished.” Tran recently
challenged the idea that the formulation of Maxwell’s equations is settled
science, even in the classical case [3]. Although alternative ways of deriving
the effect in didactic ways are promoted, for example by focusing on the de
Broglie wavelength [4], the conceptual consequences of the potentials view are
rarely taught. Berry adopts the potentials view by claiming that the “gauge-
invariant part of the vector potential was promoted to a real physical field,
not just a convenient device for summarizing certain information about the
electric and magnetic fields” [5].

To make things more precise, let me define the two views that involve an
interpretation of the potentials.

> Potentials view: the scalar potential field $\phi(\textbf{x})$ and vector
> potential field $\textbf{A}(\textbf{x})$ are physical (that is, more real
> than mathematical fictions), over and above the degrees of freedom that give
> rise to the electric field $\textbf{E}(\textbf{x})$ and magnetic field
> $\textbf{B}(\textbf{x})$. A short-hand notation for this interpretation will
> be ($\phi,\textbf{A}$)-theory.

This is in contrast with the rival view:

> Fields view: only the degrees of freedom that directly give rise to the
> electric field $\textbf{E}(\textbf{x})$ and magnetic field
> $\textbf{B}(\textbf{x})$ are physical and any additional (mathematical)
> degrees of freedom of the potentials are pure mathematical fictions. A
> short-hand notation for this interpretation will be
> ($\textbf{E},\textbf{B}$)-theory.

The fields view is the traditional interpretation, but it is logically quite
strong: its defenders share the belief that the electric and magnetic fields
are sufficient for explaining all electrodynamic phenomena. On the contrary,
within the potentials view there is ample leeway. The reason is that one has
to choose where to draw the line between physical and fictional degrees of
freedom, resulting in many different positions—none of which are
uncontroversial444Here I give a brief review of prominent interpretations of
electrodynamics in light of the AB effect. The first is the holonomies view,
advocated by Healey, in which the loop integrals of the potentials are
promoted to physical ontology [6]. Unlike the potentials, these loops are
gauge-invariant quantities; unlike the fields, they are non-locally possessed
properties which nonetheless act locally—and at least they are defined in the
region where the electrons move [7]. Another view is taken by Mattingly, who
argues that gauge fields do not commit us to any novel ontology and that the
effect should be understood in terms of the sum of 4-current fields of single
charged particles [8]. Here, ‘distributivity’ fails in the sense that the
electron is sensitive to the component fields, even while the net field
vanishes. The 4-current field would carry information but no energy-momentum
[9]. Boyer argues that the AB effect is explained by the back-reaction of the
E and B fields of the charged particles themselves, which interact with the
solenoid [10]. Vaidman argues that the potentials merely seem real due to the
stringent canonical formulation of quantum dynamics (the Schrödinger equation
necessarily deals with a potential instead of fields or forces: cf. Eq.
(3.1)). Therefore, Vaidman calls upon the community to seek a reformulation of
quantum mechanics in terms of the electric and magnetic fields and without
potentials [11, 12]. Aharonov himself, together with Cohen and Rohrlich
replied to Vaidman that the effect may be due to a local gauge potential or
due to non-local gauge-invariant fields [13, 14]. Clarity on these issues is
especially important in the light of the calculations performed by Pearle and
Rizzi, who have worked out Vaidman’s idea of including the solenoid in the AB
experiment into a fully quantum-mechanical description [15]. This ‘Vaidman-ACR
debate’, as we might dub it, is a contemporary example of a dispute in which
extra-empirical values play a crucial role in physical practice. I intend to
evaluate this particular debate in future work.—but which share commitment to
the physicality of all or some of the degrees of freedom which were considered
unphysical degrees of freedom by the traditional fields view.

In what follows, I will initially be concerned with the naive version of the
potentials view which says ‘the potentials are real’ without specifying what
we mean by that. This will involve some fuzziness about both locality and
reality criteria; but the initial focus on the naive view helps to tease out
the philosophical commitments that play a role in these controversies. Also,
this view appears to me to be the most straightforward way to give a local
explanation of the AB effect. In the second half of the paper, the focus will
shift to the more precise language of equivalence classes in order to sharpen
intuitions about locality and ontology.

I will approach the debate over the ($\phi$,A)-theory or (E,B)-theory from the
viewpoint of underdetermination, which sheds light on the role of locality and
reality in theory choice. I explore the Aharonov-Bohm effect and the
potentials formalism in sections 3-4 in such a way that all steps in the
calculations are present. Then, I emphasize that if one regards the potentials
as completely real, as many do, there must also be a preferred gauge. If one
does not choose a preferred gauge, it is impossible to avoid what I call
‘gauge-underdetermination’ (section 5). Thus, one needs either to appeal to a
criterion for the preferred gauge or to weaken the statement that the
potentials are real. This is often brushed over or misunderstood in the
literature about gauge theories, as illustrated by excerpts from influential
authors (section 6), sometimes leading to inconsistencies in the treatment of
the AB effect.

Figure 1: A spectrum of equivalence relations on the potentials and
corresponding philosophical positions with wider relations towards the right.
On the far right, one has the fields view for which only E and B are real.
Everything to the left of this posits more physical degrees of freedom and is
hence within the potentials view. The Narrow Equivalence Class, which is the
one most often used in contemporary physics, encodes fewer gauge degrees of
freedom—it coincides with the fields view in simply-connected spacetimes. One
can narrow this down further by additional constraints on the local gauge
function $\chi$, such as my suggested Lorenz gauge. Fixing all degrees of
freedom, for example in Coulomb gauge, would lead to Maudlin’s ‘One True
Gauge’ potentials theory. For completeness: such a position would still
involve a residual but trivial equivalence class, reflecting possible choices
of units. The reader is invited to find additional positions.

To find criteria that soften gauge-underdetermination, one must recognize that
cutting down on gauge-underdetermination and narrowing one’s gauge equivalence
class are two sides of the same coin. To do this, it is natural (but not
necessary) to remain close to the original motivation: locality. In section 7,
I survey several precise formulations of a locality condition, such as Bell
locality, local interaction Hamiltonians, separability and signal locality. In
section 8, I review the reasoning that led Aharonov and Bohm to infer the
reality of the potentials. I discuss how this can be read as narrowing down
the ‘old’ Wide Equivalence Class of admitted gauge symmetries
($\textbf{A}\mapsto\textbf{A}^{\prime}:=\textbf{A}+\textbf{C}$ such that
$\nabla\times\textbf{C}=0$) to the Narrow Equivalence Class
($\textbf{A}\mapsto\textbf{A}^{\prime}:=\textbf{A}+\nabla\chi$), so as to
admit some physical degrees of freedom of the potential in addition to those
that give rise to E and B. The (modern) Narrow Equivalence Class is local in a
sense, namely that the electron picks up a phase incrementally along its path.
The kind of locality I then suggest we should seek is signal locality, similar
to the locality exhibited by E and B, since they propagate with a finite
velocity. With that knowledge, I suggest that one can go a step further and
extend this narrowing of the gauge equivalence class: I argue in section 9
that the ‘Lorenz Equivalence Class’ satisfies that desideratum.

The overall argument of the paper thus consists of there being several options
on a spectrum of gauge equivalence classes, as shown in Figure 1—according to
one’s commitments to locality and reality, one can explore additional
positions to those shown.

##  3 Aharonov-Bohm: non-locality and gauge potentials

Due to their prediction of the effect that came to carry their names, David
Bohm and his PhD student Yakir Aharonov [16] argued that a “further
interpretation of the potentials is needed in quantum mechanics.” Their
solution, which was adopted by many physicists, was to promote the potentials
from mathematical fiction to something physical.555This effect had been semi-
classically calculated by Ehrenberg and Siday in 1949 [17], and might
therefore be called the Aharonov-Bohm-Ehrenberg-Siday effect. Aharonov and
Bohm clearly stressed the metaphysical importance of this result, which seems
to be the main reason why the phenomenon carries their name.

As illustrated in Fig. 2, Aharonov and Bohm envisage the following
experimental set-up, which originated as a thought experiment but was shortly
afterwards experimentally realized by Robert Chambers (who is mentioned in
their original paper). A coherent electron beam is split and directed around a
solenoid and brought together again in a region where interference can be
detected at a screen. The solenoid can be imagined as an infinite tightly-
wound coil (so that the current is strictly circular, without a perpendicular
component), which in turn ensures, by the usual magnetostatic symmetry
arguments, that the magnetic field is confined to inside the solenoid (in the
direction perpendicular to the plane of the figure). The solenoid itself is
then shielded, so that the wavefunction of the electrons is excluded from the
region occupied by the solenoid.

In a canonical formulation, external magnetic fields are coupled to the
electron via the electromagnetic potential. The Hamiltonian for an electron of
mass $m$, momentum p and charge $q$ in an external magnetic field is

$H=\frac{1}{2m}\left(\textbf{p}-q\textbf{A}\right)^{2},$ (3.1)

where A is stationary [18]. If the electron wavefunction in the absence of the
magnetic field is given by $\psi_{0}(\textbf{x})$, the presence of the
magnetic field forces us to add a phase to the electron wavefunction—the Dirac
phase factor [19]—calculated by solving the time-independent Schrödinger
equation,

$\psi(\textbf{x})=\psi_{0}(\textbf{x})\exp\left(\frac{iq}{\hbar}\int\textbf{A}\cdot
d\textbf{x}\right),$ (3.2)

where the phase factor is given by the line integral of the vector potential
over some path in the region where the electron is allowed to move (that is,
it can take any path outside the solenoid).666In the four-vector formalism one
can accommodate the time-dependent version of the AB effect with the Dirac
phase factor
$\left(q/\hbar\right)\oint_{\gamma}A_{\mu}dx^{\mu}=\left(q/\hbar\right)\oint_{\gamma}\left[\textbf{A}\cdot
d\textbf{x}-\phi dt\right]$ over some closed path $\gamma$ through spacetime.
I will steer clear of using covariant notation to (i) keep the discussion
accessible to the undergraduate level, which it conceptually surely is, and
(ii) to prevent the activation of already-possessed knowledge of the
experienced gauge-theorist, particularly the tendency to readily interpret
only gauge-invariant quantities as physical without reconsidering which
symmetry it is based on.

Figure 2: Experimental set-up to show the AB interference effect with time-
independent vector potential as a result of switching on the current through
the solenoid. The usual two-slit interference pattern is translated upward by
an amount $\Delta x=-q\lambda_{B}l\Phi/2\pi\hbar d$. Figure from [21, p.7].

Decomposing the wavefunction of the electron into a superposition of two
parts, $\psi(\textbf{x})=\psi_{1}(\textbf{x})+\psi_{2}(\textbf{x})$, one
describing the electron following a path $1$ in the clockwise direction around
the solenoid and the other the counter-clockwise path $2$, the phase
difference between these paths in the interference region is

$\frac{\Delta S}{\hbar}=-\frac{q}{\hbar}\left(\int_{\text{path
1}}\textbf{A}\cdot d\textbf{x}-\int_{\text{path 2}}\textbf{A}\cdot
d\textbf{x}\right)=-\frac{q}{\hbar}\oint_{C}\textbf{A}\cdot d\textbf{x},$
(3.3)

which is a line integral over the closed path $C$ defined by traversing path
$1$ and then, in reverse, path $2$. The two phases picked up along the paths
do not cancel, but add to give the loop integral because of the opposite
orientations of the integrals of the two paths.

But what fixes the path of integration so that we can actually evaluate the
integral? After all, a quantum particle can be regarded as taking all possible
paths. This is rather subtle because the value of A is subject to a gauge
condition. The approach is as follows. For some closed loop $\gamma$, using
the familiar but tailor-made gauge freedom
$\textbf{A}\mapsto\textbf{A}^{\prime}=\textbf{A}+\nabla\chi$, we find the
following equality

$\oint_{\gamma}\textbf{A}^{\prime}\cdot
d\textbf{x}=\oint_{\gamma}\left(\textbf{A}+\nabla\chi\right)\cdot
d\textbf{x}=\oint_{\gamma}\textbf{A}\cdot d\textbf{x},$ (3.4)

where the second equality holds due to the identical vanishing of a gradient
around a closed path. Eq. (3.4) shows that the loop integral
$\oint_{\gamma}\left(\textbf{A}^{\prime}-\textbf{A}\right)\cdot d\textbf{x}=0$
vanishes.777To visualize this better, we can, following Binney and Skinner
[20, Sec. 3.3.3], consider the form of the vector potential for an infinitely
thin solenoid (at the origin and in the $\hat{z}$-direction) in cylindrical
polar coordinates, so that $\textbf{A}=\frac{\Phi}{2\pi
r^{2}}\left(-y,x,0\right)$ for $r=\sqrt{x^{2}+y^{2}}$ and $\Phi$ the magnetic
flux (cf. Eq. (3.6)). Now choose the gauge such that this vector field
vanishes, namely $\nabla\chi=\frac{\Phi}{2\pi r^{2}}\left(y,-x,0\right)$,
which is achieved by choosing $\chi=-\frac{\Phi}{2\pi}\theta$ for the polar
angle $\theta=\arctan\left(y/x\right)$. In other words, the line integral over
$\int_{\textbf{x}_{1}}^{\textbf{x}_{2}}\left(\textbf{A}^{\prime}-\textbf{A}\right)\cdot
d\textbf{x}$ depends only on the initial position $\textbf{x}_{1}$ and final
position $\textbf{x}_{2}$, not on the path taken. The reason why I call this
gauge freedom tailor-made is that it is specifically designed to make this
integral vanish, making it narrower than a more general gauge symmetry
$\textbf{A}\mapsto\textbf{A}^{\prime}=\textbf{A}+\textbf{C}$ with
$\nabla\times\textbf{C}=0$, as I will discuss in section 5.

What this calculation implies is a shift in the interference pattern depending
on the magnetic vector potential A. When the distance $l$ between the screen
and the slits is much larger than the size of the screen in the interference
region (given by the position variable $x$), the usual two-slit interference
pattern, $\Delta S_{\text{two slit}}/\hbar=2\pi xd/\lambda_{B}l$ (where
$\lambda_{B}=h/|\textbf{p}|$ is the de Broglie wavelength of the electron and
$d$ is the separation between the slits), shifts upward (or downward,
depending on the direction of the current) by an amount888The early
experiments by Chambers and by Moellenstedt and Bayh did not involve slits.
Also, Timothy Boyer has kindly pointed out to me that Figure 2 and my phrasing
incorrectly suggests that the whole interference pattern is displaced sideways
undisturbed, whereas in reality the double-slit interference pattern changes,
not the single-slit envelope [22]: an inaccuracy or lack of nuance that he
traces back to the Feynman Lectures. Here, it is sufficient to understand the
problem through Figure 2.

$\Delta x=\frac{\lambda_{B}l}{2\pi d}\frac{\Delta
S}{\hbar}=-\frac{\lambda_{B}l}{2\pi d}\frac{q}{\hbar}\oint_{C}\textbf{A}\cdot
d\textbf{x}=-\frac{\lambda_{B}l}{2\pi d}\frac{q}{\hbar}\Phi.$ (3.5)

To identify $\Phi$ as the magnetic flux through the region that is enclosed by
the loop $C$, one uses Stokes’ theorem to rewrite this phase shift in terms of
the magnetic field B through surface area $S$ with the loop $C$ as its
boundary,

$\oint_{C}\textbf{A}\cdot
d\textbf{x}=\int\left(\nabla\times\textbf{A}\right)\cdot
d\textbf{S}=\int\textbf{B}\cdot d\textbf{S}:=\Phi.$ (3.6)

That we can write the loop integral in terms of the magnetic field is not a
remarkable result by itself, since it is, after all, expected that magnetic
fields influence electrons’ behaviour. The remarkable fact is that the
magnetic field is confined to the region inside the solenoid and is zero
outside. The electrons experience a shift even though they never move through
a region of non-zero magnetic field: but the shift still depends on the
current.

Aharonov and Bohm argue that because “in a field-free multiply-connected
region of space, the physical properties of the system still depend on the
potentials,” we should promote the potentials from fiction to being physical:

> [t]he Lorentz force
> $\left[e\textbf{E}+\frac{e}{c}\textbf{v}\times\textbf{B}\right]$ does not
> appear anywhere in the fundamental theory, but appears only as an
> approximation holding in the classical limit. It would therefore seem
> natural at this point to propose that, in quantum mechanics, the fundamental
> physical entities are the potentials, while the fields are derived from them
> by differentiations [16, p.490].

The important point is that Aharonov and Bohm appeal to some kind of
explanation of the AB phase using the tools at hand in the mathematical
formalism of the theory. Even if one is in principle willing to allow a non-
local influence of the B-field on the electron,999Indeed, David Bohm himself
would not be so bothered by non-local explanations in physics. the structure
of that B-field says nothing about how this non-local influence would work.
After all, the B-field satisfies the wave equation so that it propagates at a
finite speed $c$, so that it surely cannot be invoked as acting non-
locally.101010In the jargon I introduce in section 7, B satisfies signal
locality, which is here fulfilled by satisfying the wave equation (given that
B is physical). Hence, the (E,B)-theory puts no explanation of this phenomenon
on the table.

Hence, the AB effect directly challenges the fields view. The predictions of
the combination of classical (E,B)-theory and quantum-mechanical test
particles fail, forcing a reconsideration in either one of these theories. One
option would be to search for a non-local law in quantum mechanics, which at
first sight sounds reasonable in view of the violation of the Bell
inequalities, as will be discussed in section 7. The other option, which is
widely adopted and will now be explored in more detail, is to follow Aharonov
and Bohm’s advice in promoting the potentials to ‘fundamental ontology’.

##  4 Electrodynamics in terms of potentials

It makes a difference which parts of the mathematical formalism of a theory
are seen as physical. Two theories that share the same formalism can differ in
their assessment of the physicality of part of that formalism. Those parts
that are supposed to correspond (in some admittedly philosophically
controversial way111111According to Quine, the ontology is given by the domain
you quantify over [23]; closest to that in spirit is a primitive ontologist
who postulates an explicit fundamental ontology together with axioms [24];
structural realists say that the ontology is provided by the mathematical
structures used in the axioms [25]. The language in this paper is
naturalistically inclined: I will follow most physicists in labelling those
objects that are supposed to be in the world as ‘physical’.) to entities in
the actual world, I will call ‘fundamental ontology’: the physical building
blocks according to the theory (I say ‘according to the theory’ to prevent the
connotation of ‘ultimate building blocks’, i.e., I take fundamentality as a
relation between theories). Yet, regardless of the way the relation between
theory and the world is fleshed out, it is clear that it is entirely
possible—even the historical norm—that what parts of the formalism are
considered to be physical becomes contested in times when a new theory is
needed.

In electrostatics, the electric field can be interpreted as merely encoding
the propensity of a test charge to accelerate under the (non-local) Coulomb
force. In electrodynamics, Maxwell’s equations do not deal directly in terms
of forces, but with the fields and the source charges. Electrodynamical
phenomena can be described by Maxwell’s equations, which, in vacuum, read:
Gauss’ law, $\nabla\cdot\textbf{E}=0$; Faraday’s law,
$\nabla\times\textbf{E}=-\partial\textbf{B}/\partial t$; Gauss’ law for
magnetism, $\nabla\cdot\textbf{B}=0$; and the Maxwell-Ampère law,
$\nabla\times\textbf{B}=(1/c^{2})\partial\textbf{E}/\partial t$. These
equations lead121212Taking the curl of Faraday’s law, using the vector
identity
$\nabla\times(\nabla\times\textbf{a})=\nabla(\nabla\cdot\textbf{a})-\nabla^{2}\textbf{a}$,
invoking the constraint that is Gauss’ law and using the Maxwell-Ampère law.
to electromagnetic waves with speed $c$, which can travel for millions of
years in the absence of any nearby charges. These fields carry energy and
momentum, as expressed by the Poynting vector
$\textbf{S}\propto\textbf{E}\times\textbf{B}$, which gives the energy flux of
an electromagnetic wave. This leads one to interpret the electric and magnetic
fields as fundamental ontology, as opposed to a mere propensity of a test
charge to accelerate when subject to the Coulomb force. This interpretive move
is made in every course on electrodynamics. Besides, here I should mention the
unification of optics and electromagnetism—and it is this explanation that
convinces the student that there is something more to the vector fields E and
B than meets the eye: light.

We can attempt a similar step to favour the ($\phi$, A)-theory over the
(E,B)-theory. An alternative formulation in terms of the scalar potential
$\phi$ and vector potential A adds some extra structure to the fields, from
which the field can be obtained via the equations

$\textbf{E}=-\nabla\phi-\frac{\partial\textbf{A}}{\partial t},$ (4.1)

and

$\textbf{B}=\nabla\times\textbf{A}.$ (4.2)

These definitions are chosen such that Gauss’ law of magnetism and Faraday’s
law are trivially satisfied. With these definitions, the two dynamical
equations (the time-derivatives in Faraday’s law and the Maxwell-Ampère’s law
encode the evolution of the fields) become

$\nabla^{2}\phi=-\frac{\partial}{\partial
t}\left(\nabla\cdot\textbf{A}\right),$ (4.3)

and (using the same vector identity as in footnote 12)

$\nabla^{2}\textbf{A}-\frac{1}{c^{2}}\frac{\partial^{2}\textbf{A}}{\partial
t^{2}}=\nabla\left(\nabla\cdot\textbf{A}+\frac{1}{c^{2}}\frac{\partial\phi}{\partial
t}\right).$ (4.4)

The form is less neat than in the (E,B)-theory, but this shows that a
formulation in terms of potentials is possible without making reference to the
electric and magnetic fields at all.

Both formulations are empirically equivalent since their predictions—at least
in the known domains that have been probed—will not depend on any of the
‘extra’ structure introduced by the definitions Eqs. (4.1)-(4.2). The reason
is that the values of the electric and magnetic fields we can measure, and the
trajectories of charged particles in these fields as determined by the Lorentz
force law, remain the same regardless of the choice between the (E,B)- or
($\phi$,A)-theories.

Here is a clear case of what philosophers call ‘underdetermination of theory
by data’. So the following position can be maintained: an eccentric can claim
that we should postulate the potentials $\phi$ and A as directly related to
the ‘real’ things in the world, instead of the electric and magnetic fields.
That position is the most straightforward version of the potentials view. Note
that the (E,B)-theory and the ($\phi$,A)-theory are regarded as two different
theories because they have different fundamental ontologies. In the real
world, including special circumstances in the lab, we see the potentials
$\phi$ and A just as indirectly as the fields E and B, although we might
intuitively feel more comfortable with the latter.

On merely empirical grounds the case must remain undecided as long as one
cannot give an independent reason to take the potentials seriously as real
things. But the Aharonov-Bohm effect provides precisely such an independent
reason, since the ($\phi$,A)-theory provides local explanations of the AB
phase shift in Eq. (3.5), whereas the (E,B) theory cannot give such an
explanation. This is why many endorse the potentials view. In the next section
I will argue that, if one is to take the potentials view, then further steps
have to be considered in order to have a well-defined theory—a fact that often
goes unmentioned.

##  5 ‘Gauge-underdetermination’ and equivalence classes

Gauge symmetry is usually formalized by the transformations

$\phi\mapsto\phi^{\prime}:=\phi-\frac{\partial\chi}{\partial t},$ (5.1)

and

$\textbf{A}\mapsto\textbf{A}^{\prime}:=\textbf{A}+\nabla\chi,$ (5.2)

where $\chi(\textbf{x},t)$ is an arbitrary scalar function, depending on both
space and time coordinates. The fields E and B are left unchanged regardless
of the function $\chi$, thanks to the definitions (4.1) and (4.2). Hence, if
one is only interested in E and B, one can choose a $\chi$ that simplifies the
derivation of a solution to a given physical problem. In principle, an
infinity of gauges is possible, as the scalar function $\chi$ is arbitrary.

For example, Maxwell’s equations in the potentials formulation can be given a
simpler form by choosing the Coulomb gauge (often used in magnetostatics),

$\nabla\cdot\textbf{A}=0.$ (5.3)

In other words, the gauge parameter $\chi$ is constrained by
$\nabla^{2}\chi=0$, which has a unique solution and is therefore a complete
gauge-fix. Gauss’ law (4.3) and the Maxwell-Ampère law (4.4) then become

$\nabla^{2}\phi=0,$ (5.4)
$\nabla^{2}\textbf{A}-\frac{1}{c^{2}}\frac{\partial^{2}\textbf{A}}{\partial
t^{2}}=\frac{1}{c^{2}}\frac{\partial}{\partial t}\nabla\phi.$ (5.5)

Eq. (5.4) is simply the Laplace equation, but the Maxwell-Ampère law has mixed
space- and time-derivatives.

Alternatively, if one were to choose the Lorenz gauge,131313Not Lorentz,
although often this choice of gauge is connected to Hendrik Antoon Lorentz
instead of Ludvig Lorenz: a confusion well documented by van Bladel [26]. it
precisely cancels that mixed term, by requiring

$\nabla\cdot\textbf{A}+\frac{1}{c^{2}}\frac{\partial\phi}{\partial t}=0.$
(5.6)

In other words, the gauge parameter $\chi$ is constrained by
$\nabla^{2}\chi=1/(c^{2})\partial^{2}/dt^{2}$, which is an incomplete gauge
condition, as there is residual freedom in the form of a scalar wave (one can
choose a further fix that leaves no residual freedom, for example by setting
$\phi=0$, which is the so-called temporal gauge). In the Lorenz gauge, Eqs.
(4.3) and (4.4) reduce to

$\nabla^{2}\phi=\frac{1}{c^{2}}\frac{\partial^{2}\phi}{\partial t^{2}}$ (5.7)

and

$\nabla^{2}\textbf{A}=\frac{1}{c^{2}}\frac{\partial^{2}\textbf{A}}{\partial
t^{2}},$ (5.8)

resulting in the wave equation for both the vector and scalar potentials.
There is also a residual gauge degree of freedom, which can be fixed by an
initial condition that specifies $\phi$ at spatial infinity—which I discuss
further in section 9.

Equations (5.1)-(5.2) define an equivalence class. Thus it is said that any
choice of $\chi$ will lead to the same physics. But one has to first agree on
what the physics is. In this case, the usual argument is that this equivalence
class leaves the electric and magnetic fields invariant. This is of course
true. Yet, anticipating section 8 and turning to what I called (just below Eq.
(3.4)) the tailor-made gauge symmetry, there is, in fact, a wider equivalence
class one can consider which leaves the fields invariant. Namely, for any C
and $C_{0}$ such that $\nabla\times\textbf{C}=0$ and $\nabla
C_{0}=\partial\textbf{C}/\partial t$, one transforms

$\phi\mapsto\phi^{\prime}:=\phi-C_{0},$ (5.9)

and

$\textbf{A}\mapsto\textbf{A}^{\prime}:=\textbf{A}+\textbf{C}.$ (5.10)

This defines an equivalence class of potentials, related to each other by the
choice of gauge vector C and gauge scalar $C_{0}$. This class is wider than
the class defined by Eqs. (5.1)-(5.2), where the gauge freedom is given only
by the possible choices of the scalar $\chi$. There is more freedom in the
gauge vector C than in the gauge scalar function $\chi$, even though both
classes give the same electric and magnetic fields (for Eq. (4.2) due to the
imposed requirement that the rotation of C vanishes and for Eq. (4.1) because
the rotation of a gradient vanishes identically).

Therefore, I will speak of the Wide Equivalence Class defined by (5.9)-(5.10)
and the Narrow Equivalence Class defined by (5.1)-(5.2). Agreed: there is no
difference between the two classes in a simply-connected space. Yet there are
no local facts that allow us to figure out how our space is connected. It is
precisely the legacy of Aharonov and Bohm, whose thought experiment introduces
a multiply-connected space (the solenoid ‘punctures’ the space, as it were,
introducing a conical singularity), that one now works with the Narrow
Equivalence Class, which provides additional degrees of freedom that can in
some sense account for the AB phase shift. Section 8 is devoted to what sense
this is. This relates to the goal of this paper: to shed light on the path of
narrowing this class down even further to obtain a clear local ontology.

Back to the naive potentials view: the “eccentric” who wants to hold on to
regarding $\phi$ and A as completely real cannot claim that a choice of gauge
‘does not change the physics’. For her, the potentials are the physics and,
hence, every different $\chi$ corresponds to a different theory. Each choice
of $\chi$ leaves the fields E and B unchanged; and hence these fields
underdetermine the theory, on the potentials view. However, down this path one
quickly encounters a straightforward problem. There is no independent way to
determine the value of $\chi$, since it can no longer be seen as simply a
pragmatic tool for the purposes of calculation. In other words, the
predictions are ambiguous; the equations of motion will have non-unique
solutions and one is left with an indeterministic theory.

On the potentials view, determinism manifestly fails as long as $\chi$ is not
fixed. For one can make an argument analogous to the ‘hole argument’ that
exploits diffeomorphisms in general relativity [27]. Suppose the fields view
and the potentials view are related by a time-dependent gauge transformation
that is the identity up to a time $t$. Hence, they agree up to $t$, but
diverge afterwards. If different values of $\chi$ lead to different facts
about the real world—which the fields view does not claim but the potentials
view does—this amounts to indeterminism. For there are multiple ways that
$\phi$ and A can evolve in time.

The ($\phi$,A)-theorist should not want to endorse such
indeterminism—certainly not in response to an eminently predictable phenomenon
such as the AB effect—since the theory would be unable to make definite
predictions (although it could restrict the predictions to merely E and B).
Not even the option to assign probabilistic weights to the alternatives, that
is used in textbook quantum mechanics, is available to her. Therefore, the
($\phi$,A)-theorist should embrace that Eqs. (5.1) and (5.2) represent a set
of theories, parametrized by $\chi$. Hence, the flipside of gauge-
indeterminism is what I call ‘gauge-underdetermination’. But this
underdetermination is only problematic if one leaves $\chi$ unfixed.

The conclusion then seems unavoidable. If one claims that the potentials are
completely physical, the potentials view leads to indeterminism: the exact
same values of E and B can be predicted using different gauges which
correspond to different futures of $\phi$ and A given fixed initial
conditions. The way to solve this and recover the ability to make unique
predictions is to regard only one of the gauge choices as genuinely physical.
Maudlin141414Tim Maudlin, incidentally, does not defend this position as his
personal solution to the ontological problems posed by the AB effect. The
position originated in a response to Healey’s ‘mid-way’ position about the
reality of the potentials. has called this the ‘One True Gauge’ principle
[28]. Hence, it is not arbitrary to choose between the Lorenz, the Coulomb or
some other gauge. Different physical facts corresponds to different choices of
gauge. Once a unique choice has been made and the theory accepted, the other
gauge choices can be considered as mathematically convenient fictions, as
tools for making calculations suited to different problems. That is why the
name ‘($\phi$,A)-theory’ is not sufficient and should really be seen as a
collection of ‘($\phi$,A)-theories’, of which, for example, the Coulomb gauge
potentials theory, ($\phi$,A)${}_{\text{CG}}$-theory, and the Lorenz gauge
potentials theory, ($\phi$,A)${}_{\text{LG}}$-theory, are members.

We need not be so ambitious, however, as to consider all degrees of freedom
encoded in $\phi$ and A physical. One can define gauge equivalence classes
whose members all match each other on the degrees of freedom that one
considers physical. The above ‘collection’ of ($\phi$,A)-theories thus
corresponds to the Narrow Equivalence Class. Depending on one’s criteria, one
would narrow this equivalence class down further, which is equivalent to
adding more constraints, i.e., shrinking the space of functions for $\chi$.

In section 7, I present candidates for trimming down gauge-underdetermination
in the form of Bell locality and signal locality. In section 8, I discuss
Aharonov and Bohm’s locality concept and modern views of the local explanation
that can be given in terms of the equivalence class defined by Eqs.
(5.1)-(5.2). This class is suitable to provide an explanation in terms of
local interactions, but is deemed insufficiently suited to provide a signal-
local explanation of the AB effect in terms of a sharp ontology. Thus, I will
formulate desiderata that can lead to a further narrowing. In section 9, I
will suggest that picking the Lorenz gauge as such a constraint will narrow
the gauge equivalence class down in such a way that one has a signal-local
explanation of the AB effect in terms of travelling $\phi$ and A potential
fields, just as one has travelling E and B fields in the traditional Maxwell
theory. But before that, I discuss in the next section how and why the
considerations that go into choosing a gauge equivalence class, and in
particular the related problem of gauge-underdetermination, are so often not
engaged with in the scientific literature.

##  6 Gauge-underdetermination in the scientific literature

The fact that underdetermination automatically arises in the potentials view
is too often unacknowledged. In this section, I will criticise the
presentations of several widely-read authors. In general, the issue is that
even though these authors commit to the potentials being real, they are still
considered as gauge fields where the gauge transformation remains as free a
tool as in the fields view. This is having your cake and eating it too—using
the potentials view for an explanation of the AB effect and the fields view to
get out of gauge-underdetermination.

Norsen devotes a section of his recent (important) book to questions about
reality and locality similar to those in this paper. He considers the
possibility of accepting the electrodynamic potentials as physically real [29,
p.21]. Then he alludes to Bell’s argument (cf section 7) that the scalar
potential is non-local in the Coulomb gauge and argues, like Bell, that this
is not a problem because it is “bound up in some way with human knowledge or
conventions.”

But what is bound up with human knowledge or convention? If it is the
potentials, then we are simply rejecting the premise that the potentials are
real. Hence it is more likely that Norsen means the gauge choice. But if one
regards the potentials as physically real, the gauge choice can no longer be
considered a convention, since every ‘choice’ leads to a different description
of the state of affairs in nature.

Another instance is found in the introduction to gauge theories by Moriyasu,
who states that

> [t]he Aharonov-Bohm effect clearly contradicted the accepted notion that
> only the electric and magnetic fields could produce observable effects. More
> important, it became evident that the potential had to be treated as a
> physical field that was also directly observable. The alternative would be
> to believe that the phase shift is produced by the magnetic field “acting at
> a distance” in direct conflict with relativity [30, p.21].

But the precise way in which the magnetic field could act at a distance is not
fleshed out. More importantly, there is no mention in the book about the
consequential underdetermination that arises from taking the potentials as the
fundamental ontology; at the same time, the author speaks of making different
gauge choices as if the fields view was adhered to.

In his Lecture 9, Feynman defined locality as an intrinsic part of (real)
fields: “a field is ‘real’ if it is what must be specified at the position of
the particle in order to get the motion” [31]. He further made the
sociological observation that this conclusion had become consensus, since “E
and B are slowly disappearing from the modern expression of physical laws;
they are being replaced by A and $\phi$.” The latter was surely correct at the
time and has become even more so today, if only due to the canonical
formulation inserting the potentials into Schrödinger’s equation.
Nevertheless, also in the Feynman lectures, there is no mention of the problem
of gauge-underdetermination that arises from taking the potentials as
physical, though different gauge choices are being used. Hence it is not at
all clear what is meant be ‘real’ here.

Let us briefly survey Healey’s positive account of the AB effect, which is
called the holonomies view [6] (also see Belot [32] and Wu and Yang [33]). In
addition to assigning vectors such as E and B to points in space, this view
assigns complex numbers—called holonomies—to closed curves in space. These
holonomies are gauge-invariant complex numbers of unit modulus:

$h(\gamma)=\exp\left[i\oint_{\gamma}\textbf{A}\cdot d\textbf{x}\right].$ (6.1)

The holonomies (6.1) are gauge-invariant for the same reason that Eq. (3.4) is
gauge-invariant, using the familiar but tailor-made gauge symmetry of the
Narrow Equivalence Class. The loop integrals of the potentials are promoted to
fundamental ontology [6, section 4.4]. There are several costs to this
approach. Notably, it thus seems one cannot write down the equations of motion
in terms of the holonomies. Also, even though the holonomies act locally, they
lead to a non-local theory in the sense that it is non-separable: determining
the state in a given region involves the value of $h(\gamma)$ for every
possible loop. It is non-separable in the sense that the state of a region is
not fully determined by the conjunction of states on all its subregions (which
is the kind of non-locality or ‘non-separability’ that Einstein objected to
[34]: cf. section 7).

Healey dismisses the potentials view by claiming that “there is reason to
doubt that the magnetic vector potential is a physically real field,” since “A
is not gauge-invariant” [35, p.22]. This kind of gauge-invariance is Healey’s
desideratum. But from this it does not follow that one should doubt the
physicality of A. He argues that, since one can transform away part of the
vector potential in the region outside of the solenoid, for example gauging it
to zero at path $1$ of the electron beam (cf, footnote 7), the effect should
not derive from A itself. Healey specifically suggests that

> the potential is defined only up to a gauge-transformation, and for any
> continuous path from source to screen that does not enclose the solenoid
> there is a gauge-transformation that equates the value of A at every point
> on that path when a current is flowing to its value when no current is
> flowing. The shift interference pattern cannot therefore be produced by a
> direct interaction between individual electrons following such continuous
> paths and the magnetic vector potential A outside the solenoid. Accepting
> the physical reality of the vector potential fails to render the AB effect
> local: while denying its physical reality leaves one without any local
> explanation of the effect [35, p.22].

Here, Healey seems to be dismissing the potentials view as a sleight of
hand—denying it from the start. Healey seems to be saying that there is always
a gauge choice in which the explanation cannot be given in terms of a local
effect of the vector potential. Healey’s point of view, then, is that if
$\phi$ and A are real then they should be real in all gauges. But this prompts
the question why we use the gauge symmetries that we use.

One need not follow Healey in ignoring the possibility of additional physical
degrees of freedom of the Narrow Equivalence Class. It is unproblematic to say
that only gauge-invariant quantities can be considered physical, but this
depends on which definition of the gauge equivalence class one takes, for
example the Wide Equivalence Class
($\textbf{A}\mapsto\textbf{A}^{\prime}:=\textbf{A}+\textbf{C}$ such that
$\nabla\times\textbf{C}=0$) or the Narrow Equivalence Class
($\textbf{A}\mapsto\textbf{A}^{\prime}:=\textbf{A}+\nabla\chi$), or even a
narrower equivalence class
($\textbf{A}\mapsto\textbf{A}^{\prime}:=\textbf{A}+\nabla\chi$ plus a further
constraint). In addition to that, on the potentials view, there is in fact a
local explanation of the AB phase shift using a narrower equivalence class. I
discuss this in sections 7 and 8, and in section 9 I argue for a specific
narrower ‘Lorenz Equivalence class’, where the AB phase shift can be explained
“by a direct interaction between individual electrons following such
continuous paths and the magnetic vector potential A outside the solenoid.”

A final topic that, I suggest, could be emphasized more strongly in the
literature is how the potentials view combines with quantum mechanics. Indeed,
there are important constraints on the particular quantum theory one can
adhere to, if one simultaneously adopts the potentials view (depending on the
version of the potentials view that one is considering).

The choice between quantum theories turns, in part, on whether to include a
preferred foliation in a particular quantum theory. That is, if the collapse
of the wavefunction is a physically real process, then presumably it picks out
some preferred foliation of the spacetime, namely the hyperplane in which the
state vector projects.151515One should, however, be careful before equating
the apparent non-locality that derives from violations of the Bell
inequalities with the apparent non-locality that we have seen in the AB effect
above. One can debate whether to begin the inquiry with analogies to
violations of the Bell inequalities or the interpretation of gauge freedom.
The former position is defended by Healey [35] [36] and the latter by Maudlin
[28]. In such a case, one might pursue the strategy of taking a non-local
gauge potential theory as real and aligning the non-local foliation of that
gauge theory to the foliation suggested by one’s quantum theory. This would, I
believe, be a good strategy for reconciling particular collapse theories such
as GRW theory with the potentials view. One can also consider the Coulomb
gauge, which both picks out a relativistic structure, as seen in Eq. (5.5),
and a preferred foliation, as we have recognized through Eq. (5.4). Following
Maudlin’s suggestion, this spacetime structure serves well to implement
Bohmian mechanics [37, p.20].

##  7 Desiderata: Bell locality and signal locality

One reason why gauge-underdetermination is so often only briefly discussed, is
that it is not straightforward which gauge should be preferred. As with every
occurrence of underdetermination one needs additional principles if one wants
to cut down on possibilities. But on what criterion should the gauge be fixed?
I propose that, faced with the original motivation of formulating a local
explanation of the AB effect, the choice should be made on the same grounds.

Yet clearly locality is itself a vague concept, with several uses, and these
uses in turn tend to be vague. Einstein was concerned with the localization of
physical facts: what we would now call separability or anti-holism [34]. This
still allows for instantaneous action at a distance. The denial of action at a
distance gives one the stronger notion of signal locality. I deem signal
locality to be successful for our purposes and will discuss it below. First,
however, I discuss the more precise notion of a Bell local theory, which seems
to be the most promising version. Nevertheless, I will come to the conclusion
that Bell locality is generally not helpful. Only for some complete gauge
choices can we evaluate it.

John Bell regarded gauge choices, as (E,B)-theorists do, as conventions. His
famous example [38, p.219] is that of British sovereignty: “when the Queen of
England dies in London, the Prince of Wales becomes instantaneously King.” But
nobody is particularly bothered by this kind of non-locality, since the
property of being the British sovereign is not a local beable, nor any kind of
beable: it is a convention. In this Pickwickian sense, conventions can indeed
travel faster than light. In his 1976 paper on local beables, Bell writes

> In Maxwell’s electromagnetic theory, for example, the fields E and [B] are
> ‘physical’ (beables, we will say) but the potentials A and $\phi$ are ‘non-
> physical’. Because of gauge invariance the same physical situation can be
> described by very different potentials. It does not matter that in Coulomb
> gauge the scalar potential propagates with infinite velocity [see Eq.
> (5.4)]. It is not really supposed to be there. It is just a mathematical
> convenience [39, p.54].”

On the potentials view, however, $\phi$ and A are supposed to really be there.
In that case, it is not so straightforward to evaluate whether the
($\phi$,A)-theory is local in the sense Bell was concerned with—to be
explicated below. Different gauge choices lead to different locality
considerations, so that gauge choices lead to conceptually different theories.
Only in the context of a single clearly defined theory can we evaluate Bell
locality. When there is gauge-underdetermination, the choice of gauge
(partially) specifies what the supposed ontology (i.e., Bell’s local beables)
of the theory are.

So let us define the concept of Bell locality. Consider two spatially
separated regions, region $1$ and region $2$, and the events in their
respective past light cones. Then consider a third region, region $3$,
confined to the past light cone of region $1$, but completely excluded from
the past light cone of region $2$, as depicted in Fig. 3. The idea is that a
theory is ‘Bell local’ if and only if events in $2$ are irrelevant for
predictions about events in $1$ given that one has a full specification of
what happens in region $3$. Mathematically, the conditional probability that
an event $\mathcal{E}_{1}$ in region $1$ occurs given the set of events
$\mathcal{C}_{3}$ in region $3$ should be equal to the conditional probability
that the same event $\mathcal{E}_{1}$ occurs given the same $\mathcal{C}_{3}$
and also any additional event $\mathcal{E}_{2}$ in region $2$:

$P[\mathcal{E}_{1}|\mathcal{C}_{3}]=P[\mathcal{E}_{1}|\mathcal{C}_{3},\mathcal{E}_{2}].$
(7.1)

Of course, $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ might still be correlated
events. After all, their respective past light cones do have an overlapping
region which can contain an event that is a common causal influence on the two
events. There can also be some indirect influence of the causes of
$\mathcal{E}_{1}$ overlapping with the causes of $\mathcal{E}_{2}$. But the
point is that in a Bell local theory any such information that event
$\mathcal{E}_{2}$ might reveal about event $\mathcal{E}_{1}$ is already
contained in $\mathcal{C}_{3}$: so given $\mathcal{C}_{3}$, any information
about whether $\mathcal{E}_{2}$ occurs is redundant.161616To simplify
terminology, I use the property ‘Bell local’ as synonymous with ‘locally
causal’, interpreting Eq. (7.1) as $\mathcal{C}_{3}$ ‘completely screens off
$\mathcal{E}_{1}$ from events in $\mathcal{E}_{2}$’.

Figure 3: “Full specification of what happens in 3 makes events in 2
irrelevant for predictions about 1 in a locally causal theory.” Figure from
[38, p.225].

Note further that Bell locality is a property at the level of the theory and
does not directly mention events in reality. To link the theory to the things
in the world, Bell introduces [38, p.219] the notion of beables: the “beables
of the theory are those entities in it which are, at least tentatively, to be
taken seriously, as corresponding to something real.” Hence, beables are
theoretical entities that can be directly linked to something in the world,
depending on how one fleshes this out (cf. footnote 11). Further, beables are
local beables if they can be confined to some finite spacetime region. One can
see that Bell locality presupposes separability.

To further examine the usefulness of Bell locality in the context of the AB
effect, one should realize that quantum mechanics itself is Bell non-local,
due to the violations of the Bell inequalities. As mentioned earlier in this
section, one can compare the relation between different senses of non-locality
in the AB experiment and the experimental tests of the Bell inequalities. How
severe Bell non-locality really is remains widely contested, especially
concerning whether parameter independence or outcome independence is violated,
which depends on making Bell’s work even more precise [40]. Furthermore, to
evaluate the Bell locality of a theory, we presuppose a measurement-problem-
free theory where the ontological commitments are reasonably clear. That is,
we cannot see at the phenomenological level if outcome independence or
parameter independence is violated, since we inevitably run into the
measurement problem when we consider ‘outcomes’ [41]. Yet, this leads us
astray in our present context. In this present context the goal is to
formulate desiderata in terms of locality considerations. That is, we attempt
to use locality as a criterion for theory choice in a gauge-underdetermined
situation so as narrow down the class of admitted gauge degrees of freedom. In
any case, I maintain that matters are worsened if there is an additional
beable in the theory that violates Eq. (7.1). For example, a Coulomb gauge
potentials theory would violate it through direct action at a distance (see
below).

The electric and magnetic fields are defined in four-dimensional spacetime and
satisfy the wave equation, Eq. (7.2), which automatically confines the
propagation of causal influences to the light cones. Since the theory is
deterministic, $\mathcal{C}_{3}$ uniquely determines event $\mathcal{E}_{1}$,
since it is in the future domain of dependence $\Sigma(\mathcal{C}_{3})$ of
region $\mathcal{C}_{3}$: a sufficient set of relevant initial conditions for
$\mathcal{E}_{1}$ lie in region $3$ so that the equations of motion of all the
beables involved in $\mathcal{E}_{1}$ have a unique solution. On the fields
view (i.e., taking the electric and magnetic fields as physical), the
(E,B)-theory is Bell local and the E and B fields are local beables.

Yet, even though the (E,B)-theory is Bell local, it is unable to give a local
explanation of the AB phase shift. It would therefore be natural to evaluate
the Bell locality of the potentials view in the context of the AB effect.
However, the experimental set-up is not easily translated into the language of
Eq. (7.1). In the AB experiment the electron beam comes arbitrarily close to
the solenoid, so that two putative events $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$ involving the electron cannot be spacelike separated without
destroying the AB phase shift. This is because the electrons need to form a
closed loop and the electrons are confined to the light cone. In short: the
Aharonov-Bohm set-up involves a single system, making Bell locality hard to
apply. Therefore, Bell locality will not lend itself as a criterion to pick
out some local gauge choice. One can only take a particular potentials theory,
put it into the language of Eq. (7.1), and consider if there are violations of
Bell locality.

The sense of locality that, I will argue, can tackle gauge-underdetermination
and provide a local picture of the AB effect is that of signal locality. This
is what one usually understands by the concept ‘locality’: it means that the
causal influences that bodies exert on one another propagate at some finite
speed. So the theory is to treat causal influences as propagating at a finite
speed through (disjoint regions of) spacetime. One imagines ‘messengers’ that
take some time to travel from one event to the other in order to ‘tell’ the
actors there how they should react. If that travelling speed is infinite, one
speaks of ‘action at a distance’.

Electric and magnetic fields satisfy signal locality. Taking the the curl of
both Faraday’s law and the Maxwell-Ampère law, one derives that the electric
and magnetic fields both satisfy the wave equation:

$\nabla^{2}\textbf{E}=\frac{1}{c^{2}}\frac{\partial^{2}\textbf{E}}{\partial
t^{2}},$ (7.2)

and likewise for the B-field. The solutions are waves moving at speeds $c$,
such that electromagnetic signals travel with a finite velocity.171717This is
also the case when sources are included. Then, the wave equations acquire an
inhomogeneous part, which can be solved by the Green’s function method. A
point charge (or current) at position $\textbf{x}^{\prime}$ considered only at
one point in time ($t^{\prime}=0$) gives rise to an electric (or magnetic)
field, evolving forward in time,
$\textbf{E}(\textbf{x},t)=-\delta\left(t-|\textbf{x}-\textbf{x}^{\prime}|/c\right)/4\pi|\textbf{x}-\textbf{x}^{\prime}|\hat{r}$,
with the delta distribution $\delta(x)$ and radial unit vector $\hat{r}$. It
is interpreted as a wavefront that propagates spherically outwards, reaching
all the positions x at times $t=\textbf{x}/c$. A real source is represented by
the superposition of such charges (currents) in space and time.

In the Coulomb gauge—as we have seen in Eq. (5.4) and in Bell’s words
above—the scalar potential satisfies the Laplace equation, which is a non-
local equation also satisfied by the Newtonian gravitational potential. One
way to see this is to take the limit $c\rightarrow\infty$ in the wave equation
so that it reduces to Laplace’s equation. Hence, signal locality is violated.

If we now briefly return to the property of Bell locality, we recognize that
only now that a complete gauge is fixed, viz. the Coulomb gauge, one is in a
position to evaluate this in terms of Bell locality. It is readily seen that
Eq. (7.1) is violated since the beables at $\mathcal{E}_{1}$ are not fully
specified by region $\mathcal{C}_{3}$ and can be spacelike influenced by
$\mathcal{E}_{2}$ from outside the past light cone. Hence, as the scalar
potential is a beable in this theory, the Coulomb gauge potentials theory
($\phi$,$\textbf{A})_{\text{CG}}$ violates Bell locality in quite a crude way;
indeed, exactly as crudely as Newtonian gravitational theory does.

Alternatively, in the Lorenz gauge potentials theory ($\phi$,A)LG, both the
scalar potential $\phi$ and the vector potential A satisfy the wave equation
and propagate with velocity $c$, as exhibited by Eqs. (5.7)-(5.8). Hence, in
the Lorenz gauge, signal locality is satisfied, as can be motivated by Fig. 4.
So here is a local explanation that—unlike the (E,B)-theory—has the ability to
explain the AB phase shift locally. The next two concluding sections develop
this theme.

##  8 Local interaction and narrowing equivalence classes

Ahanorov and Bohm speak of the apparent non-locality of their experiment as
follows.

> Of course, our discussion does not bring into question the gauge invariance
> of the theory. But it does show that in a theory involving only local
> interactions (e.g., Schrodinger’s or Dirac’s equation, and current quantum-
> mechanical field theories), the potentials must, in certain cases, be
> considered as physically effective, even when there are no fields acting on
> the charged particles [16, p.490].

Hence they consider the results inexplicable in terms of local interactions,
as long as the potentials remain interpreted as mathematical fictions. For the
behaviour of the quantum probe would be influenced by the presence of B inside
the solenoid, although B vanishes at every point in space outside the solenoid
at every time that the probe is present at those spatial points.

Aharonov and Bohm argue for locality by noting that “according to our current
relativistic notions, all fields must interact only locally [16, p.490].” As
their paper pre-dates Bell’s formulation of locality, and they do not
explicate what they mean by locality, it is not clear if Aharonov and Bohm
have something like local beables in mind. It is not even clear if they have
signal locality in mind, although they can certainly be read that way. At the
very least, their notion of locality seems more in line with the idea that a
theory is local when given in terms of equations of motion where the
interactions only take place when the coordinates of the interacting systems
coincide. Call this local interactions.

Coming to what was promised at the end of section 5, one can read Aharonov and
Bohm conservatively: as rejecting the Wide Equivalence Class that leaves only
the E and B fields invariant, namely Eqs. (5.9)-(5.10), repeated here as
$\textbf{A}\mapsto\textbf{A}^{\prime}:=\textbf{A}+\textbf{C}$ and
$\phi\mapsto\phi^{\prime}:=\phi-C_{0},$ such that $\nabla\times\textbf{C}=0$
and $\nabla C_{0}=\partial\textbf{C}/\partial t$. One can now reformulate
Aharonov and Bohm’s legacy: the traditional fields view that used the Wide
Equivalence Class to encode the physical states of affairs in the world is
inadequate because the class is too wide. For, in general, two elements of
such a class differ in their physics, as is revealed by the shifted
interference patterns exhibited by probing the field around the solenoid. The
Narrower Equivalence Class joins these elements together. The ‘puncture’ in
space that is the solenoid, introduces a multiply-connected space and the
difference brought about by this non-trivial topology can be probed by a
suitable quantum particle.181818The electron in the AB experiment is described
by a quantum wave function. In some sense, however, this is only accidental.
If one had used a classical, not a quantum, probe, an effect by the potential
on the probe would still be there, but there might not be such a
straightforward way to observe that effect. In addition—even though this does
not demonstrate that the quantity is classical—note that computing the AB
phase shift does not involve any loop integrals. See also Chapter 8 of Neil
Dewar’s forthcoming book [42] and Henrique Gomes [43, p.19]. In other words,
on a non-trivial topology the Narrow Equivalence Class is not the same as an
electric and magnetic fields. The former has additional degrees of freedom,
and these are the ones that account for the AB effect in terms of local
interactions of the fields.191919Much more can be said about the specific
considerations of Bohm and Aharonov (cf. the ‘Vaidman-ACR’-debate in footnote
4 and references there).

As I have stressed, there are different senses of locality. Even someone
seeking a local explanation in a strong sense must admit that a space such as
the Moebius strip can have global features that are not caught by the
description of any local patch. Hence, locally you cannot say if the space is
simply-connected or multiply-connected. On the other hand, everybody must
accept that the value of a line integral such as
$\oint_{\gamma}\textbf{A}\cdot d\textbf{x}$ changes incrementally as the
endpoint varies. So, for the AB effect, anybody seeking a local explanation
must respect such constraints. Yet, one might wonder what happened to the
signal locality that we remember from the fields view, namely that E and B
propagate as waves with speed $c$. Are we forced to give this up or can we
have it on top of the above notion of local interaction? In other words, what
does a signal-local explanation of the AB effect look like?

Figure 4: A sketch of the AB effect. The vector potential A, in the Lorenz
gauge potentials theory ($\phi$,A)LG, propagates outward at speed $c$ after
the current has been switched on (instantaneously) at $t_{1}$, while the
magnetic field B is confined to the region occupied by the solenoid at all
times. An electron wave-packet is emitted at $t_{0}$ and is steered by
external potentials around the solenoid both into the plane of the figure or
away from the reader and out of the plane of the figure or towards the reader.
For simplicity, the slits and surface of the screen are not drawn, but all
possible paths of the electron must be imagined. If one takes the vector
potential in the Lorenz gauge as fundamental ontology, then there is a signal-
local explanation of the phase shift (while there is none using the Wide or
Narrow Equivalence Class). The resulting gauge equivalence class, here dubbed
the Lorenz Equivalence Class, has a residual gauge symmetry in the form of a
scalar wave, i.e., $\chi$ satisfying
$\nabla^{2}\chi=c^{-2}\partial^{2}\chi/\partial t^{2}$. This narrows down the
usual equivalence class
$\textbf{A}\mapsto\textbf{A}^{\prime}:=\textbf{A}+\nabla\chi$ significantly.

##  9 A suggestion: the Lorenz Equivalence Class

In the final paragraph of section 7, I pointed out that the Lorenz gauge
potentials theory ($\phi$,A)LG satisfies signal locality. It does so because
in the Lorenz gauge the equations of motion of $\phi$ and A satisfy the wave
equation (5.7)-(5.8) with propagation speed $c$. Hence, if we narrow the
equivalence class further by using the Lorenz gauge condition, the potentials
can be interpreted as travelling waves, analogous to the travelling of E and
B. Then, some degrees of freedom of the potentials—in addition to those giving
rise to E and B—travel outward from the solenoid to the electron. Thus, this
($\phi$,A)LG-theory opens the door to a signal-local explanation of the AB
phase shift, as illustrated in Fig. 4.

My proposal is, accordingly, to take the even narrower gauge equivalence class
than the Narrow Equivalence Class by adding the constraint on the gauge
function $\chi(\textbf{x},t)$ in the form of the Lorenz gauge condition (5.6).
This constraint of the last line can be readily calculated by performing a
gauge transformation (5.1)-(5.2) of the condition (5.6). It turns out that
there is a residual freedom in which $\chi$ itself obeys the wave
equation.202020Alternatively, using the gauge freedom of the Wide Equivalence
Class (5.9)-(5.10) returns four wave equations,
$\nabla^{2}C_{0}=(1/c^{2})\partial^{2}C_{0}/\partial t^{2}$ and
$\nabla^{2}\textbf{C}=(1/c^{2})\partial^{2}\textbf{C}/\partial t^{2}$, still
subject to $\partial\textbf{C}/\partial t=\nabla C_{0}$. However, this Wide
Equivalence Class + Lorenz gauge condition is not sufficient to make the line
integral (3.4) independent of the path, which is what I have in section 8
called Aharonov and Bohm’s legacy. The Lorenz Equivalence Class is then

$\displaystyle\phi\mapsto\phi^{\prime}:=\phi-\frac{\partial\chi}{\partial t},$
$\displaystyle\textbf{A}\mapsto\textbf{A}^{\prime}:=\textbf{A}+\nabla\chi,$
(9.1)
$\displaystyle\nabla^{2}\chi=\frac{1}{c^{2}}\frac{\partial^{2}\chi}{\partial
t^{2}}.$

On this view, the ‘true’ gauge degrees of freedom are the ones that leave the
scalar wave solution intact. This scalar wave solution is a result of the
Lorenz gauge-fix being incomplete or partial. The remaining freedom describes
how the potentials behave far away, i.e. towards spatial infinity. This can be
fixed by initial conditions; but, for all I have argued here, those initial
conditions seem as if they cannot be determined empirically.

Let us return to the spectrum in Figure 1. We can now recognize that the
Lorenz Equivalence Class is quite far to the left within the potentials view,
but there is still the residual gauge freedom resulting from the partial
gauge-fix, which we can safely regard as mathematical fiction. The Coulomb
gauge condition (5.3) is a complete gauge-fix, as it would result in the
Laplace equation for the gauge function, $\nabla^{2}\chi=0$, which—for given
boundary conditions that one can take to be given by the experiment—has a
unique solution. Another oft-used way to achieve this is the Lorenz gauge plus
the ‘temporal gauge’ $\phi=0$. Were one to gauge-fix completely along such
routes, the resulting philosophical position is the One True Gauge principle,
which can be seen as a gauge equivalence class with one member.

To be precise, however, one leaves room for the trivial gauge equivalence
class, which consists of transformations of the unit one expresses the
potentials in, call it the ‘Singleton Set’. Although this issue is hardly ever
discussed in connection with the AB effect and gauge theories—and rightly so,
since this variety is conceptually straightforward—nevertheless it means that
until one chooses units, even a complete gauge-fixing yields equivalence
classes each of which has many elements, corresponding to the various possible
units that one might choose to adopt. In the fields view, there is the same
freedom to change the units of E and B, indicating that also the electric and
magnetic field are not free from human choice.

##  10 Conclusion

Aharonov and Bohm showed us that there is good reason to take the
electromagnetic potentials seriously as part of the fundamental ontology of
electrodynamics. One consequence of their work was that the Narrow Equivalence
Class has become standard in contemporary textbooks, leading to some physical
degrees of freedom in the potentials over and above those that give rise to E
and B. The reality of the gauge potential demands careful treatment of one’s
commitment to the gauge symmetry of the theory. Naively taking the potentials
as ‘real’ immediately runs into an indeterminism of the equations of motion of
the potentials and hence a group of rival theories between which one cannot
choose on empirical grounds: gauge-underdetermination. This is a partial
commitment: one cannot have the reality of the potential (having the cake)
while also regarding gauge-fixing as arbitrary (eating it too). Such a move
does not only play a role in electrodynamics, but is seen in all theories
where degrees of freedom can be seen as ‘gauge’, such as all of the Standard
Model Lagrangian, diffeomorphism invariance in relativity theory, and various
approaches to quantum gravity.

One way to have signal locality is to choose the Lorenz gauge: $\phi$ and A
then obey the wave equation, just like E and B do. Agreed, besides the Lorenz
gauge there are many conceivable positions to lying on the spectrum of Figure
1, and I suggest these should be explored on their philosophical merits and
shortcomings. I have chosen to stay close to the original motivation for
taking the potentials as physical, namely locality. Emphasizing the
desideratum of signal locality, I have suggested we should use the Lorenz
gauge, because then the wave equation is satisfied by the potentials. It is
true that the dynamics of the potentials in the Lorenz gauge is automatically
Lorentz invariant, but this is also the case for other (relativistic) gauges.
In conclusion, I thus suggest we should narrow down further the Narrow
Equivalence Class to the Lorenz Equivalence Class, with a scalar wave solution
that encodes the residual gauge freedom, as the ontology of electrodynamics.

## Acknowledgments

I wish to thank Jeremy Butterfield, Guido Bacciagaluppi and Henrique Gomes for
thorough comments and corrections; and Dennis Dieks, Maaneli Derakhshani, Sam
Rijken, Vipin Chaudhary and Caspar Jacobs for helpful correspondence and
discussions; and also two anonymous reviewers for crucial observations.

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# Sentence Punctuation for Collaborative Commentary Generation in Esports
Live-Streaming

Hong Huang1, Junjie H. Xu1  Graduate School of
Comprehensive Human Sciences
University of Tsukuba
Tsukuba, Ibaraki, Japan
{s2165071<EMAIL_ADDRESS>Xiaoling Ling  Faculty of Arts
University of British Columbia
Vancouver, British Columbia, Canada
<EMAIL_ADDRESS>Pujana Paliyawan  Research Organization of
Science and Technology
Ritsumeikan University
Kusatsu, Shiga, Japan
<EMAIL_ADDRESS>
###### Abstract

To solve the existing sentence punctuation problem for collaborative
commentary generation in Esports live-streaming, this paper presents two
strategies for sentence punctuation for text sequences of game commentary,
that is, punctuating sentences by two or three text sequence(s) originally
punctuated by Youtube to obtain a complete sentence of commentary. We
conducted comparative experiments utilizing and fine-tuning a state-of-the-art
pre-trained generative language model among two strategies and the baseline to
generate collaborative commentary. Both objective evaluations by automatic
metrics and subjective analyses showed that our strategy of punctuating
sentences by two text sequences outperformed the baseline.

###### Index Terms:

Collaborative Commentary Generation, Sentence Punctuation

11footnotetext: Equal contribution

##  I Introduction

The prosperity of Esport catalyzes a significant research interest for
academics [1, 2, 3] and industrial researchers [4, 5] to pay attention to
research topics of Esport’s game commentary since the game commentator could
entertain the audiences by interactively informing them of game-related
information during live-streaming. Moreover, the task of collaborative
commentary generation aims to generate an expected follow-up commentary based
on commentary given by the human commentator to collaborate with him or her
interactively.

Although the video caption of Esport games which were collected from the
public video website that could be used for game commentary, is processed and
punctuated, we notice the punctuated text sequences are usually incomplete
game commentary (see Case Solo in Table I, game scene screenshots were
captured from Esports game videos derived from
Youtube111https://www.youtube.com/watch?v=pJRjqqajKwU222https://www.youtube.com/watch?v=buNTNOgE2Bs333https://www.youtube.com/watch?v=q7NvQUzE0FA).
Thus, we assume the use of original punctuated sentences will cause the
problem of generating incomplete commentary, resulting in the game AI
commentator output incomplete game commentary, fails to collaborate with the
human commentator.

To this end, further punctuating sentences is needed to leverage such data
effectively. In this paper, we present two strategies for sentence punctuation
for game commentary. We compared our two strategies with baseline by employing
and fine-tuning a state-of-the-art generative language model Text-to-Text
Transfer Transformer (T5) [6] to generate collaborative commentary,
respectively. Objective evaluations from automatic metrics and subjective
analyses on generated commentaries among three strategies shown our strategy
of punctuating sentences by two text sequences outperformed the baseline.

TABLE I: Examples of punctuating commentary text using 3 sentence punctuation
strategies respectively, temporary commentaries derived from Youtube given
from top to bottom in row Text. indicates such sentence is a complete
commentary and vice versa.

Scene Case Solo, originally punctuated by Youtube Duo, punctuated by every 2
sentences in Solo Tri, punctuated by every 3 sentences in Solo Text
<start>really frightens me but at the same time <end> <start>their strengths
of being able to burst <start>many differences down and they’re just <start>I
respect that you have just faker <end> down the bear <end> gonna file straight
up towards those <start>nightmares you wake up in a cold sweat <end>
<start>and they’re hoping to get rocks to pull super minions in the base you
can see is <end> <start>you’re like fakers behind me I know <end> up gets
rooted teleports coming in <end> <start>going in that pack it’s definitely
<start>right like even though I only have a mat <end> <start>curtain calling
wish have already been awkward the Nexus Kevin is gonna be out <start>on the
floor I think is in the bed <end> used a trick is running away <end> in the
bank <end>

##  II METHOD

Sentence Punctuation for Collaborative Commentary: According to the finding
mentioned in introduction, the generated commentary should be a complete
sentence of commentary rather than a sliced snippet. As shown in Table II, in
this paper we compared three strategies: the baseline Case Solo, Case Duo and
Case Tri, denoting punctuating sentences by 1, 2 and 3 text sequence(s)
originally punctuated by Youtube respectively.

Pre-trained Generative Language Models: We used and fine-tuned a state-of-the-
art generative pre-trained model, Text-to-Text Transfer Transformer (T5) for
game collaborative commentary generation.

##  III Experiments

###  III-A Experimental Settings

We collected text sequences from 100 videos of one of the most watched Esports
games [7] called League of Legend from Youtube, further processed into Case
Solo, Duo and Tri respectively. Text sequences from 90 videos were used for
fine-tuning and others were used for testing. In experiment, we used the small
(T5-small) and base (T5-base) version of T5 model and compared three
strategies of sentence punctuation for Esports collaborative commentary
generation.

###  III-B Evaluation Results

Objective Results Given by Automatic Metrics: We evaluated the generated
commentary by using BLEU [8], ROUGE-1, 2, L [9], and METEOR [10]. As expected,
the performance of ROUGE-1, 2, L among three strategies of sentence
punctuation increased, which implies fine-tuning increases the recall rate of
words in generated commentary from reference commentary, and in Case Solo the
METEOR score also increases, which stands for correlation of wording and
phrasing with generated text with reference text. We marked the highest score
of each metric for three strategies in bold, respectively.

Subjective Analyses on Generated Commentaries: We compared the generated
commentary from fine-tuned T5-base model. We observed (1) the model generated
sliced key game-related phrases and incomplete sentences for commentary that
is semantically ambiguous in all cases, especially in Case Solo. (2) Case Duo
and Case Tri model could generate the entire key game-related phrases such as
the game-related scenes narrative with the correct name of characters and
moves. (3) However, Case Tri model generates more than one commentary (we
expect only to generate one commentary for one commentary given by a human
commentator).

TABLE II: Results of conducted comparative experiments. The evaluation was given by automatic metrics on generated collaborative commentary used in this work, respectively. Model | BLEU | ROUGE-1 | ROUGE-2 | ROUGE-L | METEOR
---|---|---|---|---|---
Case Solo |
T5-small-solo | 2.88 | 0.17 | 0.01 | 0.17 | 6.44
T5-base-solo | 2.88 | 0.52 | 0.07 | 0.52 | 19.17
Fine-tuned-T5-small-solo | 2.92 | 3.15 | 0.71 | 3.15 | 7.65
Fine-tuned-T5-base-solo | 2.97 | 3.91 | 0.48 | 3.91 | 22.76
Case Duo |
T5-small-duo | 1.13 | 0.82 | 0.01 | 0.82 | 12.24
T5-base-duo | 1.01 | 1.11 | 0.01 | 1.11 | 12.02
Fine-tuned-T5-small-duo | 1.04 | 11.26 | 6.08 | 11.24 | 13.18
Fine-tuned-T5-base-duo | 1.05 | 16.60 | 8.99 | 16.52 | 11.76
Case Tri |
T5-small-tri | 1.06 | 1.12 | 0.03 | 1.11 | 12.26
T5-base-tri | 1.01 | 0.94 | 0.02 | 0.93 | 12.87
Fine-tuned-T5-small-tri | 1.03 | 9.32 | 4.61 | 9.30 | 12.84
Fine-tuned-T5-base-tri | 1.04 | 11.90 | 5.91 | 11.89 | 12.24

##  IV Conclusion

To solve the sentence punctuation problem for collaborative commentary
generation in Esports live-streaming, this paper presents two strategies for
sentence punctuation for punctuating text sequences of game commentary.We
conducted comparative experiments among baseline and two strategies on state-
of-the-art pre-trained language models for Esports collaborative commentary
generation. From experiment results, we found that (1) the fine-tuning on pre-
trained language models improves recalling the words from reference
commentary. (2) Objective evaluations from automatic metrics and subjective
analyses on generated examples on generated commentaries among three
strategies showed Case Duo, the strategy of punctuating sentences by two text
sequences outperformed the baseline. (3) From the subjective analyses on Case
Tri we found that inputting lengthy text sequences results in the AI
commentator output more than one commentary that confuses the human
commentator. Our future work will develop a more advanced sentence punctuation
strategy study using human evaluation for collaborative commentary generation
in Esports live-streaming.

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