# Ontological Clarity via Canonical Presentation: Electromagnetism and the Aharonov–Bohm Effect

## Abstract

**:**

## 1. Introduction

## 2. What is a Physical Theory?

- (1)
- The local beables or matter, which exists at delimited regions of the spatio-temporal structure.
- (2)
- The non-local beables (if any) that have no particular value at any space-time location.
- (3)
- The spatio-temporal structure, in terms of which the distinction between local and non-local beables is drawn.
- (4)
- The dynamical laws, which specify, either deterministically or probabilistically, how the various beables must or can behave.

If one asks what, irrespective of quantum mechanics, is characteristic of the world of ideas of physics, one is first of all struck by the following: the concepts of physics relate to a real outside world, that is, ideas are established relating to things such as bodies, fields, etc., which claim “real existence” that is independent of the perceiving subject—ideas which, on the other hand, have been brought into as secure a relationship as possible with the sense data. It is further characteristic of these physical objects that they are thought of as arranged in a space-time continuum. An essential aspect of this arrangement of things in physics is that they lay claim, at a certain time, to an existence independent of one another, provided these objects “are situated in different parts of space”… This principle has been carried to extremes in the field theory by localizing the elementary objects on which it is based and which exist independently of each other, as well as the elementary laws which have been postulated for it, in the infinitely small (four-dimensional) elements of space.[4] (pp. 170–171)

Philosophy is written in this grand book—I mean the universe—which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth.[6] (p. 65)

## 3. Mathematical Physics and the Canonical Presentation

- (1)
- The fundamental physical ontology of the theory, which may further be divided into the local beables (matter and space-time structure) and the non-local beables, if any (e.g., a quantum state represented by a wave function on configuration space).
- (2)
- The spatio-temporal structure of the theory.
- (3)
- The mathematical items that will be used to represent both 1 and 2, with a commentary making clear which degrees of freedom in the mathematics are gauge and which are not.
- (4)
- The nomology of the theory, which will be represented by equations couched in terms only of the items mentioned in (3).
- (5)
- Mathematical fictions—these are mathematically defined quantities that are not intended to directly represent any part of the physical ontology. Such fictions can play an important practical role when trying to calculate with the theory.
- (6)
- Derivative ontology—these are items that are taken to be physically real but not fundamental. They must be definable in terms of the fundamental ontology and nomology.

## 4. The Canonical Presentation of Classical Electromagnetic Theory

^{3}for each value of t. t ranges over some interval of the real numbers.

^{3}and zero outside it. One might immediately assume that this is the mathematical representation of a uniform sphere of matter at rest. However, a moment’s thought reveals that it could just as well represent a uniform sphere of matter rotating on an axis, or performing any other motion that an incompressible continuous fluid might. In short, the physical content expressed by the function $\overrightarrow{\mathrm{v}}$(x, y, z, t) outruns the physical content expressed by $\mathsf{\mu}$(x, y, z, t). If one really wants to make a matter density on a continuum a fundamental part of the physical ontology, then one must also accept that there is a velocity function assigned to the matter density that does not supervene on the distribution of the matter density over all of space for all of time.

_{2}O. That is certainly true, and necessarily true, but does not have the characteristics of a physical or chemical law. We do not think that there are as many distinct chemical laws as there are chemical species: that would lead to millions of laws of chemistry. Water is H

_{2}O is rather the answer to the fundamental philosophical question: What is it? It tells of the metaphysical nature of water, what water fundamentally is. Although different in some respects, “There are no magnetic monopoles” is a similar sort of claim: it specifies part of the fundamental nature of magnetic charges.

_{m}, with q

_{m}representing a magnetic charge, together with the negative ontological claim that magnetic charges do not exist.

_{i}as the divergence of the electric field at a point, we can define it as the limit of the flux over surfaces that enclose the point as the maximum distance from the point to the surface goes to zero. In a similar spirit, define the q

_{i}-adjusted electric field E

_{qi}at a point p as the electric field at p minus $\frac{{q}_{i}}{{r}^{2}}\widehat{r}$, where r is the distance from the location of particle i to p and $\hat{\mathrm{r}}$ is the unit vector in the direction from the location of the particle to p. Let $\overline{{\mathrm{E}}_{\mathrm{qi},\text{}\mathsf{\Sigma}}}$ be the average of E

_{qi}over the surface $\mathsf{\Sigma}.$ Finally, define the electric field at a point on the worldline of particle i to be the limit of $\overline{{\mathrm{E}}_{\mathrm{qi},\text{}\mathsf{\Sigma}}}$ as the distance of the points on $\mathsf{\Sigma}$ from the point on the worldline goes to zero. It is that value of E that is used in the Lorentz force law.

## 5. The Aharonov–Bohm Effect

In classical electrodynamics, the vector and scalar potentials were first introduced as a convenient mathematical aid for calculating the fields. It is true that in order to obtain a classical canonical formalism, the potentials are needed. Nevertheless, the fundamental equations of motion can always be expressed directly in terms of the fields alone.In the quantum mechanics, however, the canonical formalism is necessary, and as a result, the potentials cannot be eliminated from the basic equations. Nevertheless, these equations, as well as the physical quantities, are all gauge invariant; so that it may seem that even in quantum mechanics, the potentials themselves have no independent significance.In this paper we shall show that the above conclusions are not correct and that a further interpretation of the potentials is needed in quantum mechanics.

^{2}X = Y.

## 6. Adjusting the Spatiotemporal Structure

_{i}to zero, all that is left of Maxwell’s theory is the homogeneous Maxwell equations. In the theory that makes the potentials fundamental, choosing the Lorenz gauge condition converts the nomology to ${\square}^{2}\varphi =\text{}0$ and ${\square}^{2}\overrightarrow{A}=0$. As noted above, if this where the whole story ended then the theory would be suggesting that it lives naturally in a Minkowski space. The d’Almerbertian is easily and naturally definable in Minkowski space-time, where it is Lorentz invariant. Indeed, the Lorentz invariance of the nomology in Lorenz gauge is one of the reasons so many people refer to Lorenz gauge as Lorentz gauge. Even more importantly, the manifest Lorentz invariance of the theory cast in Lorenz gauge provides an easy argument to the conclusion that Maxwellian electro-magnetic is a Lorentz invariant theory. One should then switch the spatio-temporal structure to Minkowski, as it is simpler.

## 7. Conclusions

In classical mechanics, we recall that potentials cannot have such significance because the equation of motion involves only the field quantities themselves. For this reason, the potentials have been regarded as purely mathematical auxiliaries, while only the field quantities were thought to have a direct physical meaning.In quantum mechanics, the essential difference is that the equations of motion for a particle are replaced by the Schrödinger equation for a wave. This Schrödinger equation is obtained from a canonical formula, which cannot be expressed in terms of the fields alone, but which also requires the potentials. Indeed, the potentials play a role, in Schrödinger’s equation, which is analogous to that of the index of refraction in optics. The Lorentz force [eE+ (e/c)v$\times $H] does not appear anywhere in the fundamental theory, but appears only as an approximation appearing 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 by differentiations.([1], p. 490)

## Funding

## Conflicts of Interest

## References

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Theory | Physical Ontology; Spatiotemporal Structure | Mathematical Representation of Physical Ontology | Purely Mathematical Facts | Nomology | Derivative Ontology; Mathematical Fictions |
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Classical E & M, Mass Density Version | Electric Field Magnetic Field Charge Density Mass Density Lorentz Force; Time 3-D Euclidean Absolute Space | $\overrightarrow{{E}}$(x, y, z, t) $\overrightarrow{{B}}$(x, y, z, t) $\mathsf{\rho}$(x, y, z, t) $\mathsf{\mu}$(x, y, z, t) $\overrightarrow{{{F}}_{{L}}}$(x, y, z, t) t $\in \text{}\mathbb{R}\text{}$ (x, y, z)$\text{}\in \text{}\mathbb{R}\text{}$ ^{3} | If Curl $\overrightarrow{{C}}$ = 0 on a simply connected space, then $\overrightarrow{{C}}=\mathrm{Grad}\left(\mathsf{\xi}\right)$ for some $\mathsf{\xi}$. If Div $\overrightarrow{{B}}$ = 0 on a simply connected space, then $\overrightarrow{{B}}=\mathrm{Curl}\left(\overrightarrow{{A}}\right)$ for some $\overrightarrow{{A}}$ Gauge transformations $\overrightarrow{{A}\prime}=\text{}\overrightarrow{{A}}+\text{}\mathrm{grad}\mathsf{\xi}$ ${\mathsf{\varphi}}^{\prime}=\text{}\mathsf{\varphi}-\frac{\partial \mathsf{\xi}}{\partial {t}}$ | $\mathrm{Div}\left(\overrightarrow{{E}}\right)=\text{}\mathsf{\rho}$ $\mathrm{Div}\left(\overrightarrow{{B}}\right)=\text{}0$ $\mathrm{Curl}\left(\overrightarrow{{E}}\right)+\text{}\frac{\partial \overrightarrow{{B}}}{\partial {t}}=0$ $\mathrm{Curl}\left(\overrightarrow{{B}}\right)-\text{}\frac{\partial \overrightarrow{{E}}}{\partial {t}}=\mathsf{\rho}\overrightarrow{{v}}$ $\overrightarrow{{{F}}_{{L}}}=\text{}\mathsf{\rho}\left(\overrightarrow{{E}}+\left(\overrightarrow{{v}}\text{}\times \text{}\overrightarrow{{B}}\right)\right)$ $\overrightarrow{{{F}}_{\mathrm{net}}}=\text{}\mathsf{\mu}\frac{{d}\overrightarrow{{v}}}{{d}{t}}$ | Derivative Ontology: $\overrightarrow{{J}}=\text{}\mathsf{\rho}\overrightarrow{{v}}$ $\overrightarrow{{{F}}_{\mathrm{net}}}$ = vector sum of all forces on a body at a point Mathematical Fictions: Let $\overrightarrow{{B}}=\mathrm{Curl}\left(\overrightarrow{{A}}\right)$ Then $\mathrm{Curl}\left(\overrightarrow{{E}}+\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}\right)=0$ so $\overrightarrow{{E}}+\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}=-\mathrm{Grad}\left(\mathsf{\varphi}\right)$ or $\overrightarrow{{E}}=-\mathrm{Grad}\left(\mathsf{\varphi}\right)$ $\u2013\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}$ |

Theory | Physical Ontology; Spatiotemporal Structure | Mathematical Representation of Physical Ontology | Purely Mathematical Facts | Nomology | Derivative Ontology; Mathematical Fictions |
---|---|---|---|---|---|

Classical E & M, Particle Version | Electric Field Magnetic Field Point Particles Particle Charge Particle Mass Lorentz Force; Time 3-D Euclidean Absolute Space | $\overrightarrow{{E}}$(x,y,z,t) $\overrightarrow{{B}}$(x,y,z,t) $\overrightarrow{{{x}}_{{i}}}$(t) = (x _{i},y_{i},z_{i}) q _{i}$\in $ R m _{i} $\in $ R > 0 $\overrightarrow{{{F}}_{{L}}}$(x,y,z,t) t $\in \text{}\mathbb{R}\text{}$ (x,y,z)$\text{}\in \text{}\mathbb{R}\text{}$ ^{3} | If Curl $\overrightarrow{{C}}$ = 0 on a simply connected space, then $\overrightarrow{{C}}=\mathrm{Grad}\left(\mathsf{\xi}\right)$ for some $\mathsf{\xi}$. If Div $\overrightarrow{{B}}$ = 0 on a simply connected space, then $\overrightarrow{{B}}=\mathrm{Curl}\left(\overrightarrow{{A}}\right)$ for some $\overrightarrow{{A}}$ Gauge transformations $\overrightarrow{{A}\prime}=\text{}\overrightarrow{{A}}+\text{}\mathrm{grad}\mathsf{\xi}$ ${\mathsf{\varphi}}^{\prime}=\text{}\mathsf{\varphi}-\frac{\partial \mathsf{\xi}}{\partial {t}}$ | $\mathrm{Div}\left(\overrightarrow{{E}}\right)=\text{}{{q}}_{{i}}$ $\mathrm{Div}\left(\overrightarrow{{B}}\right)=\text{}0$ $\mathrm{Curl}\left(\overrightarrow{{E}}\right)+\text{}\frac{\partial \overrightarrow{{B}}}{\partial {t}}=0$ $\mathrm{Curl}\left(\overrightarrow{{B}}\right)-\text{}\frac{\partial \overrightarrow{{E}}}{\partial {t}}={{q}}_{{i}}\overrightarrow{{{v}}_{{i}}}$ $\overrightarrow{{{F}}_{{L}{i}}}=\text{}{{q}}_{{i}}\left(\overrightarrow{{E}}+\left(\overrightarrow{{{v}}_{{i}}}\text{}\times \text{}\overrightarrow{{B}}\right)\right)$ $\overrightarrow{{{F}}_{\mathrm{net}{i}}}=\text{}{{m}}_{{i}}\frac{{{d}}^{2}\overrightarrow{{{x}}_{{i}}}\text{}\left({t}\right)}{{d}{{t}}^{2}}$ | Derivative Ontology: $\overrightarrow{{{v}}_{{i}}}=\frac{{d}\overrightarrow{{{x}}_{{i}}}\text{}\left({t}\right)}{{d}{t}}$ $\overrightarrow{{{J}}_{{i}}}=\text{}{{q}}_{{i}}\overrightarrow{{{v}}_{{i}}}$ $\overrightarrow{{{F}}_{\mathrm{net}}}$ = vector sum of all forces on a particle Mathematical Fictions: Let $\overrightarrow{{B}}=\mathrm{Curl}\left(\overrightarrow{{A}}\right)$ Then $\mathrm{Curl}\left(\overrightarrow{{E}}+\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}\right)=0$ so $\overrightarrow{{E}}+\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}=-\mathrm{Grad}\left(\mathsf{\varphi}\right)$ or $\overrightarrow{{E}}=-\mathrm{Grad}\left(\mathsf{\varphi}\right)$ $-\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}$ |

Theory | Physical Ontology; Spatiotemporal Structure | Mathematical Representation of Physical Ontology | Purely Mathematical Facts | Nomology | Derivative Ontology; Mathematical Fictions |
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Classical E & M, Particle Version With Derived Charges | Electric Field Magnetic Field Point Particles Particle Mass Lorentz Force; Time 3-D Euclidean Absolute Space | $\overrightarrow{{E}}$(x, y, z, t) $\overrightarrow{{B}}$(x, y, z, t) $\overrightarrow{{{x}}_{{i}}}$(t) = (x _{i}, y_{i}, z_{i}) m _{i}$\in $ R > 0 $\overrightarrow{{{F}}_{{L}}}$(x, y, z, t); t $\in \text{}\mathbb{R}\text{}$ (x, y, z)$\text{}\in \text{}\mathbb{R}\text{}$ ^{3} | If Curl $\overrightarrow{{C}}$ = 0 on a simply connected space, then $\overrightarrow{{C}}=\mathrm{Grad}\left(\mathsf{\xi}\right)$ for some $\mathsf{\xi}$. If Div $\overrightarrow{{B}}$ = 0 on a simply connected space, then $\overrightarrow{{B}}=\mathrm{Curl}\left(\overrightarrow{{A}}\right)$ for some $\overrightarrow{{A}}$ Gauge transformations $\overrightarrow{{A}\prime}=\text{}\overrightarrow{{A}}+\text{}\mathrm{grad}\mathsf{\xi}$ ${\mathsf{\varphi}}^{\prime}=\text{}\mathsf{\varphi}-\frac{\partial \mathsf{\xi}}{\partial {t}}$ | $\mathrm{Curl}\left(\overrightarrow{{E}}\right)+\text{}\frac{\partial \overrightarrow{{B}}}{\partial {t}}=0$ $\mathrm{Curl}\left(\overrightarrow{{B}}\right)-\text{}\frac{\partial \overrightarrow{{E}}}{\partial {t}}={{q}}_{{i}}\overrightarrow{{{v}}_{{i}}}$ $\overrightarrow{{{F}}_{{L}{i}}}=\text{}{{q}}_{{i}}\left(\overrightarrow{{E}}+\left(\overrightarrow{{{v}}_{{i}}}\text{}\times \text{}\overrightarrow{{B}}\right)\right)$ $\overrightarrow{{{F}}_{\mathrm{net}{i}}}=\text{}{{m}}_{{i}}\frac{{{d}}^{2}\overrightarrow{{{x}}_{{i}}}\text{}\left({t}\right)}{{d}{{t}}^{2}}$ | Derivative Ontology: $\mathrm{Div}\left(\overrightarrow{{E}}\right)=\text{}{{q}}_{{i}}$ $\mathrm{Div}\left(\overrightarrow{{B}}\right)=\text{}{{q}}_{{m}}$ $\overrightarrow{{{v}}_{{i}}}=\frac{{d}\overrightarrow{{{x}}_{{i}}}\text{}\left({t}\right)}{{d}{t}}$ $\overrightarrow{{{J}}_{{i}}}=\text{}{{q}}_{{i}}\overrightarrow{{{v}}_{{i}}}$ $\overrightarrow{{{F}}_{\mathrm{net}}}$ = vector sum of all forces on a particle Mathematical Fictions: Let $\overrightarrow{{B}}=\mathrm{Curl}\left(\overrightarrow{{A}}\right)$ Then $\mathrm{Curl}\left(\overrightarrow{{E}}+\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}\right)=0$ so $\overrightarrow{{E}}+\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}=-\mathrm{Grad}\left(\mathsf{\varphi}\right)$ or $\overrightarrow{{E}}=-\mathrm{Grad}\left(\mathsf{\varphi}\right)$ $-\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}$ |

Theory | Physical Ontology; Spatiotemporal Structure | Mathematical Representation of Physical Ontology | Purely Mathematical Facts | Nomology | Derivative Ontology; Mathematical Fictions |
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Classical E & M, Particle Version With Derived Charges | Electric Field Magnetic Field Point Particles Particle Mass Lorentz Force; Time 3-D Euclidean Absolute Space | $\overrightarrow{{E}}$(x, y, z, t) $\overrightarrow{{B}}$(x, y, z, t) s.t. $\mathrm{Div}\left(\overrightarrow{{B}}\right)=\text{}0$ $\overrightarrow{{{x}}_{{i}}}$(t) = (x _{i}, y_{i}, z_{i}) m _{i}$\in $ R > 0 $\overrightarrow{{{F}}_{{L}}}$(x, y, z, t) t $\in \text{}\mathbb{R}\text{}$ (x, y, z)$\text{}\in \text{}\mathbb{R}\text{}$ ^{3} | If Curl $\overrightarrow{{C}}$ = 0 on a simply connected space, then $\overrightarrow{{C}}=\mathrm{Grad}\left(\mathsf{\xi}\right)$ for some $\mathsf{\xi}$. If Div $\overrightarrow{{B}}$ = 0 on a simply connected space, then $\overrightarrow{{B}}=\mathrm{Curl}\left(\overrightarrow{{A}}\right)$ for some $\overrightarrow{{A}}$ Gauge transformations $\overrightarrow{{A}\prime}=\text{}\overrightarrow{{A}}+\text{}\mathrm{grad}\mathsf{\xi}$ ${\mathsf{\varphi}}^{\prime}=\text{}\mathsf{\varphi}-\frac{\partial \mathsf{\xi}}{\partial {t}}$ | $\mathrm{Curl}\left(\overrightarrow{{E}}\right)+\text{}\frac{\partial \overrightarrow{{B}}}{\partial {t}}=0$ $\mathrm{Curl}\left(\overrightarrow{{B}}\right)-\text{}\frac{\partial \overrightarrow{{E}}}{\partial {t}}={{q}}_{{i}}\overrightarrow{{{v}}_{{i}}}$ $\overrightarrow{{{F}}_{{L}{i}}}=\text{}{{q}}_{{i}}\left(\overrightarrow{{E}}+\left(\overrightarrow{{{v}}_{{i}}}\text{}\times \text{}\overrightarrow{{B}}\right)\right)$ $\overrightarrow{{{F}}_{\mathrm{net}{i}}}=\text{}{{m}}_{{i}}\frac{{{d}}^{2}\overrightarrow{{{x}}_{{i}}}\text{}\left({t}\right)}{{d}{{t}}^{2}}$ | Derivative Ontology: $\mathrm{Div}\left(\overrightarrow{{E}}\right)=\text{}{{q}}_{{i}}$ $\overrightarrow{{{v}}_{{i}}}=\frac{{d}\overrightarrow{{{x}}_{{i}}}\text{}\left({t}\right)}{{d}{t}}$ $\overrightarrow{{{J}}_{{i}}}=\text{}{{q}}_{{i}}\overrightarrow{{{v}}_{{i}}}$ $\overrightarrow{{{F}}_{\mathrm{net}}}$ = vector sum of all forces on a particle Mathematical Fictions: Let $\overrightarrow{{B}}=\mathrm{Curl}\left(\overrightarrow{{A}}\right)$ Then $\mathrm{Curl}\left(\overrightarrow{{E}}+\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}\right)=0$ so $\overrightarrow{{E}}+\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}=-\mathrm{Grad}\left(\mathsf{\varphi}\right)$ or $\overrightarrow{{E}}=-\mathrm{Grad}\left(\mathsf{\varphi}\right)$ $-\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}$ |

Theory | Physical Ontology; Spatiotemporal Structure | Mathematical Representation of Physical Ontology | Purely Mathematical Facts | Nomology | Constitutive Principles of Derivative Ontology |
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Vector and scalar potentials, Mass density, Newtonian Space and Time | Vector Potential Scalar Potential Charge density Mass density Lorentz Force; Time 3-D Euclidean Absolute Space | $\overrightarrow{{A}}$(x, y, z, t) $\mathsf{\varphi}$(x, y, z, t) $\mathsf{\rho}$(x, y, z, t) $\mathsf{\mu}$(x, y, z, t) $\overrightarrow{{{F}}_{{L}}}$(x, y, z, t) t $\in \text{}\mathbb{R}\text{}$ (x, y, z)$\text{}\in \text{}\mathbb{R}\text{}$ ^{3} | The nomology does not fix the history of $\overrightarrow{{A}},\text{}\mathrm{or}\text{}\mathsf{\varphi}$, given complete initial values. Radical indeterminism $\overrightarrow{{A}\prime}=\text{}\overrightarrow{{A}}+\text{}\mathrm{Grad}\mathsf{\xi}$ ${\mathsf{\varphi}}^{\prime}=\text{}\mathsf{\varphi}-\frac{\partial \mathsf{\xi}}{\partial {t}}$ | ${\nabla}^{2}\mathsf{\varphi}-\text{}\frac{\partial \text{}\mathrm{Div}\text{}\overrightarrow{{A}}}{\partial {t}}=-\text{}\mathsf{\rho}$ ${\nabla}^{2}\overrightarrow{{A}}-\frac{{\partial}^{2}\overrightarrow{{A}}}{\partial {{t}}^{2}}-\text{}\mathrm{Grad}\left(\mathrm{Div}\overrightarrow{{A}}+\text{}\frac{\partial \mathsf{\varphi}}{\partial {t}}\right)=\mathsf{\rho}\overrightarrow{{v}}$ $\overrightarrow{{{F}}_{\mathrm{net}}}=\text{}\mathsf{\mu}\frac{{d}\overrightarrow{{v}}}{{d}{t}}$ $\overrightarrow{{{F}}_{{L}}}=\mathsf{\rho}\left(-\mathrm{Grad}\mathsf{\varphi}-\frac{{d}\overrightarrow{{A}}}{{d}{t}}+\mathrm{Grad}\left(\overrightarrow{{v}}\text{}\xb7\text{}\overrightarrow{{A}}\right)\right)$ | $\overrightarrow{{j}}=\text{}\mathsf{\rho}\overrightarrow{{v}}$ $\overrightarrow{{B}}=\mathrm{Curl}\text{}\overrightarrow{{A}}$ $\overrightarrow{{E}}=-\mathrm{Grad}\left(\mathsf{\varphi}\right)$$-\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}$ |

Theory | Physical Ontology; Spatiotemporal Structure | Mathematical Representation of Physical Ontology | Purely Mathematical Facts | Nomology | Constitutive Principles of Derivative Ontology |
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Vector and scalar potentials in Lorenz Gauge, Mass density, Charge density Newtonian Space and Time | Vector Potential Scalar Potential Charge density Mass density Lorentz Force; Time 3-D Euclidean Absolute Space | $\overrightarrow{{A}}$(x, y, z, t) $\mathsf{\varphi}$(x, y, z, t) $\mathsf{\rho}$(x, y, z, t) $\mathsf{\mu}$(x, y, z, t) $\overrightarrow{{{F}}_{{L}}}$(x, y, z, t) t $\in \text{}\mathbb{R}\text{}$ (x, y, z)$\text{}\in \text{}\mathbb{R}\text{}$ ^{3} | Definition of Lorenz Gauge: $\mathrm{Div}\overrightarrow{{A}}=-\text{}\frac{\partial \mathsf{\varphi}}{\partial {t}}$ $\mathsf{\varphi}$ is fixed by initial boundary conditions, e.g., requiring it to zero sufficiently fast at $\infty $. | ${\nabla}^{2}\mathsf{\varphi}-\text{}\frac{{\partial}^{2}\text{}\mathsf{\varphi}}{\partial {{t}}^{2}}=-\text{}\mathsf{\rho}$ ${\nabla}^{2}\overrightarrow{{A}}-\text{}\frac{{\partial}^{2}\text{}\overrightarrow{{A}}}{\partial {{t}}^{2}}=-\mathsf{\rho}\overrightarrow{{v}}$ or, using the d’Alembertian: ${\square}^{2}\text{}\mathsf{\varphi}=-\text{}\mathsf{\rho}$ ${\square}^{2}\overrightarrow{{A}}=-\mathsf{\rho}\overrightarrow{{v}}$ $\mathsf{\mu}\frac{{d}\overrightarrow{{v}}}{{d}{t}}=\mathsf{\rho}\left(-\mathrm{Grad}\mathsf{\varphi}-\frac{{d}\overrightarrow{{A}}}{{d}{t}}+\mathrm{Grad}\left(\overrightarrow{{v}}\text{}\xb7\text{}\overrightarrow{{A}}\right)\right)$ (if $\overrightarrow{{{F}}_{{L}}}=\overrightarrow{{{F}}_{\mathrm{net}}}$) | $\overrightarrow{{j}}=\text{}\mathsf{\rho}\overrightarrow{{v}}$ $\overrightarrow{{B}}=\mathrm{Curl}\text{}\overrightarrow{{A}}$ $\overrightarrow{{E}}=-\mathrm{Grad}\left(\mathsf{\varphi}\right)$$-\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}$ |

Theory | Physical Ontology; Spatiotemporal Structure | Mathematical Representation of Physical Ontology | Purely Mathematical Facts | Nomology | Constitutive Principles of Derivative Ontology |
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Vector and scalar potentials in Lorenz Gauge, Mass density, Derived Charge Density Newtonian Space and Time | Vector Potential Scalar Potential Mass density Lorentz Force; Time 3-D Euclidean Absolute Space | $\overrightarrow{{A}}$(x, y, z, t) $\mathsf{\varphi}$(x, y, z, t) $\mathsf{\mu}$(x, y, z, t) $\overrightarrow{{{F}}_{{L}}}$(x, y, z, t) t $\in \text{}\mathbb{R}\text{}$ (x, y, z)$\text{}\in \text{}\mathbb{R}\text{}$ ^{3} | Definition: Lorenz Gauge $\mathrm{Div}\overrightarrow{{A}}=-\text{}\frac{\partial \mathsf{\varphi}}{\partial {t}}$ $\mathsf{\varphi}$ is also gauge-fixed if it has to go to zero sufficiently fast at $\infty $. | ${\nabla}^{2}\overrightarrow{{A}}-\text{}\frac{{\partial}^{2}\text{}\overrightarrow{{A}}}{\partial {{t}}^{2}}=-\mathsf{\rho}\overrightarrow{{v}}$ or, using the d’Alembertian: ${\square}^{2}\overrightarrow{{A}}=-\mathsf{\rho}\overrightarrow{{v}}$ $\mathsf{\mu}\frac{{d}\overrightarrow{{v}}}{{d}{t}}={\square}^{2}\mathsf{\varphi}\left(-\mathrm{Grad}\mathsf{\varphi}-\frac{{d}\overrightarrow{{A}}}{{d}{t}}+\mathrm{Grad}\left(\overrightarrow{{v}}\text{}\xb7\text{}\overrightarrow{{A}}\right)\right)$ (if $\overrightarrow{{{F}}_{{L}}}=$$\overrightarrow{{{F}}_{\mathrm{net}}}$) | $\overrightarrow{{j}}=\text{}\mathsf{\rho}\overrightarrow{{v}}$ $\overrightarrow{{B}}=\mathrm{Curl}\text{}\overrightarrow{{A}}$ $\overrightarrow{{E}}=-\mathrm{Grad}\left(\mathsf{\varphi}\right)$$-\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}$ ${\nabla}^{2}\mathsf{\varphi}-\text{}\frac{{\partial}^{2}\text{}\mathsf{\varphi}}{\partial {{t}}^{2}}=-\text{}\mathsf{\rho}$ or $\mathsf{\rho}=-{\square}^{2}\mathsf{\varphi}$ |

Theory | Physical Ontology; Spatiotemporal Structure | Mathematical Representation of Physical Ontology | Purely Mathematical Facts | Nomology | Constitutive Principles of Derivative Ontology |
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Vector and scalar potentials in Coulomb Gauge, Mass density, Charge Density Newtonian Space and Time | Vector Potential Scalar Potential Mass Density Charge Density Lorentz Force; Time 3-D Euclidean Absolute Space | $\overrightarrow{{A}}$(x, y, z, t) $\mathsf{\varphi}$(x, y, z, t) $\mathsf{\mu}$(x, y, z, t) $\mathsf{\rho}$(x, y, z, t) $\overrightarrow{{{F}}_{{L}}}$(x, y, z, t) t $\in \text{}\mathbb{R}\text{}$ (x, y, z)$\text{}\in \text{}\mathbb{R}\text{}$ ^{3} | Definition of Coulomb Gauge $\mathrm{Div}\overrightarrow{{A}}=0$ | $\text{}\frac{{\partial}^{2}\text{}\overrightarrow{{A}}}{\partial {{t}}^{2}}-{\nabla}^{2}\overrightarrow{{A}}=\overrightarrow{{j}}-\text{}\mathrm{Grad}\left(\text{}\frac{\partial \mathsf{\varphi}}{\partial {t}}\right)$ or, using the d’Alembertian: $-{\square}^{2}\overrightarrow{{A}}=\overrightarrow{{j}}-\text{}\mathrm{Grad}\left(\text{}\frac{\partial \mathsf{\varphi}}{\partial {t}}\right)$ $\mathsf{\mu}\frac{{d}\overrightarrow{{v}}}{{d}{t}}=-{\nabla}^{2}\mathsf{\varphi}\text{}\left(\mathrm{Grad}\left(\mathsf{\varphi}-\left(\overrightarrow{{v}}\text{}\xb7\text{}\overrightarrow{{A}}\right)\right)+\frac{{d}\overrightarrow{{A}}}{{d}{t}}\right)$ (if $\overrightarrow{{{F}}_{{L}}}=$$\overrightarrow{{{F}}_{\mathrm{net}}}$) $\mathsf{\varphi}\left(\overrightarrow{{x}},{t}\right)={{\displaystyle \int}}^{\text{}}\frac{\mathsf{\rho}\left(\overrightarrow{{x}\prime},{t}\right)}{|\overrightarrow{{x}}-\overrightarrow{{x}\prime}|}{{d}}^{3}\overrightarrow{{x}\prime}$ | $\overrightarrow{{j}}=\text{}\mathsf{\rho}\overrightarrow{{v}}$ $\overrightarrow{{B}}=\mathrm{Curl}\text{}\overrightarrow{{A}}$ $\overrightarrow{{E}}=-\mathrm{Grad}\left(\mathsf{\varphi}\right)$$-\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}$$-{\nabla}^{2}\mathsf{\varphi}=\text{}\mathsf{\rho}$ |

Theory | Physical Ontology; Spatiotemporal Structure | Math Representation of Physical Ontology | Purely Mathematical Facts | Nomology | Constitutive Principles of Derivative Ontology |
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Vector and scalar potentials in Coulomb Gauge, Mass density, Derived Charge Density Newtonian Space and Time | Vector Potential Scalar Potential Mass density Lorentz Force; Time 3-D Euclidean Absolute Space | $\overrightarrow{{A}}$(x, y, z, t) $\mathsf{\varphi}$(x, y, z, t) $\mathsf{\mu}$(x, y, z, t) $\overrightarrow{{{F}}_{{L}}}$(x, y, z, t) t $\in \text{}\mathbb{R}\text{}$ (x, y, z)$\text{}\in \text{}\mathbb{R}\text{}$ ^{3} | Definition of Coulomb Gauge $\mathrm{Div}\overrightarrow{{A}}=0$ ${{\displaystyle \int}}^{\text{}}\frac{\mathsf{\rho}\left(\overrightarrow{{x}\prime},{t}\right)}{|\overrightarrow{{x}}-\overrightarrow{{x}\prime}|\prime}{{d}}^{3}\overrightarrow{{x}\prime}$ | $\text{}\frac{{\partial}^{2}\text{}\overrightarrow{{A}}}{\partial {{t}}^{2}}-{\nabla}^{2}\overrightarrow{{A}}=\overrightarrow{{j}}-\text{}\mathrm{Grad}\left(\text{}\frac{\partial \mathsf{\varphi}}{\partial {t}}\right)$ or, using the d’Alembertian: $-{\square}^{2}\overrightarrow{{A}}=\overrightarrow{{j}}-\text{}\mathrm{Grad}\left(\text{}\frac{\partial \mathsf{\varphi}}{\partial {t}}\right)$ $\mathsf{\mu}\frac{{d}\overrightarrow{{v}}}{{d}{t}}=-{\nabla}^{2}\mathsf{\varphi}\text{}\left(\mathrm{Grad}\left(\mathsf{\varphi}-\left(\overrightarrow{{v}}\text{}\xb7\text{}\overrightarrow{{A}}\right)\right)+\frac{{d}\overrightarrow{{A}}}{{d}{t}}\right)$ (if $\overrightarrow{{{F}}_{{L}}}=$ $\mathsf{\varphi}\left(\overrightarrow{{x}},{t}\right)=$ $\overrightarrow{{{F}}_{\mathrm{net}}}$) | $\overrightarrow{{j}}=\text{}\mathsf{\rho}\overrightarrow{{v}}$ $\overrightarrow{{B}}=\mathrm{Curl}\text{}\overrightarrow{{A}}$ $\overrightarrow{{E}}=-\mathrm{Grad}\left(\mathsf{\varphi}\right)$ $-\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}$$-{\nabla}^{2}\mathsf{\varphi}=\text{}\mathsf{\rho}$ |

Theory | Physical Ontology; Spatiotemporal Structure | Mathematical Representation of Physical Ontology | Purely Mathematical Facts | Nomology | Constitutive Principles of Derivative Ontology |
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Vector and scalar potentials in Coulomb Gauge, Particles with Mass and Charge, Newtonian Space and Time | Vector Potential Scalar Potential Point Particles Particle Charge Particle Mass Lorentz Force Time 3-D Euclidean Absolute Space | $\overrightarrow{{A}}$(x, y, z, t) $\mathsf{\varphi}$(x, y, z, t) $\overrightarrow{{{x}}_{{i}}}$(t) = (x _{i}, y_{i}, z_{i}) q _{i}$\in $ R m _{i} $\in $ R > 0 $\overrightarrow{{{F}}_{{L}}}$(x, y, z, t) t $\in \text{}\mathbb{R}\text{}$ (x, y, z)$\text{}\in \text{}\mathbb{R}\text{}$ ^{3} | Definition: Coulomb Gauge $\mathrm{Div}\overrightarrow{{A}}=0$ | $\text{}\frac{{\partial}^{2}\text{}\overrightarrow{{A}}}{\partial {{t}}^{2}}-{\nabla}^{2}\overrightarrow{{A}}=\overrightarrow{{j}}-\text{}\mathrm{Grad}\left(\text{}\frac{\partial \mathsf{\varphi}}{\partial {t}}\right)$ or, using the d’Alembertian: $-{\square}^{2}\overrightarrow{{A}}=\overrightarrow{{J}}-\text{}\mathrm{Grad}\left(\text{}\frac{\partial \mathsf{\varphi}}{\partial {t}}\right)$ ${{m}}_{{i}}\frac{{d}\overrightarrow{{{v}}_{{i}}}}{{d}{t}}={{q}}_{{i}}\left(\mathrm{Grad}\left(\mathsf{\varphi}-\left(\overrightarrow{{{v}}_{{i}}}\text{}\xb7\text{}\overrightarrow{{A}}\right)\right)+\frac{{d}\overrightarrow{{A}}}{{d}{t}}\right)$ (if $\overrightarrow{{{F}}_{{L}}}=$$\overrightarrow{{{F}}_{\mathrm{net}}}$) $\mathsf{\varphi}\left(\overrightarrow{{x}},{t}\right)=\sum _{{n}=1}^{{n}={N}}\text{}\frac{{{q}}_{{i}}}{|\overrightarrow{{x}}-\overrightarrow{{x}\prime}|\prime}$ omitting points on particle worldlines | $\overrightarrow{{j}}=\text{}\mathsf{\rho}\overrightarrow{{v}}$ $\overrightarrow{{B}}=\mathrm{Curl}\text{}\overrightarrow{{A}}$ $\overrightarrow{{E}}=-\mathrm{Grad}\left(\mathsf{\varphi}\right)$$-\text{}\frac{\partial \overrightarrow{{A}}}{\partial {t}}$ $\overrightarrow{{v}}=\text{}\overrightarrow{{{v}}_{{i}}}=\frac{{d}\overrightarrow{{{x}}_{{i}}}\text{}\left({t}\right)}{{d}{t}}$ $\overrightarrow{{{J}}_{{i}}}=\text{}{{q}}_{{i}}\overrightarrow{{{v}}_{{i}}}$ |

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**MDPI and ACS Style**

Maudlin, T.
Ontological Clarity via Canonical Presentation: Electromagnetism and the Aharonov–Bohm Effect. *Entropy* **2018**, *20*, 465.
https://doi.org/10.3390/e20060465

**AMA Style**

Maudlin T.
Ontological Clarity via Canonical Presentation: Electromagnetism and the Aharonov–Bohm Effect. *Entropy*. 2018; 20(6):465.
https://doi.org/10.3390/e20060465

**Chicago/Turabian Style**

Maudlin, Tim.
2018. "Ontological Clarity via Canonical Presentation: Electromagnetism and the Aharonov–Bohm Effect" *Entropy* 20, no. 6: 465.
https://doi.org/10.3390/e20060465