# Ontologies in physics—6: A basic problem: How the mainstream QM views the variables in Schrodinger’s equation

1. Prologue:

From this post, at last, we begin tackling quantum mechanics! We will be covering those topics from the physics and maths of it which are absolutely necessary from developing our own ontological viewpoint.

We will first have a look at the most comprehensive version of the non-relativistic Schrodinger equation. (Our approach so far has addressed only the non-relativistic version of QM.)

We will then note a few points concerning the way the mainstream physics (MSMQ) de facto approaches it—which is remarkably different from how engineers regard their partial differential equations.

In the process, we will come isolate and pin down a basic issue concerning how the two variables $\Psi$ and $V$ from Schrodinger’s equation are to be seen.

We regard this issue as a problem to be resolved, and not as just an unfamiliar kind of maths that needs no further explanation or development.

OK. Let’s get going.

2. The $N$-particle Schrodinger’s equation:

Consider an isolated system having $3D$ infinite space in it. Introduce $N$ number of charged particles (EC Objects in our ontological view) in it. (Anytime you take arbitrary number of elementary charges, it’s helpful to think of them as being evenly spread between positive and negative polarities, because the net charge of the universe is zero.) All the particles are elementary charges. Thus, $-|q_i| = e$ for all the particles. We will not worry about any differences in their masses, for now.

Following the mainstream QM, we also imagine the existence of something in the system such that its effect is the availability of a potential energy $V$.

The multi-particle time-dependent Schrodinger equation now reads:

$i\,\hbar \dfrac{\partial \Psi(\vec{R},t)}{\partial t} = - \dfrac{\hbar^2}{2m} \nabla^2 \Psi(\vec{R},t) + V(\vec{R},t)\Psi(\vec{R},t)$

Here, $\vec{R}$ denotes a set of particle positions, i.e., $\vec{R} = \lbrace \vec{r}_1, \vec{r}_2, \vec{r}_3, \dots, \vec{r}_N \rbrace$. The rest of the notation is standard.

3. The mainstream view of the wavefunction:

The mainstream QM (MSMQ) says that the wavefunction $\Psi(\vec{R},t)$ exists not in the physical $3$-dimensional space, but in a much bigger, abstract, $3N$-dimensional configuration space. What do they mean by this?

According to MSQM, a particle’s position is not definite until it is measured. Upon a measurement for the position, however, we do get a definite $3D$ point in the physical space for its position. This point could have been anywhere in the physical $3D$ space spanned by the system. However, measurement process “selects” one and only one point for this particle, at random, during any measurement process. … Repeat for all other particles. Notice, the measured positions are in the physical $3D$.

Suppose we measure the positions of all the particles in the system. (Actually, speaking in more general terms, the argument applies also to position variables before measurement concretizes them to certain values.)

Suppose we now associate the measured positions via the set $\vec{R} = \lbrace \vec{r}_1, \vec{r}_2, \vec{r}_3, \dots, \vec{r}_N \rbrace$, where each $\vec{r}_i$ refers to a position in the physical $3D$ space.

We will not delve into the issue of what measurement means, right away. We will simply try to understand the form of the equation. There is a certain issue associated with its form, but it may not become immediately apparent, esp. if you come from an engineering background. So, let’s make sure to know what that issue is:

Following the mainstream QM, the meaning of the wavefunction $\Psi$ is this: It is a complex-valued function defined over an abstract $3N$-dimensional configuration space (which has $3$ coordinates for each of the $N$ number of particles).

The meaning of any function defined over an abstract $3ND$ configuration space is this:

If you take the set of all the particle positions $\vec{R}$ and plug them into such a function, then it evaluates to some single number. In case of the wavefunction, this number happens to be a complex number, in general. (Remember, all real numbers anyway are complex numbers, but not vice-versa.) Using the C++ programming terms, if you take real-valued $3D$ positions, pack them in an STL vector of size $N$, and send the vector into the function as an argument, then it returns just one specific complex number.)

All the input arguments (the $N$-number of $3D$ positions) are necessary; they all taken at once produce the value of the function—the single number. Vary any Cartesian component ($x$, $y$, or $z$) for any particle position, and $\Psi$ will, in general, give you another complex number.

Since a $3D$ space can accommodate only $3$ number of independent coordinates, but since all $3N$ components are required to know a single $\Psi$ value, it can only be an abstract entity.

Got the argument?

Alright. What about the term $V$?

4. The mainstream view of $V$ in the Schrodinger equation:

In the mainstream QM, the $V$ term need not always have its origin in the electrostatic interactions of elementary point-charges.

It could be any arbitrary source that imparts a potential energy to the system. Thus, in the mainstream QM, the source of $V$ could also be gravitational, magnetic, etc. Further, in the mainstream QM, $V$ could be any arbitrary function; it doesn’t have to be singularly anchored into any kind of point-particles.

In the context of discussions of foundations of QM—of QM Ontology—we reject such an interpretation. We instead take the view that $V$ arises only from the electrostatic interactions of charges. The following discussion is written from this viewpoint.

It turns out that, speaking in the most fundamental and general terms, and following the mainstream QM’s logic, the $V$ function too must be seen as a function that “lives” in an abstract $3ND$ configuration space. Let’s try to understand a certain peculiarity of the electrostatic $V$ function better.

Consider an electrostatic system of two point-charges. The potential energy of the system now depends on their separation: $V = V(\vec{r}_2 - \vec{r}_1) \propto q_1q_2/|\vec{r}_2 - \vec{r}_1|$. But a separation is not the same as a position.

For simplicity, assume unit positive charges in a $1D$ space, and the constant of proportionality also to be $1$ in suitable units. Suppose now you keep $\vec{r}_1$ fixed, say at $x = 0.0$, and vary only $\vec{r}_2$, say to $x = 1.0, 2.0, 3.0, \dots$, then you will get a certain series of $V$ values, $1.0, 0.5, 0.33\dots, \dots$.

You might therefore be tempted to imagine a $1D$ function for $V$, because there is a clear-cut mapping here, being given by the ordered pairs of $\vec{r}_2 \Rightarrow V$ values like: $(1.0, 1.0), (2.0, 0.5), (3.0, 0.33\dots), \dots$. So, it seems that $V$ can be described as a function of $\vec{r}_2$.

But this conclusion would be wrong because the first charge has been kept fixed all along in this procedure. However, its position can be varied too. If you now begin moving the first charge too, then using the same $\vec{r}_2$ value will gives you different values for $V$. Thus, $V$ can be defined only as a function of the separation space $\vec{s} = \vec{r}_2 - \vec{r}_1$.

If there are more than two particles, i.e. in the general case, the multi-particle Schrodinger equation of $N$ particles uses that form of $V$ which has $N(N-1)$ pairs of separation vectors forming its argument. Here we list some of them: $\vec{r}_2 - \vec{r}_1, \vec{r}_3 - \vec{r}_1, \vec{r}_4 - \vec{r}_1, \dots$, $\vec{r}_1 - \vec{r}_2, \vec{r}_3 - \vec{r}_2, \vec{r}_4 - \vec{r}_2, \dots$, $\vec{r}_1 - \vec{r}_3, \vec{r}_2 - \vec{r}_3, \vec{r}_4 - \vec{r}_1, \dots$, $\dots$. Using the index notation:

$V = \sum\limits_{i=1}^{N}\sum\limits_{j\neq i, j=1}^{N} V(\vec{s}_{ij})$,

where $\vec{s}_{ij} = \vec{r}_j - \vec{r}_i$.

Of course, there is a certain redundancy here, because the $s_{ij} = |\vec{s}_{ij}| = |\vec{s}_{ji}| = s_{ji}$. The electrostatic potential energy function depends only on $s_{ij}$, not on $\vec{s}_{ij}$. The general sum formula can be re-written in a form that avoids double listing of the equivalent pairs of the separation vectors, but it not only looks a bit more complicated, but also makes it somewhat more difficult to understand the issues involved. So, we will continue using the simple form—one which generates all possible $N(N-1)$ terms for the separation vectors.

If you try to embed this separation space in the physical $3D$ space, you will find that it cannot be done. You can’t associate a unique separation vector for each position vector in the physical space, because associated with any point-position, there come to be an infinity of separation vectors all of which have to be associated with it. For instance, for the position vector $\vec{r}_2$, there are an infinity of separation vectors $\vec{s} = \vec{a} - \vec{r}_2$ where $\vec{a}$ is an arbitrary point (standing in for the variable $\vec{r}_1$). Thus, the mapping from a specific position vector $\vec{r}_2$ to potential energy values becomes an $1: \infty$ mapping. Similarly for $\vec{r}_1$. That’s why $V$ is not a function of the point-positions in the physical space.

Of course, $V$ can still be seen as proper $1:1$ mapping, i.e., as a proper function. But it is a function defined on the space formed by all possible separation vectors, not on the physical space.

Homework: Contrast this situation from a function of two space variables, e.g., $F = F(\vec{x},\vec{y})$. Explain why $F$ is a function (i.e. a $1:1$ mapping) that is defined on a space of position vectors, but $V$ can be taken to be a function only if it is seen as being defined on a space of separation vectors. In other words, why the use of separation vector space makes the $V$ go from a $1:\infty$ mapping to a $1:1$ mapping.

5. Wrapping up the problem statement:

If the above seems a quizzical way of looking at the phenomena, well, that precisely is how the multi-particle Schrodinger equation is formulated. Really. The wavefunction $\Psi$ is defined on an abstract $3ND$ configuration space. Really. The potential energy function $V$ is defined using the more abstract notion of the separation space(s). Really.

If you specify the position coordinates, then you obtain a single number each for the potential energy and the wavefunction. The mainstream QM essentially views them both as aspatial variables. They do capture something about the quantum system, but only as if they were some kind of quantities that applied at once to the global system. They do not have a physical existence in the $3D$ space-–even if the position coordinates from the physical $3D$ space do determine them.

In contrast, following our new approach, we take the view that such a characterization of quantum mechanics cannot be accepted, certainly not on the grounds as flimsy as: “That’s just how the math of quantum mechanics is! And it works!!” The grounds are flimsy, even if a Nobel laureate or two might have informally uttered such words.

We believe that there is a problem here: In not being able to regard either $\Psi$ or $V$ as referring to some simple ontological entities existing in the physical $3D$ space.

So, our immediate problem statement becomes this:

To find some suitable quantities defined on the physical $3D$ space, and to use them in such a way, that our maths would turn out to be exactly the same as given for the mainstream quantum mechanics.

6. A preview of things to come: A bit about the strategy we adopt to solve this problem:

To solve this problem, we begin with what is easiest to us, namely, the simpler, classical-looking, $V$ function. Most of the next post will remain concerned with understanding the $V$ term from the viewpoint of the above-noted problem. Unfortunately, a repercussion would be that our discussion might end up looking a lot like an endless repetition of the issues already seen (and resolved) in the earlier posts from this series.

However, if you ever suspect, I would advise you to keep the doubt aside and read the next post when it comes. Though the terms and the equations might look exactly as what was noted earlier, the way they are rooted in the $3D$ reality and combined together, is new. New enough, that it directly shows a way to regard even the $\Psi$ field as a physical $3D$ field.

Quantum physicists always warn you that achieving such a thing—a $3D$ space-based interpretation for the system-$\Psi$—is impossible. A certain working quantum physicist—an author of a textbook published abroad—had warned me that many people (including he himself) had tried it for years, but had not succeeded. Accordingly, he had drawn two conclusions (if I recall it right from my fallible memory): (i) It would be a very, very difficult problem, if not impossible. (ii) Therefore, he would be very skeptical if anyone makes the claim that he does have a $3D$-based interpretation, that the QM $\Psi$ “lives” in the same ordinary $3D$ space that we engineers routinely use.

Apparently, therefore, what you would be reading here in the subsequent posts would be something like a brand-new physics. (So, keep your doubts, but hang on nevertheless.)

If valid, our new approach would have brought the $\Psi$ field from its $3N$-dimensional Platonic “heaven” to the ordinary physical space of $3$ dimensions.

“Bhageerath” (भगीरथ) [^] ? … Well, I don’t think in such terms. “Bhageerath” must have been an actual historical figure, but his deeds obviously have got shrouded in the subsequent mysticism and mythology. In any case, we don’t mean to invite any comparisons in terms of the scale of achievements. He could possibly serve as an inspiration—for the scale of efforts. But not as an object of comparison.

All in all, “Bhageerath”’s deed were his, and they anyway lie in the distant—even hazy—past. Our understanding is our own, and we must expend our own efforts.

But yes, if found valid, our approach will have extended the state of the art concerning how to understand this theory. Reason good enough to hang around? You decide. For me, the motivation simply has been to understand quantum mechanics right; to develop a solid understanding of its basic nature.

Bye for now, take care, and sure join me the next time—which should be soon enough.

A song I like:

[The official music director here is SD. But I do definitely sense a touch of RD here. Just like for many songs from the movie “Aaraadhanaa”, “Guide”, “Prem-Pujari”, etc. Or, for that matter, music for most any one of the movies that the senior Burman composed during the late ’60s or early ’70s. … RD anyway was listed as an assistant for many of SD’s movies from those times.]

(Hindi) “aaj ko junali raat maa”
Music: S. D. Burman
Lyrics: Majrooh Sultanpuri

History:
— First published 2019.10.13 14:10 IST.
— Corrected typos, deleted erroneous or ill-formed passages, and improved the wording on home-work (in section 4) on the same day, by 18:29 IST.
— Added the personal comment in the songs section on 2019.10.13 (same day) 22:42 IST.

# Ontologies in physics—5: Energy-based analysis of EM force-fields

1. A recap of the physics of EM force-fields:

Let me recap the most salient parts of the discussion we’ve so far had, in this series, concerning the physics of the electrostatic forces and fields. As before, we will continue to ignore the specifically dynamical effects in EM. Thus, the positions of charges are fixed in space at any instant.

1.1 Coulomb’s Law:

Suppose there are two charges. Then there arise two forces: $\vec{F}_{12}$ which is exerted by $q_1$ on $q_2$ at $\vec{r}_2$, and $\vec{F}_{21}$ which is exerted by $q_2$ on $q_1$ at $\vec{r}_1$. They happen to be equal in magnitude but opposite in directions, exactly as if they were a pair of the direct contact-forces described in the NM ontology. They obey the inverse-square relation for separation, exactly as in gravity.

1.2 A mathematical function of hypothetical forces distributed over space:

We keep any one of the two charges (say $q_1$) fixed, and imagine what would happen when the other charge (say $q_T$) were to be placed at all different points in the infinite space, one position at a time. We thus generate an infinite set of $\vec{r}_2 \leftrightarrow \vec{F}_{1T}$ pairs. Another name for this mapping is: the mathematical force-field, a function of space. This function is only a mathematical convenience. It represents a mathematical generalization of Coulomb’s law to cover the entire space.

Let the force-field function for (i.e. associated with) $q_1$ be denoted as $\vec{F}_{1}(\vec{r} - \vec{r}_1)$. Here, the variable $\vec{r}$ successively assumes all points in the entire space; it is inspired from but only abstractly represents, the position vector for the second charge. It does not represent the actual position of an actual charge unless we fix the position of the second charge via the initial/boundary conditions of a problem, and substitute such a position vector in place of $\vec{r}$. So, in the absence of BC/IC’s of a problem, $\vec{r}$ remains just a dummy variable whose sole purpose to allow us to define the field function. The field is, of course, determined by Coulomb’s law:
$\vec{F}_{1}(\vec{r} - \vec{r}_1) = \dfrac{1}{4\,\pi\,\epsilon_0}\;\dfrac{q_1\,q_T}{r^2}\,\hat{r}$, where $r = |\vec{r} - \vec{r}_1|$ and $\hat{r} = \dfrac{(\vec{r} - \vec{r}_1)}{r}$.

1.3 Maxwell’s field idea:

Maxwell took the idea of the mathematical field functions from the Continental physicists, and synthesized a set of coupled differential equations that together captured all the known laws of electromagnetism (in a form that didn’t have Lorentz’ force law as a separate equation). At the same time, Maxwell also followed the conceptual lead provided by Faraday, and asserted that the field function wasn’t just a mathematical device; it represented something that existed physically. Accordingly, the field was to be seen as a continuously existing condition existing in a mechanical aether. Thus, the ontological change is from “mathematical field function” to “physical force fields”.

1.4 Lorentz’ idea of the EM aether:

The aether is electrical in nature, not mechanical. The actually existing field quantity is the electric vector, $\vec{E}$, which is defined as $\vec{E}_{1}(\vec{r} - \vec{r}_1) = \dfrac{\vec{F}_{1}(\vec{r} - \vec{r}_1)}{q_T}$. He also recast Maxwell’s original equations into a simplified form of a set of four equations, and further added an equation (known by his name) so as to make classical EM description complete.

1.5 Our modification to Lorentz’ idea:

The charges $q_1$ and $q_2$ are not arbitrary; their magnitudes are always equal to the electronic charge (within the algebraic sign). The quantity $q_T$ appearing in field definition itself is always $e$, the fundamental constant of electronic charge. Hence, there is no need to scale $\vec{F}$ to $q_T$. Accordingly, the field which actually exists in “empty space” is that of $\vec{F}$.

I have left out most of the ontological points from this recap.

2. Superposition of force-fields:

Fields due to multiple charges superpose: their net effect is given by the algebraic sum of the independent fields respectively produced by the charges in question. Thus, if there are two charges $q_1$ and $q_2$ present in an isolated system, then we have these two fields in it:
$\vec{F}_{1}(\vec{r} - \vec{r}_1) = \dfrac{1}{4\,\pi\,\epsilon_0}\;\dfrac{q_1\,q_T}{r^2}\,\hat{r}$, where $r = |\vec{r} - \vec{r}_1|$ and $\hat{r} = \dfrac{(\vec{r} - \vec{r}_1)}{r}$,
and
$\vec{F}_{2}(\vec{r} - \vec{r}_2) = \dfrac{1}{4\,\pi\,\epsilon_0}\;\dfrac{q_2\,q_T}{r^2}\,\hat{r}$, where $r = |\vec{r} - \vec{r}_2|$ and $\hat{r} = \dfrac{(\vec{r} - \vec{r}_2)}{r}$.
The total field in the system is then given by:
$\vec{F}_{\text{sys}} = \vec{F}_{1}(\vec{r} - \vec{r}_1) + \vec{F}_{2}(\vec{r} - \vec{r}_2)$.
Notice that both the terms on the right hand-side denote a field that actually exists, but whose magnitude is to be calculated in reference to $q_T$. Thus, now, the test charge $q_T$ is the third charge.

The first two charges form the actual system; the third charge is a device of calculations that provide the measure of the forces which anyway exist even in its absence.

3. Force-fields as physical, not mathematical entities—same maths but different ontology:

In going from the mathematical field function to Maxwell’s fields, the maths remains exactly the same, but there is a remarkable change in the ontology. The difference in the two ideas can perhaps be better illustrated via an error of thought.

3.1 A wrong imagination:

Suppose we define a field for a single charge. Let’s reproduce the equation for convenience, but let’s note the test charge explicitly:
$\vec{F}_{1}(\vec{r}_T - \vec{r}_1) = \dfrac{1}{4\,\pi\,\epsilon_0}\;\dfrac{q_1\,q_T}{r^2}\,\hat{r}$, where $r = |\vec{r}_T - \vec{r}_1|$ and $\hat{r} = \dfrac{(\vec{r}_T - \vec{r}_1)}{r}$.

Since Coulomb’s law is symmetrical, it might be tempting to introduce a similar force field function, now for the test charge $q_T$; it could be given as:
$\vec{F}_{T}(\vec{r}_1 - \vec{r}_T) = \dfrac{1}{4\,\pi\,\epsilon_0}\;\dfrac{q_1\,q_T}{r^2}\,\hat{r}$,
where $r = |\vec{r}_1 - \vec{r}_T|$, $\hat{r} = \dfrac{(\vec{r}_1 - \vec{r}_T)}{r}$, and $\vec{r}_1$ is the variable position of the first charge, which now acts as a test charge for the test charge [sic] $q_T$.

Further, it may then also be tempting to think of a net or total force field for the system. We might try to give it as the vector sum of the two:
$\vec{F}_{\text{sys}} = \vec{F}_{1}(\vec{r} - \vec{r}_1) + \vec{F}_{T}(\vec{r} - \vec{r}_T)$

Inasmuch as the above two equations basically refer to an actual presence of an actual test charge, the whole idea becomes plain wrong! Let alone the physical fields, even the purely mathematical idea of fields does not allow for an actual existence of a test charge. The mathematicians begin to derive their (field-) “functions” using the device of a test charge, but they also unhesitatingly drop it from all considerations once it has served the purpose of giving them their prized equations. They feel no guilt about it, because they almost never notice the practice of context-dropping which is so wide-spread in their community.

But of course, even purely mathematically, the whole scheme is anyway weird because if the test charge were to not exist, you couldn’t define the mathematical field function. But if it were to regarded as actually existing, then it would have to have its own field too, but such a field would not remain single-valued. It would go on acquiring different values as you shifted $q_T$ to different locations. So, the total (system-wide) field wouldn’t any longer be a $1:1$ mapping. It would become a $1:\infty$ mapping.

Thus, despite its neat appearance, the “equations” involving the test charge are invalid. They do not define a definite function; something definite doesn’t come out of this procedure.

We need not take the errors or the context-drops further, because Faraday and Maxwell anyway had something entirely different—far better—in mind anyway!

3.2 Faraday and Maxwell’s idea regarding fields:

Following Faraday and Maxwell’s field idea, what physically exists are only (i) the first charge $q_1$, and (ii) the field condition $\vec{F}_{1}(\vec{r} - \vec{r}_1)$ it produces in the aether, that’s all!

The test charge is not there even as a mathematical device; it’s not present even just “virtually” (whatever it might mean). Following their logic (even if not their original writings), the test charge is simply non-existent in their description.

If we are to follow Maxwell’s scheme, the variable $\vec{r}_T$ becomes $\vec{r}$; it actually ranges over the entire space in defining the field function. The variable $q_T$ is factored out by division of $\vec{F}$ to $\vec{E}$. As to the force $\vec{F}$ itself, it is not a force on a charge, but a force that a small CV within the aether exerts on its neighbouring portions.

Thus, when we make a transition from the mathematical field function to the physical field, the test charge ceases to have any existence in reality—and any relevance in mathematics. Instead, each small CV of the domain itself is described to carry some such a “forceful” or “force-producing” condition at all times. If a second charge (say a test charge) is then introduced in that CV, it merely feels the condition which is already present there.

The force-condition represented by the Maxwellian field is always present. Its existence is not conditioned upon the existence of the second charge. In fact it’s the other way around. The force on the second charge is conditioned upon the existence of the field in its neighbourhood, and the field everywhere in space is produced by the distant first charge.

So, a test charge $q_T$ begins to appear in the Maxwellian description only from the time it is physically introduced at one (and only one) point in the field. The charge then passively senses the already existing field condition. The field condition was always existing there even in the absence of the test charge. There is no special status to the test charge—it is as good as any other charge.

4. Ontological implications of the fact that in the fields-based view, charges don’t interact with each other—at all!

In the field-theoretical view, the charges never interact with each other—directly or indirectly. They interact only with a field—in fact, with only that local portion of a field which is directly adjacent to them. The interaction of a charge and a field, or vice versa, occurs only via the direct contact.

I don’t know if Maxwell or Lorentz thought this way, but taking their idea to its logical end, if two charges are “somehow” brought in a direct contact, I think, they wouldn’t even force each other at the point of touch. They simply wouldn’t “know” what to do with the other charge—through the touch. Nature has “taught” them to deal with (i.e., either force or be forced by) only a field—in fact only that surface portion of the field which is in direct contact with them. The charges don’t “know” how to force (or be forced by) anything other than a field—be it a chargeless EM object, a charged EC object, or even a CV of the field that is not in direct touch to them. A direct charge-to-charge interaction is not at all defined in the fields idea of Maxwellian EM. Neither is an action that skips the intervening portions of the aether.

The logical consequence is this:

If the charges are finite-sized, e.g. spherical in shape, then they would touch each other only at one point. They will continue exchanging forces with the field at all other points on the the spherical boundary because  these other points remain in touch with the field. But the point of touch would contribute nothing.

If charges are point-particles, and if they are brought progressively closer, then they would exert ever increasing forces (attractive or repulsive) on each other. The forces would even approach infinity as the separation goes on decreasing. However, at the point of an actual “touch”, the forces should simply disappear, because the intervening field no longer can fit in between them.

If the charges in question are two point-particles, then a direct contact can only occur when both are literally at the same point-position. The implication is that one electron and one proton, when placed at the same point, could possibly exert no forces on each other. Further, since both their fields are singularly anchored in their respective positions, and since the electric charge they carry also is identical in magnitude though opposite in sign, their respective fields would cancel each other at every point in space. Thus, the net field would be zero—the entire infinitely spread field would simply disappear. The MIT notes [^] illustrate this situation via a simulation.

Thus, we have two points of view here:

1. According to the mainstream physics (as in the MIT notes), when two opposite point-charges occupy the same location, there is no net force left anywhere in the entirety of space. The isolated point where both the point-charges are present is excluded from analysis anyway.
2. Additionally, we can say something more following our ontological insights: There would be no force between the two point-charges either. That’s because charges interact only with fields, and no space is left for intervening field to occupy if both charges are “on top of each other.”

Funny.

Just one more point. When the interacting point-charges are elementary (as electrons and protons are), in both of the aforementioned viewpoints, the charges do not get discharged even when if they are of opposite polarities and even if they are present at the same point. However, inasmuch as the only way for a charge to make its presence felt is via its interactions with a field, two opposite charges existing behave as if they were temporarily discharged.

On the question of whether charges, when temporarily discharged as described above, continue to retain their attribute of inertia or not. To say that the inertia does not get affected is to ascribe a non-electrical attribute to the charges. I gather that Lorentz had put forth some idea of the entire mass being only electrical in nature. I have not thought about it so far, and so, do not take any definitive position about this issue.

This paragraph inserted via an update on 2019.10.07 11:39 IST: I do tend to think that inertia does exist as an attribute separate from the electrical charge, and so, even when two opposite charges occupying the same location get temporarily “discharged,” they still retain their respective inertias. That’s because I think that explaining mass in electrical terms alone has a certain weakness. Think: Once the charges are effectively “discharged,” the whole space ceases to have a net force-field. This implies a discontinuity in the existence of the internal energy too (at least in a naive argument about it all), which would violate the first law of thermodynamics for that state. Further, suppose that the two charges do re-emerge from the discharged state. The question is: Why should their “re-charging” occur only in the neighbourhood of the point of the “discharge”? In an infinite space, they could have emerged anywhere else too. All in all, the whole thing gets more and more complicated and unsatisfactory. So, it seem better to regard the electrical charge and the mechanical inertia (the so-called “rest mass” of special relativity) as two independent attributes. Anyway, please disregard the more speculative discussions like these. (Update on 2019.10.07 11:39 IST over.)

To summarize what we can definitely say:

In the fields idea, every single charge causes local force conditions to come to exist in all parts of aether. On the other side of the equation, any given charge also gets forced due to the field generated by all the other charges; due to Gauss’ theorem, it cannot however feel the force of the field generated by itself. The field in its neighbourhood is all that a given charge “knows” about. It has no other mechanism to come to “know” if there are any other charges in existence anywhere or not; it even can’t “tell” whether the same aether has some other parts at some other locations or not. A lone charge doesn’t require a second charge (as a test charge) in order to cause its field to come into existence (which is of $\vec{E}$ for arbitrary charges, and of $\vec{F}$ for elementary charges). A lone charge too causally and inevitably creates a physical field purely out of its own independent existence, and vice versa. A charge and its field always go together, even if the field is the attribute of the aether, and not of the EC Object which causes it. A charge-less field is an ontological impossibility, and so it a field-less charge. For the latter reason, fields cannot be specified arbitrarily.

With that extra clarification, let’s resume our coverage of the energy-analysis aspects, building on what we saw the last time [^].

5. Potential energy of a spring-mass system as a variable but single number:

5.1 Analysis with one ball fixed:

Consider two NM objects, say two steel balls, connected by a spring. We idealize the balls as point-particles having mass, and the spring as having stiffness but no mass.

For convenience, keep the left ball fixed in space; the ball on the right can be moved. This mass-spring arrangement forms our thermodynamic system. Let the variable $\vec{r} = \vec{x} - \vec{x}_0$ denote the separation of the movable ball from its neutral position $\vec{x}_0$.

The resistive force that the spring (within the system) exerts on the second ball is given by $\vec{F}_{12} = -k \vec{r}$. Work is done by some other force in the environment on the system when the second ball is moved against this resistive force. The action results in an increase in the internal energy of the system, given by:
$\text{d}U = -\text{d}W = \vec{F}_{\text{env}} \cdot \text{d} \vec{r} = (-\ \vec{F}_{12}) \cdot \text{d} \vec{r}$
Notice that we’ve put a negative sign in front of $\vec{F}_{12}$ because $\vec{F}_{\text{env}}$ acts in opposite direction to the spring force $\vec{F}_{12}$. Using Hooke’s law for the spring force, we get:
$\text{d}U = -\text{d}W = k \vec{r} \cdot \text{d} \vec{r} = k r \text{d}r$
After integrating between two arbitrary $r$ values $r_1$ and $r_2$, we find that
$U(r_2) - U(r_1) = \dfrac{1}{2}k \left( r_2^2 - r_1^2 \right)$ where $U(r_i)$ is the energy at the position indexed by $i$.

Now, we note that $\vec{F}_{12} = 0$ for $r = 0$. This is because the neutral position $x_0$ remains fixed in space because the left hand-side ball is fixed.

Further, for convenience, we also choose to set $r_1 = 0$.

Finally, and separately, we also choose to have $U(r_1) = 0$.

Notice that $\vec{F}_{12}$ is physically zero at $r = 0$, whereas $r_1$ being set to zero is just a mathematically convenient choice for the origin of the coordinate frame. But these preconditions still do not mandate that $U(r_1)$ also has to be regarded as zero. If $\vec{F}_{12}$ is zero, only $\text{d}U$ is zero in the infinitesimal neighbourhood of $\vec{r}_1$. But $\text{d}U$ being zero at $r_1$ does not mean that $U$ also has to be zero there. (There is an infinity of parallel lines all having the same non-zero slope such that none of them passes through the origin.) In short, the constant of integration could have been any number. But we deliberately choose it to be zero, purely out of convenience of arithmetical manipulations.

With these choices, we now get to the simpler expression:
$U(r) = \dfrac{1}{2}k r^2$

Obviously, by differenting the above expression, we can see that:
$\vec{F}_{12}(r) = -\ \nabla U(r)$

The spring-mass system is a $1D$ example, but similar examples can be constructed where a conservative force varies in $3D$ space. Hence the more general expression of $\nabla$.

At any specific position $x$ of the second ball, there is a single unique number for the potential energy of the system $U$. This fact holds for all conservative forces.

(A conservative force is a position-dependent force such that if the particle undergoes arbitrary displacements, and then is brought back to the original position, not only the force it experiences at that position but also its entire dynamical state (including its velocity, acceleration and all the infinity of the higher derivatives) is exactly as it was initially. In this consideration, we neglect all the other agents, and consider only one force at a time. Friction is not a conservative force. A conservative force is frictionless.)

5.2 Analysis for the case when both balls undergo arbitrary displacements:

The above description can be generalized in the situation in which both the balls are movable. Here is an outline.

Let $x_{L_0}$ and $x_{R_0}$ respectively denote the absolute positions of the balls on the left and the right in the undeformed state, and let their positions in the deformed state respectively be $x_{L_1}$ and $x_{R_1}$. The two forces exerted by the spring on the two balls are given as: $\vec{F}_L = -k(x_{L_1} - x_{L_0})$ and $\vec{F}_R = -k(x_{R_1} - x_{R_0})$. An environmental force does work on the system by acting on the left ball (i.e., against $\vec{F}_L$), and another environmental force does work on the system by acting on the right side-ball (i.e., against $\vec{F}_L$). The quantum of the two work done on the system are: $-\ \text{d} W_L = -(\vec{F}_{L}) \cdot \text{d} \vec{x}_L$, where $\text{d}x_L$ is the infinitesimal change in the position of the left hand-side ball. Similarly, $-\ \text{d} W_R = -(\vec{F}_{R}) \cdot \text{d} \vec{x}_R$. The total work done on the system is then given by their algebraic addition: $-\ \text{d} W_{\text{sys}} = -\ \text{d} W_L + (-\ \text{d} W_R)$. Integrating, we get the increase in the internal energy $U_{\text{sys}}$ associated with the two finite displacements of the balls.

Proving that for such a system (both balls movable), the internal energy $U$ so obtained is identical to the case in which only one ball is movable, is left as an exercise for the interested reader.  (You need to just conduct the integrations and note that the separation between two balls is the algebraic sum of the individual displacements at each end of the spring. Thus, the basic idea is that the potential energy of the system is a number which is a function of only the separation vector of the two balls taken together.)

The above sketchy outline for a two-ball system was very general in the sense that the environmental forces $-\vec{F}_{L}$ and $-\vec{F}_{R}$ had to be equal, but they didn’t have to displace the respective balls through an equal distance. (Think of an additional spring on the left and right of our system, and assume that their stiffnesses are neither equal to each other nor to the stiffness of the middle spring. In this case, the forces on our (middle) spring will remain equal, but the two balls would have been displaced to different extents.)

If we further assume that both the balls also displace through the same distance, then we can split the total increase in the internal energy of the system into two equal components, one each for a moving boundary at a ball. Thus,

$x_{L_1} - x_{L_0} = x_{R_1} - x_{R_0} = \Delta x$, $U_{\text{sys}} = U_L + U_R$ and $U_L = U_R$, so that

$U_{\text{sys}} = \dfrac{1}{2}\sum\limits_{i=1}^{2} k (\Delta x)^2$.

The assumption that each ball in a pair displaces through the same distance is justifiable if the spring itself is without inertia and so its motion doesn’t involve transmission of momentum, so that the interactions of the spring with the environmental forces (through the massive point-particles of balls) remains independent of the position of the spring or the state of deformation it is in.

6. Electrostatic potential energy of a pair of EC objects as a variable but single number (a global attribute):

6.1 Preliminaries:

EM forces superpose. Therefore, for most ontological purposes, discussions in terms of a single pair of charges is enough. But note that at least two charges must be considered. Having just one charge (and its force-field) won’t do, because a physically isolated single charge experiences no forces—not from its own field. (Here, we don’t have to make an appeal to a spherical symmetry; it’s enough to invoke Gauss’ divergence theorem; looking up the proof is left as an exercise for the reader.)

6.2 Potential energy of a pair of charges:

Consider a two charge system once again. As before, the first charge remains fixed in space; this is purely for convenience. The second charge can be placed at different points. The thermodynamic boundary exists at the two points and nowhere else.

As the position of the second charge changes, we obtain a different configuration. Work must be done on the system to move the second charge. (Force must be exerted by the environment and on the system in order to keep the first charge where it is. However, since the first charge does not undergo displacement, no work is done on the system by this force.) In EM systems, the sign of the work can get further confusing, because forces can be both attractive or repulsive. This is in addition to the standard thermodynamic sign convention. Read the description below accordingly. (Best is to work with the algebraic term $q$ all throughout, and then to put specific charge values only in the end.)

By the first law (and neglecting heat), $\text{d}U = -\delta W$, where the $\delta W$ is the work done by the system. Work crosses system boundary because when a charge shifts, the boundary fixed at it gets shifted too. (This is electro-statics. So, the shift is in the variational sense. There is no motion, no displacement in time; just the fact of a difference between two separation vectors.)

Work done by the system is defined through $\text{d}W = \vec{F}_{12} \cdot \text{d}\vec{r}$. So, if the second charge is brought nearer, the work done on the system i.e. the increase in the internal energy is:
$\text{d}U = -\ (\vec{F}_{12}) \cdot (-\text{d} \vec{r})$
After integrating we find that
$U_f - U_i = \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_1\,q_2}{r}$

As the absolute position of the second charge $\vec{r}_2 \rightarrow \infty$, the difference in the internal energies in two infinitesimally close configurations $\text{d}U \rightarrow 0$. In other words, $U_f$ and $U_i$ approach the same value. Note, it’s the difference which approaches zero, not the respective values of the individual $U_f$ and $U_i$ terms. If one of them is specified as a boundary condition “at” infinity, the other value would be infinitesimally close to the same value.

Here, we introduce a convention: As the second charge goes infinitely away, the force is anyway dropping to zero. So it makes sense (a good convention) to choose $U_{\infty} = 0$ rather than any other number. This logic is very similar to how, in the spring-mass system too, we chose a zero potential energy at the spatial position where the force became zero. In case of spring, zero force was achieved at a definite point of space. Here, the zero force occurs in a limiting process. But the idea behind choosing a $0$ value for $U$ is similar: $U$ is zero “where” $\vec{F}$ is zero. Note, this is only a mathematical convention, not a physical fact. We simply don’t have any evidence to know what specific energy value there is as the distances become very large.

With the above convention, the increase in the internal energy of the system in bringing both the charges from infinity to their present positions ($\vec{r}_1$ and $\vec{r}_2$) becomes the same as the absolute internal energy of the system (and not a change in it). This internal energy is given by: $U_r = \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_1\,q_2}{r}$, where $r = |\vec{r}_2 - \vec{r}_1|$ and hence $U_r$ are variable, but at any instant, both their values are fixed. Thus the potential energy of a system is just a single number at any point of time, no matter where the charges are.

6.3 Other points:

• What if the two charges are of different polarities? Working out whether the internal energy increases or decreases (from its zero at the infinitely large separation) is left as an exercise for philosophers. (No, this is not an insult. This exercise is about physics, not philosophy.)
• By the way, note a fact established by the known physics (but not by mathematics):
The net charge of the universe is not only conserved, it also is zero.
• If you imagine mechanical supports so as to ensure fixed positions for the two charges (the supports shift with the charges), then what about the changes in the internal energy of the system due to the work done against or by these support forces? This is left as an exercise to XII standard students/JEE aspirants.

7. Field of electrostatic potential energy ($U$):

7.1 Mathematical function for potential energy:

Following the same logic as for superposition of forces, the mathematical function for potential energy of a system containing two charges can be given as:
$U_1(\vec{r}_T) = \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_1\,q_T}{r}$, $r = |\vec{r}_T - \vec{r}_1|$, and
$U_2(\vec{r}_T) = \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_2\,q_T}{r}$, $r = |\vec{r}_T - \vec{r}_2|$

The mathematical function for the total potential energy of the system is then given by the simple algebraic sum of the two.
$U_{\text{sys}}(\vec{r}_T) = U_1(\vec{r}_T) + U_2(\vec{r}_T)$
The same logic can be extended to $n$ number of charges.

Similarly, force can be obtained from the mathematical energy function as: $\vec{F}_{12} = -\ \nabla U(r)$

7.2 Potential energy field as physically existing:

Then, once again, we can follow Faraday and Maxwell’s lead, and assert that what the mathematical expression for potential energy field gives is a quantity of something that actually exists in the physical world out there.

The infinitely extended field is sensitive (and unique) to each specific physical configuration of charges. If a single charge under consideration is fixed in space, then a unique potential energy can be assigned to each point of space. If it moves in space, so does the force-field and the potential energy-field associated with it.

7.3 Potential energy field of a single charge:

Again, following the logic seen in the case of spring and two masses, we can say that electrostatic potential energy for a pair can be split into two equal components, one each specific to a charge in the pair. Thus:

$U_{\text{sys}} = \dfrac{1}{2}\sum\limits_{i=1}^{2} \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_1\,q_2}{r}$.

7.3 Potential energy field of a $n$ charges:

Generalizing, it is easy to prove that for a system of $n$ charges:

$U_{\text{sys}} = \dfrac{1}{2}\sum\limits_{i=1}^{n} \sum\limits_{j \neq i; j=1}^{n} \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_i\,q_j}{r}$, where $r = |\vec{r}_j - \vec{r}_i|$

This form is especially useful in building simulations like the molecular dynamics.

8. An ontologically very interesting point—differences in internal energy storage, work actually done or extractable, and potential energy:

We must note a very important point here. It concerns the issue of the ontological understanding of what precisely the terms potential energy and internal energy mean.

8.1 In direct-contact systems, a moving boundary can sweep over the entire volume where the internal energy is stored:

Strictly speaking, as touched upon in the last post, the term potential energy means that part of the internal energy which can be converted into work.

Work done on a system increases its internal energy; some part of the stored internal energy can also be later on converted into work; when the forces are conservative, the increase in the internal energy of a system due to work done on it is, in its entirety, available for conversion into work.

When we talk of a work done on a system (or by it), we are basically referring only to the system boundaries—not at all to its internals. Work is not at all defined at any points other than at the instantaneous position(s) of system boundary(ies). This fact introduces a tricky issue.

In idealized systems like the ideal cylinder-piston arrangement or ideal spring-mass system, two features come into play: (i) being idealized, the forces can be treated as being conservative, and (ii) the moving objects, and hence the system boundaries, during their motions, can come to sweep the entirety of the spatial region where internal energy is stored. It is the second factor which needs special commenting.

These systems are such that during a single work-extraction process, the moving boundary can trace over and exhaust the entirety of the spatial region where the internal energy is defined. During such a sweep of the boundaries, the entirety of the earlier increases in the internal energy of the system can be converted back to work (assuming conservative forces i.e. neglecting friction).

Therefore, when work has been maximally extracted from such systems (e.g. when the spring returns to its neutral length), there is no further storage of internal energy at all left in such a system.

8.2 In fields-based systems with point-particles, a moving boundary cannot sweep over the entire volume where the internal energy is stored:

The fields-based systems show a remarkable difference in this second respect. They store far more energy than can ever be converted into work. If the fields are singular, they must be seen as storing infinitely more energy than what can ever be extracted from them.

The basic reason for this characteristic is that no matter where the objects (point-boundaries) are kept or how they are shifted, there always is an infinitely greater portion of the domain all parts of which are still left carrying energy, and this entire portion is left untouched because the boundaries are properly defined only at points (at most at surfaces), but not over the entire volume.

To repeat, a system of two (or more) distinct EC Objects, shows the following three feature: (i) work can be exchanged only at the system boundaries, (ii) the system boundary exists only at the two (or more) EC objects (point-particles), (iii) but the force- or energy-field logically “internal” to the system definition exists at all points of the infinite space at all times. Since the work-extraction process can happen only at the points where the charges are, such a process can never come to have covered, through a finite and completed process of work extraction, the entirety of space.

Therefore, there will always be far more internal energy than can ever be converted to work. In other words:

The total internal energy stored in a system via the EM force-field is far greater than the quantity of potential energy (i.e. the change in the internal energy) which is acquired by the system when two movable EC objects themselves are displaced.

8.3 Putting it mathematically:

Mathematically, the whole matter can be put very simply:

In case of the ideal spring-mass system (point-masses, massless springs, no friction), the total increase in the internal energy of the system $\Delta U$ equals the volume integral of the internal energy density over the entire swept volume (i.e. the regions where the masses can move) $\iiint\limits_{\Omega} \text{d}\Omega \rho$, where $\rho$ is the internal energy density.

In contrast, in case of fields-based systems (whether gravitational or static EM), $\Delta U \neq \iiint\limits_{\Omega} \text{d}\Omega\ \rho$, where $\rho$ is the density of the local increases in the internal energy due to the presence of an EM field; in fact, the left hand-side is far smaller (infinitely smaller if fields are singular inside the system) as compared to the right hand-side.

So, in the EM systems (as also in gravitational systems), internal energy exists everywhere in space. However, the potential energy of the two EC Objects—and hence of the system—refers to only those local energies which exist in their immediate neighbourhood, the energies which are acquired by these discrete objects. Hence, only this much energy is available for any conversion to work at the moving system boundaries.

8.4 Consequences in calculations—and in further physics:

Now, what do we do with this additional energy which is exists “internally” to the system but is not available even potentially for extraction into work?

Well, in most calculations, we can just ignore it. As they say, the datum for any potential energy is arbitrary. That’s because the datum for the changes in the internal energy are arbitrary. What we are interested in are the changes in the internal energy of the system, not in its absolute value or datum. So, the unextractable portion of internal energy is perfectly OK to have. In EM, the unextractable portion of $U$ makes for a minor fact which is necessary just to bring logical completeness to the physical description. In fact, it is a trivial fact because in EM, ultimately, the fields have a relatively indirect role to play; the actions of the fields are of no consequence unless they result in some action on the charges. (EC Objects have inertia; the field doesn’t.)

However, the situation changes very significantly when we it comes to quantum mechanics. We will pursue the QM ontology in the next post.

9. Electrostatic potential ($V$) of an EC object:

9.1 The field of the “voltage”:

The 19th century physicists wanted to get you confused even further.

• Even for the action-at-a-distance systems (i.e. the fields-based systems), they happily equated the internal energy with the potential energy.
• Further, they defined not just a potential energy field, but also an electrostatic potential field—i.e. the one which has the word “energy” dropped from its name.
• Finally, they (and everyone else) used (and continue to use) the same symbol $V$ to denote both.

The potential field $V$ is the equivalent, in energy-based analysis, of the electric vector field $\vec{E}$. The potential function is defined by dividing the potential energy function due to a single charge by the magnitude of that charge so that what we have is a “pure” function that is independent of any other charge (i.e., a field defined for a shiftable unit charge), as shown below:
$V_1 = \dfrac{U_1}{q_2} = \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_1}{r}$

The motivation of defining such a thing? Well, if the second charge is not the unit charge, then the potential for the first charge ($V_1$) still remains the same.

Of course the potential energy function for the second charge does depend on its charge:
$U_2 = V_1 q_2$

9.2 A personal comment:

As we saw, if the charges are elementary, then $\vec{E}$ is not necessary. Further, if we assume that fields are not arbitrary but are always anchored into point-particle positions, then it is always possible to split up the total internal energy $U$ into contributions arising from internal charges; see the equation in section 7.3 of this post. The whole business of having yet another field—of the electric potential—is thereby avoided.

9.3 Multiplying confusions:

QM text-books further help compound the confusions. [In an update on 2019.10.07 12:15 IST, I deleted a misleading statement here.]

Notation-wise, $V$ should be reserved for electric potential (or its field), because $V$ is also the SI symbol for the unit of potential i.e. voltage (after Volta).

In turn, something like $U$ might be used to indicate the total internal energy field, defined as the total energy content of the aether due to all the charges present in the system. The potential energy field affecting the dynamics of a given charge $q_{i}$ (and arising due to the other charges) may be denoted using something like $\Pi$; this quantity has the relation $\Pi_{q_{i}} = U_{\text{sys}} - U_{q_{i}}$, where $U_{q_{i}}$ is the field-component contributed to the internal energy field of the system by $q_{i}$. The potential energy of the system, seen as a global attribute, can continue to remain a single number that depends on the set of point-positions of charges (their configuration).

Using a Greek letter for the potential energy is not a very good choice. But we can’t use $E$ for the potential energy because the symbol is already taken by the electric force field. We can’t use $P$ because it stands for pressure, and also for protons—again confusing.

So, all in all, Schrodinger’s equation should be re-written to have $\Pi$ in place of $V$ in the expression of its Hamiltonian. It’s a field—a $3D$ field. It is different for each charge in the QM system.

Regardless of the notation, remember, $V$ in the Schrodinger equation has the unit of joule, not of volt.

10. Physics of the electric field: “stresses” and “strains” in the aether:

No, we are not done covering the planned topics for this post yet. The next point concerns some plausible mechanism whereby the Lorentz Aether (LE) might be imagined as storing the internal energy associated with the physically existence of the EM fields.

I will not go into the specifics of it except for noting that we imagine the electrostatic electric force field ($\vec{E}$ or $\vec{F}$) at a point as a result of a kind of a stress field in the LE. This imagination involving a stress field needs some commentary.

First of all, notice that, strictly speaking, you can’t have point-forces in a continuum. That’s because if each point of a continuum were to have a finite force, since there are an infinity of points within any finite volume, every finite portion of the aether would end up having an infinitely large force. (The mathematical argument here is identical in spirit to why Born’s rule applies to small CVs, but not to points.)

That is the reason why the only force-like quantities permissible within a continuum are: (i) a volume density of force, or (ii) a surface intensity of force (as in pressure, a type stress).

In between the volume- and surface-defined quantities, I believe that the $\vec{E}$ field (rather, the $\vec{F}$ field) arises out of internal surface intensities (i.e. stresses) rather than volume force densities. My reasoning is the following:

The EC Object is ontologically a different kind of an object, and forces are exchanged between the EC Object and the Lorentz Aether. Stronger: The existence of an EC Object is essential (even tantamount) to having a force-field within the aether; an EM force-field cannot arise without there being a charge which may be seen as causally producing it—or vice versa, if you will. The two always go together; none can exist without the other.

Now, two different objects that are in a direct contact can be separated from each other only by a surface, not a volume. So, if a $\vec{F}$ arises at a point, it must be seen as the limit of a surface-integral of some surface-intensity of a force on the boundaries of a small CV (control volume). That’s nothing but a stress; it’s not a body force.

So, though from a mathematical viewpoint, internal surface forces are perfectly interchangeable with internal volume forces, from an ontological perspective, the surface force-intensities are primary.

Accordingly, the field energy $\dfrac{1}{2} \epsilon_0 E^2$ noted in the MIT notes [^] may be seen as the strain-energy density, with $\epsilon_0$ playing a role analogous to that of Young’s modulus in elasticity.

But carefully note that all said and done, this analogy still must be regarded as only a convenience in visualization. As of today, we don’t know the physics of exactly how the EM force fields come to be, even if we know that they must be there. There must be some physics to them, but it’s summarily unknown to us as of today.

Of course, personally, I do visualize a stress field producing the electrostatic $\vec{F}$ field—it’s easy to keep in mind the caveat noted just above.

OK. Enough is enough. (With ~7,500 words, this has become the longest post ever written on this blog.)

11. A preview of the things to come:

Since there was a gap in writing of this series (due to flash floods in Pune which affected us too), I happened to lose the “flow” of my thoughts. So, there must be some odd points or notings here and there that got slipped out of the mind during writing this post. Not an issue. I anyway plan to pull this entire series together and convert it into a stand-alone essay (after due revisions, rearrangement of material, etc). I am in fact toying with the idea of sending the resulting PDF to some philosophy journal too, eventually. But a journal article isn’t an immediate goal. May be in 2020, after the first paper on my new approach on QM gets written.

Coming back to this series: Much of our work in this series is already over. We will require, I guess, just one more post, for covering the next topic, viz. the ontology of QM. The reason that the QM ontology will get over so quickly is because there are hardly any tricky issues still left to be touched on. Most of them actually got covered right during the progression from the NM ontology to the EM ontology. So, just one more post should be able to cover the QM ontology.

The only portion still remaining then would be: a few general philosophical remarks regarding the necessity of having a physical background object—the Lorentz Aether, and, as my small but new contribution, a few additional mathematics- and physics-based points that go in the same direction. We will see if it would be possible to cover this portion too right in the next post (on the QM ontology). If yes, we will be free to discuss visualization for my new approach to QM.

Alright then, bye for now, take care, and see you the next time.

A song I like:

(Hindi) “toraa mana darpana kahalaaye”
Lyrics: Sahir Ludhianvi
Music: Ravi
Singer: Asha Bhosale
[Credits listed in a random order.]

PS (2019.10.05 13:30 IST): Minor modifications/additions or streamlining may get affected, but the basic points will remain as they are.

History: First published on 2019.10.05 13:30 IST. Some minor portions added/deleted (noted inline) on 2019.10.07, by 12:29 IST. Further clarified a bit on 2019.10.07 21:53 (the ‘net connection was very flaky throughout the day). Now will (really) leave this post as is.

# Ontologies in physics—4: Minor changes in the ontology of EM force-fields. Understanding potential energy.

OK. After posting the last entry in this series, a thought—actually, a problem—occurred to me right the next day. I’ve noted it via an update at the end of the last post; see if you didn’t check it out again [^]. This is not the only change though. The entirety of the last post has undergone substantively revised wording, though I left the positions untouched. …All my posts are always are transient—drafts subject to change without notice. (No, I don’t care for you. Really.)

But anyway, coming back to the problem with the ontology as presented in the past post: I also saw a way to get out of that problem right on the same day, just before going to sleep at around 9 PM or so. However, this issue being tricky, I decided to wait for a few days before giving my solution. Further, since the last post had already become so huge (5,000+ words), I decided to include the solution not there, but here, in this post. Let’s begin with it.

1. The ontological problem in regarding $\vec{E}$ as physically existing:

1.1 The problem in essence:

The ontological problem which comes up with the description as given in the update to the last post is the following.

If we accept a physically existing field for the electrostatic phenomena, precisely of what quantity is its force field? Is it a field of the electric field vector $\vec{E}_{1} = \dfrac{\vec{F}_{1T}}{q_T}$, or is it of electric force itself $\vec{F}_{1T}$?

That’s the question to be settled.

1.2 The reasons why it is a problem:

If it’s a field of $\vec{F}_{1T}$, then we haven’t succeeded in isolating a field quantity that depends on $q_1$ alone, because $\vec{F}_{1T}$ additionally depends also on $q_T$. On the other hand, if it’s a field of $\vec{E}$, then the physics of the interaction becomes quizzical. It is not clear precisely what happens physically so that a local field of $\vec{E}_{1}$ comes to produce the effect of $\vec{F}_{1T}$?. It is not clear what physical process must occur at $\vec{r}_T$ such that it corresponds to the required multiplication of the local $\vec{E}_{1}$ by $q_T$.

Note, mathematically the difference is trivial. In fact, physicists (and even engineers) routinely do dimensional re-scaling of their equations. Dimensional scaling is just a handy trick that keeps the underlying physics the same. However, note, such a scaling procedure modifies the entirety of an equation; it affects all terms in the equation. In contrast, the key issue here pertains to only one term in the equation; it pertains to a difference of physics. The quantities $\vec{E}_{1}$ and $\vec{F}_{1T}$ are of different units, and this difference will come to exist as is in every dimensional re-scaling. It will remain intact in any unit system you choose, whether Gaussian, imperial, SI, or any other.

It is clear that experimentally, we measure only $\vec{F}_{1T}$, not $\vec{E}_{1}$. Therefore, if you assume a physical existence for $\vec{E}_{1}$, converting it to the observed $\vec{F}_{1T}$ quantity requires a detailed model, a separate physical mechanism to be given.

There is simply no way to resolve this issue using the EM ontology as we have presented it thus far. The reason is that in classical EM, the charge magnitudes are arbitrary. This is the key reason why we need to factor out $q_T$ in the first place.

The only way out of the tricky situation, I think, is to make reference to a physical fact discovered after the systematization of EM theory had already occurred, and to bring the import of this fact into our ontological description, thereby revising all the relevant parts of it. The experimentally discovered fact which provides the way out is this:

In nature, there exist only elementary charges (of electrons and protons), and their charge is the same in magnitude, just opposite in sign. The electronic charge $e$ is a fundamental constant of the universe.

Therefore, in a most fundamental description, i.e., in ontology, we never have to worry about multiplication by arbitrary charge-magnitudes. We can say that the actually existing field is only of the force $\vec{F}$, and never of the electric vector $\vec{E}$. Each charge causes an exactly identical field to be developed in the background object, and any other charge at a distance which interacts with a given field also must possess the same universal magnitude of $e$ (within the algebraic sign).

In short, Nature never does multiplications. We have to make this change to our ontological description.

Thus, in our revised ontology, $\vec{E}$ is only a convenience of mathematics. This quantity is useful when we want to treat a system of a large number of charges as if it were a single charge. In classical EM, we in fact don’t even require $q$ to be an integer multiple of $e$; through homogenization, we allow $q$ to have any real value. Once again, this too is only a mathematical approximation, a convenience to adopt at a higher level of abstraction. But in terms of what actually exist out there, there are only the $\vec{F}$ fields produced by an integer number of elementary charges. The ontology as such has only $\vec{F}$ as a physically existing condition.

1.4 Another way to look at the problem and our solution:

Realize, this solution also takes care of another problem that I didn’t note but spotted later on.

The field is an attribute of the background object, even if it is caused by, or having effects on, the EC objects. Therefore, ideally, a physically existing field should show an equal abstractness from both the charges.

However, the electric vector quantity $\vec{E}_1$ comes with the magnitude of the first charge built into it. That’s why $\vec{E}_1$ is not “symmetrical”: since the multiplication by $q_1$ is built into it, we are then left looking for the physics of multiplication by $q_2$ at the other end.

Adopting a $\vec{F}$ field as the actual existent makes the description at both “ends” of the field symmetrical: the same process can govern the production of a field by a charge, and the production of a force on a charge.

1.5 We know nothing about how a charge causes a force-field or vice versa:

But how exactly does a charge produce a force-field in its vicinity? Ontologically, there is no evidence to make this an issue of any significance. In fact, any answer you try to think of will end up replacing a more general phenomenon by a more specific model.

For instance, you may put forth a model which says that a charge acts like a press-fitted ball inside a continuum. However, this “model” doesn’t have much value, because the background object is basically not mechanical in nature (it is not an NM Object), and so, its interactions with EC objects also can’t mechanical in nature. In fact, a purely mechanically inspired explanation carries a definite risk that we may end up taking it too literally and forget the actual phenomena.

2. The points about the history of EM fields which I learnt only this week:

After I had thus figured out what should be the proper EM ontology in this $\vec{E}$ vs. $\vec{F}$ issue, I made a few rapid references to books and papers on the history of EM. I came to know a few very valuable points for the first time in my life. (My earlier readings of history were mostly focused on developments in continuum mechanics, variational calculus, and to some extent also QM, but not on EM.) Here are the main points I learnt this week:

2.1 Maxwell had a dielectric aether:

I already knew that Maxwell had a mechanical aether; that he ascribed NM-object properties to his aether. But I didn’t know that he in fact also had dielectric properties ascribed to the aether—which was rather unexpected to me.

2.2 Lorentz beat me to it by 90+ years:

All along I had thought that the kind of “aether” which I had in mind was a new development.

I had thought of a non-mechanical aether way back, probably while in UAB (following a remark by Ayn Rand in her seminar in the ITOE book, second edition). Soon later, I had also used it in my first attempts in resolving the QM wave-particle duality, which started from the early 1990s. Much later on, I had also used it during my PhD-time research on QM. In the Abstract of my PhD thesis, I had said:

“The present view of aether differs from all the previous views in the sense that the author takes luminiferous aether to be a physically existing but non-material substance.”

It was only in this week that I came to know that H. A. Lorentz (the 1902 Nobel laureate) had already figured it out right, some 90 years earlier. It’s just that people never highlighted the difference of his view of aether from Maxwell’s in any discussion on aether, so I couldn’t even suspect that I was not the first.

2.3 The only (minor) difference in Lorentz’s idea of the aether, and our background object:

The only difference between Lorentz’s ontological ideas and mine seems to be that implicitly, by always visualizing a singularity for the electrostatic field (albeit only as an approximation for the unknown physics of a charge of a finite size), I naturally came to imagine that no field would be present at the location of the charged object itself. Even if it’s not singular, ontologically, it’s in an entirely different category from the aether.

In contrast, Lorentz thought that what we call the EC Object had some structural detail.

Initially, he sought a dynamical description, inspired from mechanical models, even though he was clear that his aether was not mechanical in nature. But he did think of a charge has having a structure; he thought in terms like changes in shape that are undergone by a charge when it is in motion. His motivation was to supply a mechanism for his relativitistic contractions.

Now, Lorentz did advocate the position that the electron is to be regarded as the source of all electromagnetic phenomena. However, to him, a charged body still was a composite object, with both charges (elementary ones) and aether present within its volume—as happens in ions. (He had explicitly used ions in his initial thinking.) This is the reason why the $\vec{E}$ field would be present “inside” his “charge” too—like gravitational field is present inside the earth too, not just outside.

Apparently, Lorentz didn’t notice our above-mentioned difficulty (regarding the physics of multiplication). However, such differences between Lorentz’ view and ours amount to nothing as far as the basic ontological scheme is concerned.

2.4 Lorentz’ work is grossly under-rated:

Just think: How much independence of thought and conceptual skills Lorentz must have required in originally (and correctly) isolating the aether from the charges—and then, even using them to simplify Maxwell’s (very) complicated theory.

Note, other people were still working with only a mechanical aether, even people of the stature of Maxwell, Stokes (who formulated the aether-wind idea), Helmholtz, Hertz, FitzGerald, and others—all first-rate thinkers.

Further, some other brilliant people like Poincare had already begun becoming fully skpetical of the entire idea of the aether.

Situated in this intellectual climate, Lorentz still managed to reach an essentially right ontological position to be had, and then held fast to it, saying that

the universe was basically electromagnetic in nature, not mechanical; but there still had to an aether, although it was of an electrical nature, not mechanical.

All in all, a unique position in every respect.

2.5 My “Maxwell’s equations” always were in Lorentz’s/Heaviside’s form:

The second fact I learnt was that it is Lorentz’s reformulation of the electrodynamical equations which are now taught as Maxwell’s equations. This fact is important.

The basic form in which these equations come itself has been guided (“informed”) by Lorentz’ ontology. Therefore, if you just “stare at” these equations long enough, the right ontological structure implied by them begins to become clear to you. Such a thing would have been plain impossible using the form of Maxwell’s original equations.

So, in a way, it is not at all surprising that starting with Maxwell’s equations as recast in Lorentz’s mould, I should have been able to slowly but correctly trace the logic back, and thereby arrive at the same ontology. The form of the equations itself makes the task so easy for us today.

The only thing you require here is to learn to ignore teachers like Feynman on points like the aether or the proper origins of any mathematical concepts. It also helps that other teachers like Resnick and Halliday cover the same material very carefully, in a manner that develops real understanding.

Addendum on 2019.09.24 17:53 IST: Added Heaviside’s name. The form of the equations as taught today might be following a notation which is based on what was put forth by Heaviside. However, I also know that Lorentz was very clear that the aether had to be electrical i.e. non-mechanical in nature, but I am not sure where Heaviside stood with respect to aether (though he might have suggested or worked on the contractions). … Would need to dig through the history to be able to identify exactly who contributed what, and priority. Don’t have the time to go into it all right now.

3. Summarizing the changes to our EM ontology:

For obvious reasons, what we have so far called the background object ($B^{0}$) shall henceforth be called Lorentz’s Aether in our ontology.

Also, in view of the difficulty with $\vec{E}$, we now declare that our entire EM ontology has been changed in such a way that any EC Object will be regarded, in any fundamental ontological discussion (unless otherwise stated), as being only an elementary charge, having the value such that $-|q| = e$.

We now return to the planned topics. However, it’s obvious that we won’t be able to finish them all right this time around.

4. The potential energy as an internal energy of a thermodynamic (and mechanical) system:

We begin by regarding two or more EC objects as forming a thermodynamic system. We first want to make sure that we understand the potential energy of a system.

The archetype of thermodynamic systems is the usual piston-cylinder arrangement which exchanges heat and work with the rest of the universe, i.e., with the (given system’s) environment.

4.1 Two transient forms of energy that appear only on system boundaries: heat and work:

Heat and work are the two transient forms of energy that come to appear on the system boundary—and at no other point in space. Their quantities are nonzero only for the time when an actual exchange of energy (of their respective form) is actually ongoing.

Heat cannot be stored in a volume (say, the volume of a system) because heat is not a quantity defined in reference to a volume. It is defined only at the boundary surfaces of a system. We call heat that energy which passes a system boundary because of a single reason: the temperature gradient as it exists at the boundary surface. (Temperature gradients purely internal to the system or purely external in the environment don’t count—only that at the boundary does. If concerned with internal temperature gradients, you have to divide the system into as many sub-systems as you wish, and then account for them but only at the boundaries, again!)

To repeat, heat cannot be stored. If you want to say “heat energy stored in a system,” just gulp the word “heat” and utter the word “thermal” in its place, and you will be perfectly OK—it will keep professors like me happy. (Thermal energy is due to a non-zero kelvin temperature and can be stored; heat energy is due to a difference in temperatures and cannot be stored.)

Work cannot be stored in a volume (say, the volume of a system) because work is not a quantity defined in reference to a volume—only for the boundary surface, and only during the time when the boundary is physically undergoing a displacement.

Heat doesn’t require a displacement of boundary, but work does. If a system volume is forced to remain constant, then the system cannot exchange work with its surroundings.

To repeat, work cannot be stored. If you want to say “work stored in a system,” just say “increase in the internal energy of a system due to work done on it,” and you will be OK. However, since most professors themselves say things like “work done on a system gets stored in it,” guess you are only technically wrong when you say things like “work stored in a system.” It’s an informal expression that does not explicitly state all the essentials by adding “as its internal energy”. Professors too use the curtailed expressions. That’s why, even if you are technically wrong, you too shall come to pass your viva. [Your professors did the same.]

Sign convention: Here, it’s useful to keep in mind the steam engine and what we do with it. Heat supplied to the system (by its environment) is algebraically positive, and the work done by the system (on its environment) is algebraically positive.

So, the positive quantities are: heat supplied to, and work done by, the system.

4.2 Internal energy:

The total energy stored inside a system is called its internal energy.

4.3 The first law of thermodynamics:

A supply of heat to the system increases its internal energy. A supply of work by the system (because the system boundary moves with the motion of the piston due to the expansion of gas) reduces the internal energy.

The first law of thermodynamics says that the preceding description is complete. For instance, aliens don’t get to change the total energy content of a system. Accordingly, the first law of thermodynamics can be stated as:

$\text{d}U = \delta Q - \delta W$.

Notice that, strictly speaking, the first law does not make a statement about the quantity of energy stored in the system. It makes a statement only about changes in the stored quantity.

The $\delta$ in front of $Q$ and $W$ denotes an inexact differential—an infinitesimally small quantity whose value depends on the path taken in a state-space like the $PVT$ diagram. The $\text{d}$ in front of $U$ denotes an exact differential. The internal energy of a system is a state variable; it doesn’t depend on the particulars of how the energy changes. Yes, here, the difference in two path-dependent infinitesimals is a path-independent infinitesimal.

In the basic EM theory, we don’t care for thermal considerations, and so for our purposes here [why did I type so much?], the first law becomes:

$\text{d}U = - \delta W$.

Effectively, in EM theory, there are no temperature gradients at the boundary between an EM system and its environment. Everything in the EM theory stays at the same temperature. There are some important points regarding what is regarded as the boundary of a basic EM system. We will come to it in a short while.

4.4 Work done on a system, internal energy, and potential energy:

The work done by the system is positive, i.e., $+\delta W$. Hence, work done on the system is negative, say, $-\delta W$.

When work is done on a system, there is a corresponding increase in its internal energy. This increase in $U$ makes the system acquire a certain potentiality—the potentiality to produce some amount of work during its interaction with its environment.

If heat is supplied to a system, there again is a corresponding increase in its internal energy. This increase in $U$ too can make the system acquire a certain level of potentiality to produce some amount of work. In EM discussions, we neglect heat, but remember, $U$ can increase via heat supply at the boundary too.

Thus, the actuality of some energy already supplied to a system imparts to it the potentiality that it can later on produce some amount of work. As Aristotle said (and Dr. Binswanger once pointed out in one of his lectures), actuality precedes potentiality.

The potential energy of a system is that part in the increases in its internal energy which it can later on convert into work.

The potential energy, thus, is a form of internal energy. However, if a system cannot produce work, it may still have a reservoir of an internal energy, but you can’t call this stored energy a potential energy.

4.5 Example mechanisms to store energy:

But what is it which allows a system to at all store energy? What precisely happens physically when the internal energy of a system increases? Here, we have to look at the structural details of the system, its physical internals.

In a piston-gas system, there is a gas that gets compressed when you do work on the system. The gas consists of molecules; pressure is the momentum they impart during their collision with the cylinder and piston walls. So, when you compress a gas, something about its internal configurational state changes. The arrangement of the NM Objects contained in the system changes. The increase in the internal energy comes about solely because of the changes in the configuration of the NM Objects constituting the gas. So, the internal energy here is the energy of a configuration internal to a system.

You might as well have a mechanical spring in place of the gas in a piston-cylinder arrangement. Same thing. The “changes in the configurational states of the spring,” etc.

You might as well forget to have an actual piston-cylinder arrangement around it. You can have just the spring. Same thing. The spring can still store internal energy via its deformation. There is no cylinder, no piston, but only forces at its two ends, arising out of a direct contact with some NM objects in the environment. In short, the piston and the cylinder can be imaginary; the spring alone still qualifies to be called a thermodynamic system.

You might as well have just two NM Objects interacting with gravity. Same thing. Fix the system boundaries at the NM Objects (say at the centers of mass of the earth and the moon), but let the boundary be movable. Then, this arrangement sure forms a thermodynamics system. This system too is able to have (“store”) internal energy. It too can produce work at its moving “boundaries”.

4.6 The thermodynamic boundary for the earth–moon system:

Notice the fact that earth’s gravity affects not only the moon but also a satellite when it’s on the dark side of the moon. The logical system “boundary” for the earth-moon system still remains fixed at their respective centers of mass. So, the logical “boundary” of a thermodynamic system—the place where work-exchange occurs—is not necessarily a surface (let alone an enclosing one); it can also very well be a set of isolated point-positions in the physical space.

5. The electrostatic potential energy ($\Delta U$):

5.1 Electrostatic interactions as a mechanism to store energy:

Instead of two gravitating objects, you might as well have just two EC Objects interacting electrostatically.

Fix the movable system boundaries at the EC Objects (the elementary charges). This system too has an internal energy associated with the configuration—the relative positions—of the two charges, so long as the charges are able to occupy different positions. That’s what the electrostatic internal energy is. This system too can produce work at its moving “boundaries.”

The “boundary” of the EC system too is identified with point-locations, and not with an enclosing surface.

5.2 The electrostatic potential energy is $\Delta U$:

In basic EM theory, $U$ doesn’t increase because of a heat supply—we ignore heat. Hence, work is the only mode available to cause increases or decreases in the internal energy. It so turns out that at the most fundamental level, any EM system is frictionless. So, any EM system should be able to produce just as much work as was put into it. This presumption is correct.

All the work done in moving EC objects indeed forms an increase in the internal energy of the EM system comprised of them, and this entire increase in the internal energy is available for producing work at any later time. We call it the electrostatic potential energy.

The electrostatic potential energy is the increase in the internal energy of an EM system due to the work done on it in shifting the EC objects (point-particles of elementary charges) contained in it.

5.3 The electrostatic potential energy is not the same as the internal energy $U$:

Note very carefully. It is the increase in $U$ due to shifting of EC Objects which is being called the electrostatic potential energy, not the total energy content (i.e. the internal energy) denoted as $U$.

This distinction is important, and we will have a definite occasion to use it a bit later.

Enough for today. Got tired typing and re-arranging material. (Still, 4,100+ words already!) I will cover the rest of the planned portion in the next post, due soon. As usual, minor changes may be effected to this post (including even misleading formulations), but any significant change (especially that in positions/arguments) will be prominently highlighted.

Bye for now, and take care… See you the next time (if still left interested in this topic).

A song I like:

(Marathi) “maavaLatyaa dinakaraa…”
Lyrics: B. R. Tambe
Music: Hridaynath Mangeshkar
Singer: Lata Mangeshkar

History:
Originally published: 2019.09.21 18:39 IST
Added a note about Heaviside in the section 2.5:  2019.09.24 17:53 IST