# Ontologies in physics—8: Correct view of the EM “V” in the Schrodinger equation. Necessity of aether.

0. Prologue:

The EM textbook view of the electric vector field ($\vec{E}$, in volt per meter or newton per coulomb) of a charge, and so the electric potential field ($P$, in volt) is an ontologically misleading construct, even if happens to be mathematically consistent with the physics of EM. In this post, we will point out the ontologically correct view to take of the electrostatic phenomena. Corrections are implied at suitable places throughout our earlier description of the EM ontology.

In developing our new ontological view of EM, we will also be using the evidence which came to light after the Maxwellian EM was already formulated and recast by people like Lorentz at al.

A basic point to keep in mind for this post is what we discovered on the fly, right in this series: Nature does no multiplications (without there being some physical mechanism for it).

Let’s get going.

1. Some basic facts pertaining to the EM physics:

The first basic fact we note is the following:

The total amount of charge contained in the universe is zero.

We reach this ontologically important conclusion by generaling from the available physical evidence—not from “purely” mathematical considerations such as “symmetry”.

One direct outcome: We can’t associate an ontology based on a preferential direction for some imaginary polarity-conversion process. In fact, going by another bit of empirical evidence, we don’t even have to entertain any polarity-conversion process in the first place. The polarity of a charge is immutable.

The second basic fact to be noted is this:

Elementary charges have the same absolute magnitude; they differ only in their signs.

Putting these two general facts together, we can say, as a general inference, that

The number of elementary charges is equally split into the two polarities: positive and negative.

We don’t know any particular count for the number of charges in the universe. (And, we don’t have to know, given the $1/r^2$ nature of their forces, which decays rather rapidly so that even the spatial extent of the already known itself is far too big for an electron’s force-field to stay numerically relevant at such large scales). But, yes, we do know the fact that there as many positive charges as there are negative charges.

Hence, if a system description is to be a good representation of the behaviour of the universe, it should be regarded as having an equal number of positive and negative charges.

So, as the first quantitative implication:

For an arbitrary system having a total $N$ number of charges, the number $N$ should always be so selected as to be an even number. In such a case, there will be $N/2$ number of positive charges, and $N/2$ negative charges. Thus, there will be $N(N-1)/2$ number of pairs of interacting charges in all, and $N(N-1)$ number of different separation vectors.

Homework: Given such a system, find the total number of the different possible separation vectors that connect: (i) two unlike charges, (ii) two positive charges, (iii) two negative charges.

As another basic fact, we note that:

The electrostatic interaction forces being conservative in nature, they can be superposed.

Therefore, the forces in a system can be quite generally analyzed in reference to just a single arbitrary pair of charges of arbitrary polarities.

2. The three steps to reach the correct ontological view of the electrostatic fields:

Take two elementary charges of arbitrary polarities $q_i$ and $q_j$ respectively at $\vec{r}_i$ and $\vec{r}_j$.

2.1. Step 1: The empirical context:

Start with Coulomb’s law which gives the two forces, respectively acting on the two charges, with each acting in the direction of a separation vector:

$\vec{f}_{ij} = \dfrac{q_i\,q_j}{r_{ij}^2} \hat{r}_{ij}$
and
$\vec{f}_{ji} = \dfrac{q_j\,q_i}{r_{ji}^2} \hat{r}_{ji}$

In ontology of physics, it’s always very sensible and fully valid—and also equally lovely—to begin with forces, and not with the Lagrangian or the Hamiltonian. Forces “force” you to think of the individual objects that do the forcing, or of the changes which are made to the dynamical states of individual objects due to the forcing. So, you just can’t escape identifying the actual physical objects involved in forceful interactions. If you start with forces, you just can’t escape into some abstract system-wide defined numbers, and then find it easy to cut your tie from reality. That’s why. (As to any possible non-forceful interactions, tell me, who really worries about them in physics? Certainly not Noether’s theorem.)

From this point on, I will work out with just one of the forces, viz. $\vec{f}_{ij}$, and leave the other one for you as homework.

2.2. Step 2: Assign to the aether the role played by the attribute of the charge of the EC Object:

Following the textbook treatment of EM, the ontology for the EC Object which we developed has been the following: It was essentially the NM Object now with an additional attribute of the electric charge. Thus, an EC Object is a massive point-particle that carries an elementary charge as its additional attribute. Qua attributes of a point-existent, both mass and charge must be seen as being located at all times at the same point where the EC Object is—and nowhere else.

We must now modify a part of this notion.

We realize that the idea of the electric charge is helpful only inasmuch as it helps in formulating the quantitative force law of Coulomb’s.

The charge, in the text-book treatment, is associated with a massive particle. But there is no direct empirical evidence to the effect that a quantity which captures the forcing effects arising due to the attribute of the charge, therefore, has to be a property of the EC Object itself.

Only a physics / ontology which says that there has to be an absolutely “empty space” in the universe (devoid of any existent in it), can require the charge to be attributed to the massive point-particles. In our new view, this is an instance of misattribution.

Accordingly, as our second step,

Remove the charge-attribute from the EC Object at $\vec{r}_j$ and re-assign it to the elemental CV of the aether around it.

To get rid of any possible notational confusion, introduce $q_A$ in place of $q_j$ into the statement of Coulomb’s law. Here, $q_A$ is the magnitude of that quality or attribute of the aether which allows the aether itself to electrostatically interact with the first charge $q_i$ and to experience $\vec{f}_{ij}$ at the specific point $\vec{r}_j$. The subscript $_A$ serves to remind the Aether.

Accordingly,

$\vec{f}_{ij} = \dfrac{q_i\,q_A}{r_{ij}^2} \hat{r}_{ij}$

Notice, the maths has remained the same. However, the ontology has changed for both the EC Object, as well as for the single elementary CV around the point $\vec{r}_j$.

The EC Object now has no classical $q_j$ charge on it. The elementary CV too is without a point-charge—if the term is taken in the text-book sense of the term. That is to say, two adjacent elementary CVs do not exert very high electrostatic forces on each other as if they were sources of Coulombic forces. $q_A$ captures a charge-like attribute only on the receiving side.

Instead of the EC Object, now, the aether itself is seen to carry some attribute whereby it experiences the same electrostatic force as a point-charge $q_j$ of the textbook description would. From the viewpoint of Coulomb’s law, the relevant measure of this attribute therefore is $q_A = q_j$.

2.3 Step 3: Generalize to all space:

Now, as the third step, generalize.

Since $q_A$ now is an attribute of the aether, all parts of it must possess the same attribute too. After all, the entire aether is ontologically a single object. The idea of the aether as a single object is valid because in places where the aether is not, we would have to have some still other physical object present there. Further, there is no evidence which says that the force-producing condition be present only at $\vec{r}_j$ but at no other parts of the aether.

Accordingly, generalize the above equation from $\vec{r}_j$ to any arbitrary location in the aether $\vec{r}_A$:

$\vec{\mathcal{F}}_{iA} = \dfrac{q_i\,q_A}{r_{iA}^2} \hat{r}_{iA}$

where $\vec{r}_A$ is a variable that at once applies to all space.

With this generalization, we have obtained a field of $q_A$ in the aether—one that is uniform everywhere, being numerically equal to $q_j$ (complete with the latter’s sign). As a consequence, a local force is produced by $q_i$ at every arbitrary elemental CV of the aether $\vec{r}_A$. Accordingly, there is a field of local forces too.

Since this is a big ontological change, we have changed the symbol on the left hand-side too. Thus, $\vec{\mathcal{F}}_{iA}$ represents a field of force whereas $\vec{f}_{ij}$, which appeared in the original Coulomb’s law, has been just a point-force at one specific location.

We call $\vec{\mathcal{F}}_{iA}$ the aetherial force-field. It can be used to yield the force on the second EC Object when it is present at any arbitrary position $\vec{r}_j$.

3. Implications for the ontologies of EC Objects and the aether:

With that change, what is now left of the original EC Object $q_j$?

3.1. An EC Object suffers force not due to its own charge but because of the electric aether:

Well, metaphorically speaking, the EC Object now realizes that even though its very near (and possibly dear) charge has now left it. However, it also realizes that it still is being forced just the same way.

Earlier, in the textbook EM, the poor little chappie of the EC Object $q_j$ was a silent sufferer of a force; it still remains so. But the force it receives now is not due to a point-concentration of charge with it, but due to a force imparted to it by the aether—which now has that extra attribute which measures to $q_A$.

All in all, the EC Object (or the classical “charge”) now comes to better understands itself and its position in the world. It now realizes that the causal agent of its misery was not a part of its own nature, of its own identity; it always was that (“goddamn”) portion of the aether in direct contact with it.

So, from now on, it will never forget the aether. It has grown up.

3.2. An EC Object causes a force-field to come into the aether too

But not everything is lost. An EC Object is not all that miserable, really speaking. The force field $\vec{\mathcal{F}}_{iA}$ was anyway created by an EC Object—the one at $q_i$.

Thus, the same EC Object fulfills two roles: as creator of a field, and as a sufferer of a force-fields created by all other EC Objects.

3.3. Evey EC Object still remains massive:

Every EC Object still gets to retain its mass just as before. So, if unhindered, it can even accelerate in space just as before. Hey, no one has cut down on its travelling allowance, alright?

So, regardless of this revision in the ontology of the EC Object, it still remains a massive particle. Being a “charge-less” object, in fact, makes it more consistent from an ontological perspective: the EC Object now interacts only with the aether, not directly with the other charge(s) (through action-at-a-distance), as we’ve always wanted.

3.4. The aether remains without inertia:

As to the aether: Though having a quality of charge, it still remains without any inertia. At least, it doesn’t have that inertia which comes “up” in its electrostatic interactions.

Hence, even though it is the one that primarily experiences the force $\vec{\mathcal{F}}_{iA}$ at $\vec{r}_{iA}$, this force does not translate into its own acceleration, velocity, or displacement. It just stays put where it is. But if a massive particle strays at its location, then that particular aetherial CV has no option but to pass along this force, by direct contact, to that massive particle.

3.5. The aether allows the EC Object to pass through it:

As we shall see in the section below, the aether poses no drag force to the passage of an EC Object. The aether also is all pervading, and no part of it undergoes displacements—that is, it does not move away to make a way for the EC Object to go through (the way the public makes way for a ministers caravan, in India).

We might not be too mistaken if we believe that the reason for this fact is that the aether has no inertia coming into picture in its electric interactions.

Thus, we have to revise our entire ontology of what exactly an EC Object is, what we mean by charge, and what exactly the aether is.

4. Ontological implications arising out the divergence of force fields:

4.1. Zero divergence everywhere except for at the location of the forcing charge:

It can be shown that an inverse-square force-field like $\vec{\mathcal{F}}_{iA}$ (or $\vec{\mathcal{F}}_{jA}$) has zero divergence everywhere, except around the point $\vec{r}_i$ (or $\vec{r}_j$ respectively) where the field is singular. There, the divergence equals the charge $q_i$ (or $q_j$, respectively).

Notice, the $\vec{\mathcal{F}}_{iA}$ field has a zero divergence even around the forced object $\vec{r}_j$—which was used in defining it. Make sure you understand it. The other field, viz., $\vec{\mathcal{F}}_{jA}$ does have a singularity at $\vec{r}_j$ and a divergence equal to $q_j$. But $\vec{\mathcal{F}}_{iA}$ doesn’t—not at $\vec{r}_j$. Similarly for the other field.

4.2. Non-zero forces everywhere:

However, notice that the elemental CV at the location $\vec{r}_j$ of the forced charge still carries a finite force at that point—exactly as everywhere else in the aether.

Remember: Divergence is about how a force-field changes in the infinitesimal neighbourhood of a given CV; not about what force-field is present in that CV. It is about certain kind of a spatial change, not about the very quantity whose change it represents.

4.3. Static equilibrium everywhere:

Since the divergence of the force field $\vec{\mathcal{F}}_{iA}$ is zero everywhere in the aether (excepting for the single point of the singularity at $\vec{r}_i$), no CV in the aether—finite or infinitesimal—exchanges a net surface-force with a CV completely enclosing it. (A seed of a fruit is completely enclosed by the fruit.) Thus, every aetherial CV is in static equilibrium with its neighbours. The static equilibrium internal to the aether always prevails, regardless of how the EC Objects at $\vec{r}_i$ or $\vec{r}_j$ move, i.e., regardless of how the fields $\vec{\mathcal{F}}_{iA}$ or $\vec{\mathcal{F}}_{jA}$ move. No finite change of local force conditions is able to disturb the prevailing static equilibrium internal to the aether

However, a pair of equal and opposite local surface-intensities of forces do come to exist at every internal surface in the aether. Hence, a state of stress may be associated with the aether. These stresses are to be taken by way of analogy alone. Their nature is different from the stresses in the NM Ontological continuous media.

Note again, the force-field is non-uniform (it varies as the inverse-square of separation)—i.e. non-zero. So there still is a non-zero force being exerted on an aetherial CV—even if there is no net force on any surface between any two adjacent CVs.

4.4. The direct contact governs the force exchanges internal to the aether:

A direct consequence of the inverse-square law and the divergence theorem also is that the force field must be seen as arising due purely to a direct contact between the neighbouring aetherial CVs.

There is no transfer of momentum from one CV to another distant CV via any action-at-a-distance directly between the two—by “jumping the queue” of the other parts of the intervening aether, so to speak.

4.5. The direct contact governs the force exchanges between the aether and the forced EC Object:

With the presence of a non-zero $\vec{\mathcal{F}}_{iA}$ force acting on it, an aetherial CV which is in direct contact with a massive EC object, transmits a surface-force to the EC Object via the internal surface common to them.

So, while the CV itself does not move, it does force the EC Object.

4.6. Inertia-less aether implies no drag force on the EC Object:

The aetherial force-fields are conservative, and the description provided by Coulomb’s law is logically complete for electrostatics. Given these two premises, the aether must act as a drag-free medium for the passage of an EC Object.

An aetherial CV does not exert any resistive or assistive forces, over and above the forces of the $\vec{\mathcal{F}}_{iA}$ field, on the massive EC Object at $q_j$.

There is a force through direct contact between an aether and an EC Object too—just as in NM ontology. However, quite unlike in NM ontology, there also is no force for the passage of an EC Object through the aether. (Can this be explained because the aether has no inertia to show at the level of electric phenomena?)

All in all, the only difference between the forces at two neighouring points in space are the two local point-forces of the field. Hence, if the $\vec{\mathcal{F}}_{iA}$ field is non-uniform (and the Coulombic fields anyway are non-uniform), the massive point-particle of the forced EC Object (the one at $\vec{r}_j$) always slides “down-hill” of $\vec{\mathcal{F}}_{iA}$.

Note, our description here differs from the textbook description. The textbook description implies that a negative charge always climbs up the hill of a positive $\vec{E}$ field, whereas a positive charge climbs down the same hill. In our description, we use $\vec{\mathcal{F}}_{iA}$ in place of $\vec{E}$, and the motion of the EC object always goes downhill.

4.7. The electric aether as the unmoved (or unmovable) mover:

Since an aetherial CV surrounding a given EC Object forces it, but doesn’t move itself, we may call the EM aether the unmoved (or unmovable) mover.

Aristotle, it would seem, had an idea or two right at the fundamental levels also of physics—not just in metaphysics or logic. This idea also seems to match well with certain, even more ancient, Upanishadic (Sanskrit: उपनिषदीय) passages as well.

But all that is strictly as side remarks. We couldn’t possibly have started with those ancient passages and come to build the force-fields of the precisely required divergence properties, without detailed investigations into the physical phenomena. Mystically oriented physicists and philosophers are welcome to stake a claim to the next Nobel in physics, if they want. But they wouldn’t actually get a physics Nobel, because the alleged method simply doesn’t work for physics.

For building theoretical contents of physics, philosophical passages can be suggestive at best. The actual function of philosophy in physics is to provide broad truths, and guidelines. For instance, consider the fact that there has to be an ontology, at least just an implied one, for every valid theory of physics. This piece of truth itself comes from, and is established in, only philosophy—not in physics. So, philosophy is logically required. It can also be useful in being suggestive of metaphors. But even then, physics is a special science that refers to scientific observations, and uses experimental method, and quantitative laws.

5. No discontinuity in the $\vec{\mathcal{F}}_{iA}$ field around $\vec{r}_j$:

Oh, BTW, did you notice that the force-field $\vec{\mathcal{F}}_{iA}$ is continuous everywhere—including at the location $\vec{r}_j$ of the forced EM Object? Looks like our problem from the last post has got solved, does’t it? Well, yes, it is!

Even if $\vec{\mathcal{F}}_{iA}$ is discontinuous at its singularity, this singular point happens to be at the other (forcing) EC Object’s location $\vec{r}_i$. We can ignore it from our analysis because the maths of differential equations anyway excludes any singular point. Our difficulty, as noted in the last post, was not at the proton’s position (i.e. the singularity at $\vec{r}_i$) but with the discontinuity at the electron’s position (i.e. $\vec{r}_j$).

Now we can see that, ignoring the singularity of $\vec{\mathcal{F}}_{iA}$ at $\vec{r}_i$, this aetherial force field keeps on forcing the EC Object $q_j$ continuously everywhere in the field.

No matter where the $j$-th EC Object goes, it can’t hide from $\vec{\mathcal{F}}_{iA}$, and must get forced from that position too. Similarly, no matter where the $i$-th EC Object goes, it can’t hide from $\vec{\mathcal{F}}_{jA}$, and must get forced from that position too. That is, following electrostatics alone.

(EM Dynamics keeps the electrostatic description as is, but also adds the force of magnetism, which complicates the whole thing. QM the electrostatic description as is, removes the magnetic fields, but introduces $\Psi$ field, which raises such issues that we ended up writing this very lengthy series on just the ontologically important parts of them!)

6. Overall framework: Pairs of charges, and hence of force-fields:

An isolated charge by itself does not exist in the universe. There always are pairs of them.

Mathematically, a $\vec{\mathcal{F}}_{iA}$ field acquires its particular sign, which depends on the specific polarities of the respective charges forming a pair in question, right from the time the two are “brought” “from” infinity. Ontologically, this is a big difference between our view and that of the textbook EM.

The textbook EM captures interactions via $\vec{E}$ field which is found in reference to a positive test charge of unit magnitude. The force-field is thus severed from the sign of the forced charge; it reflects only the forcing charge. In our view, both the charges have equal say in determining both $\vec{\mathcal{F}}_{iA}$ and $\vec{\mathcal{F}}_{jA}$ fields.

Physically, both these fields from a pair of charges have always existed, exist, and will always exist, at all times. There is no way to annihilate only one of the two.

This feature of the EM physical reality is remarkably similar to the necessity of there physically being only a pair of forces, and not an isolated physical force in the NM ontology, following Newton’s third law.

If there are two charges in an isolated system, there are two force fields. If there are three charges, there are six force-fields. The number of force-fields equals the number of separation vectors. See the homework above.

Due to the conservative nature of the Coulombic forces, all the force-fields superpose at every point in the aether.

The massive particles of EC Objects merely accelerate under the action of the net field present at their respective positions. That’s on the acceleration side, i.e., the role that a given EC Objects plays as a forced charge. However, the same EC Object also plays a role as a forcing charge. The locus of this role moves too.

Thus,

All singularities in all the force-fields also move when the EC Objects where they are present, move.

7. An essentially pairs-wise description of the fields still remains objective:

Each of the two force-fields $\vec{\mathcal{F}}_{iA}$ and $\vec{\mathcal{F}}_{jA}$ represents an interaction between two objects, sure. However, this fact does not make the description devoid of objectivity; it does not make it inherently relativistic, Machian, or anything of that sort.

The force-fields are due to interaction between pairs of charges, and not due to mere presence of the individual charges. Yet,

The individual charges still retain the primary ontological status.

Force-fields do not have a primary standing. Their ontological standing is: (i) as attributes of the aether—which is a primary existent, and (ii) as effects produced by the EC Objects—which again are the primary existents.

Force fields acquire signs as per the properties of the two EC Objects taken together. But the same EC Object contributes exactly the same sign in every conceivable pair it forms with the other charges in the universe. Thus, the sign is an objective property of a single EC Object, not of a pair of them.

Each singularity resides in a point-particle of the EC Object. Given the same forced charge $q_j$, there are $N-1$ number of $\vec{\mathcal{F}}_{iA}$ force-fields pulling or pushing $q_j$ in different directions. Each of these $N-1$ singularities resides at specific point-positions $q_i$s of the forcing EC Objects. Further, each forced charge also acts as a forcing EC Object in the same pair. Thus, the two force-fields $\vec{\mathcal{F}}_{iA}$ and $\vec{\mathcal{F}}_{jA}$ imply two point-phenomena: the two singularities.

Each singularity at the forcing charge $q_i$ also moves when the object at its location moves under the action of the other force field $\vec{\mathcal{F}}_{jA}$ acting on it.

The ontologically mandated simultaneous existence of the two force-fields is akin to the simultaneous existence of the action-reaction pair from the good old Newtonian mechanics. The fact that forces due to direct contact come only in pairs does not imply that “everything is relative,” or properties that can be objectively isolated and attributed to individual objects cease to exist just because two objects participate in an interaction. For more on what causality means and what interaction means, see my earlier post [^] in this series.

8. Potential energy numbers as aspatial attributes of a system:

8.1. A note on the notation:

I would have liked to have left this topic for homework, but there is a new notation to be introduced here, too. So, let’s cover this topic, although as fast as possible.

So, first, a bit about the notation we will adopt here, and henceforth.

Since we are changing the ontology of the EM physics, we should ideally make changes to the notation used for the potential energies too. However, I want to minimize the changes in notation when it comes to writing down the Schrodinger equation.

So, I will make the appropriate changes in the discussion of the energy analysis that precedes the Schrodinger equation, but I will keep the notation of the Schrodinger equation intact. (I don’t want reviewers to glance at my version of the Schrodinger equation, and throw it in the dust-bin because it doesn’t follow the standard textbook usage.)

So, let’s get going. We will make the notations as we go along.

8.2. The single number of potential energy of two point-charges as an aspatial attribute:

Let $\Pi( q_i, q_j)$ be the potential energy of the system due to two charges $q_i$ and $q_j$ being present at $\vec{r}_i$ and $\vec{r}_j$, respectively.

This is a system-wide global number, a single number that changes as either $q_i$, or $q_j$, or both, are shfited. It is numerically equal to the work done on the system in variationally shifting the two EC Objects from infinity to their stated positions. Using Coulomb’s law, and the datum of zero potential energy “at” infinity, it can be shown that this quantity is given by:

$\Pi( q_i, q_j) = \dfrac{1}{4\,\pi\,\epsilon_0}\dfrac{q_i\,q_j}{r_{ij}}$.

Even if the above formula makes reference to the magnitude of the separation vector, there is nothing in it which will allow us to locate it at any one point of space. So, $\Pi( q_i, q_j)$ is an aspatial attribute. You can’t point out a specific location for it in space. It is a device of analysis alone.

8.3 Potential energy numbers obtained by keeping one charge fixed:

Let $\Pi(q_j)$ be the potential energy (a single number) imparted to the system due to the work done on the system in variationally shifting $q_j$ from infinity to its current position $\vec{r}_j$, while keeping $q_i$ fixed at $q_i$.

Notice, the charge in the parentheses is the movable charge. When only one charge is listed, the other one is assumed fixed. Since here $q_i$ is fixed, there is no work done on the system at $\vec{r}_i$. Hence, the single number that is the system potential energy, increases only due to a variational shifting of $q_j$ alone.

Similarly, let $\Pi(q_i)$ be the potential energy (a single number) imparted to the system due to the work done on the system in variationally shifting $q_i$ from infinity to its current position $\vec{r}_i$, while keeping $q_j$ fixed at $q_j$.

(It’s fun to note that it doesn’t matter whether you bring any of the charges from $+\infty$ or $-\infty$. Set up the integrals, evaluate them, and convince yourself. You can also take a short-cut via the path-independence property of the conservative forces.)

9. Obtaining a spatial field for the “potential” energy of Schrodinger’s equation:

It can be shown that:
$\Pi( q_i, q_j ) = \Pi( q_i) = \Pi(q_j)$.

Consider now the problem of the last post, viz., the hydrogen atom.

OK. It is obvious that:
$\Pi(q_j) = \dfrac{1}{4\,\pi\,\epsilon_0}\dfrac{q_i\,q_j}{r_{ij}}$

Follow the earlier procedure of ontologically re-assigning the effects due to a point-charge to the the local elemental CV of the aether at $q_j$, thereby introducing $q_A$ in place of $q_j$; and then generalizing from $\vec{r}_j$ to $\vec{r}_A$, we get to a certain field of an internal energy. Let’s call give it the symbol $V(q_j)$

Thus,
$V(q_j) = \dfrac{1}{4\,\pi\,\epsilon_0}\dfrac{q_i\,q_A}{r_{iA}}$

We thus got the “potential” energy field we wanted for use in Schrodinger’s equation. It’s continuous even at $\vec{r}_j$.

Following the mainstream QM, we will continue calling the $V(q_j)$ field function the “potential” energy field for $q_j$.

However, as mentioned in a previous post, the field denoted by $V(q_j)$ means the same as the system’s potential energy only when the $3D$ field is concretized to its value for a specific point $\vec{r}_j$. But taken in its entirety, what $V(q_j)$ denotes is an internal energy content that is infinitely larger than that part which can be converted into work and hence stands to be properly called a potential energy.

It is true that the entire internal energy content moves out of the system when both the charges are taken to infinity. However, such a passage of the energy out of the system does not imply that all of it gets exchanged at the moving boundaries, because the boundaries here are point positions.

So, strictly speaking, the $V(q_j)$ field does not qualify to be called a potential energy field. Yet, to avert confusions from an already skpetical physicist community, we will keep this technical objection of ours aside, and call $V$ the potential energy field.

If both the proton and the electron are to be regarded as movable, then we have to follow a procedure of splitting up the total, as shown below:

Split up $\Pi( q_i, q_j)$ into two equal halves:

$\Pi( q_i, q_j) = \dfrac{1}{2} \Pi( q_i, q_j ) + \dfrac{1}{2} \Pi( q_i, q_j )$

Substitute on the right hand-side the two single-movable-charge terms:
$\Pi( q_i, q_j) = \dfrac{1}{2} \Pi( q_j ) + \dfrac{1}{2} \Pi( q_i )$

Now first aetherize and then generalize $\Pi( q_j )$ to $V(q_j)$, and similarly go from $\Pi( q_i)$ to $V(q_i)$.

We thus get:
$V( q_i, q_j) = \dfrac{1}{2} \dfrac{1}{4\,\pi\,\epsilon_0}\dfrac{q_i\,q_A}{r_{iA}}+ \dfrac{1}{2} \dfrac{1}{4\,\pi\,\epsilon_0}\dfrac{q_j\,q_A}{r_{jA}}$

In the short and sweet form:
$V( q_i, q_j) = \dfrac{1}{2} V( q_j ) + \dfrac{1}{2} V( q_i )$
where all $V$s are field-quantities (in joule, not volt).

We thus have solved the problem of discontinuity in the potential energy fields.

10. A detailed comment on $\vec{E}(q_i)$ vs. $\vec{\mathcal{F}}_{iA}$:

Mathematically (in electrostatics), Lorentz’ law says:
$\vec{\mathcal{F}}_{iA} = q_j \vec{E}(q_i)$

But ontologically, there is no consistent interpretation for the textbook EM term of $\vec{E}(q_i)$ (the electric vector field). It is supposed, in the textbook EM, to be a property of $q_i$ alone. But such a thing is ontologically impossible to interpret. The only physically consistent way in which it can be interpreted is to regard $\vec{E}(q_i)$ as that hypothetical force-field which would arise due to $q_i$ if a positive unit charge $q_j$ were to be present, one at a time, at each point in the universe. We would then add these point-forces defined at all different points together and treat it as a field. So, the best physical interpretation for it is a hypothetical one. This also is a reason why such a formulation definitely implants a seed of a doubt regarding the very physical-ness of the fields idea.

As a hypothetical field, $\vec{E}$ tries to give the “independent” force-field of $q_i$ and so, its sign depends on the sign of $q_i$. Which leads to the trouble of discontinuity with the associated potential energy field too.

In contrast, $\vec{\mathcal{F}}_{iA}$ and $\vec{\mathcal{F}}_{jA}$ are easy to intepret ontologically. They are the two aetherial fields which simultaneously exist due to the existence of $q_i$ and $q_j$—and vice-versa. $\vec{\mathcal{F}}_{iA}$ doesn’t exist without there also being $\vec{\mathcal{F}}_{iA}$ and vice-versa.

The aetherial field $\vec{\mathcal{F}}_{jA}$ has a singularity at $q_i$. So, it forces $q_j$, but not $q_i$, due to symmetry of the field around the latter. Similarly, the aetherial field $\vec{\mathcal{F}}_{iA}$ has a singularity at $q_j$. It forces $q_i$ but not $q_j$ due to symmetry of the field around the latter.

Based on these observations, similar statements can be made for $V(q_i)$ and $V(q_j)$ as well.

The total atherial field due to a pair of charges, $V(q_i, q_j)$ has two singularities, one each at $\vec{r}_i$ and $\vec{r}_j$. It changes with any change in the position of either of the two charges. It too ontologically exists. However, in calculations, we almost never have the occasion to take the total field. It’s always the force on, or the partial potential energy due to, a single charge at a time.

11. The $V$ field “lives” in the ordinary $3D$ space for any arbitrary system of $N$ charges:

Oh, BTW, did you notice one thing that might have gone almost unnoticed?

Even if $\Pi(q_i, q_j)$ is defined on the space of separation vectors, and even if these separation vectors are defined only on the six-dimensional configuration space existing in the mystical Platonic “heaven”, we have already brought it down to our usual $3D$ world. We did it by expressing $\Pi(q_i, q_j)$ as (i) a concretization at $\vec{r}_i$ and $\vec{j}$ of (ii) a sum of two instantaneous $3D$ fields (each having the right sign, and each without any point-sharp discontinuity),

So, given a pair of charges, what ontologically exist are the two $3D$ fields: $\vec{\mathcal{F}}_{iA}$ + $\vec{\mathcal{F}}_{jA}$. They move when $\vec{r}_i$ and $\vec{r}_j$ move, respectively.

To think of a $6D$ configuration space of all possible values for $\vec{r}_i$ and $\vec{r}_j$ is to think of the set of all possible locations for the two singularities of the two $3D$ fields.

Just the fact that each singularity can physically be in any arbitary place does not imply that there is no $3D$ field associated with it.

The two descriptions (one on the configuration space and the other using $3D$ fields) are not only equivalent in the sense they reproduce the same consequences, our description is richer in terms of ontological clarity and richness: it speaks of $3D$ fields each part of which can interact with another $3D$ field, namely $\Psi(\vec{r},t)$, thereby forming a conceptual bridge to the otherwise floating abstractions of the mainstream QM too.

12. The aether as a necessity in the physics of EM, and hence, also of QM:

One last point before we close for today.

In making the generalization from $\vec{r}_A$ defined only at the elemental CV in the aether surrounding $q_j$‘s position $\vec{r}_j$ to the entire space, we did not justify the generalization procedure itself on the ontological grounds.

The relevant physics fact is simple: The aether (the non-inertial one, as first put forth by Lorentz, and also one that appeared very naturally in our independent development) is all pervading.

12.1 Philosophical reason for the necessity of the aether:

Philosophically, the aether, qua a physical existent, replaces the notion of the perfectly “empty” space—i.e. a notion of the physical space that, despite being physical, amounts to a literal nothing. Such a theory of physics, thereby, elevates the nothing—the zero, the naught—to the same level as that of something, anything, everything, the existence as such.

But following ancient reasoning, a nothing is only a higher-level abstraction, not a physical existent, that denotes the absence of something. It has no epistemological status as apart from that whose absence it denotes. It certainly cannot be regarded as if it were a special something. If it were to be a special something, in physics theory, it would be an ontological object.

So, something has to be there where the point-phenomena of massive particles of EC Objects are not.

That’s the philosophical justification.

12.2 Mathematical considerations supporting the idea of the aether:

Now let’s look at a few mathematical considerations which point to the same conclusion.

12.2.i. Discontinuity in force-field if the aether is not there:

As seen right in this post, if aether is removed, then the force-experiencing and the force-producing aspects of an EC Object—viz. the charge $q_j$ has to be attributed to the EC Object itself.

However, this introduces the discontinuity of the kind we discussed at length in the last post. The only way to eliminate such a discontinuity is to have a field phenomenon, viz. $q_A$, in place of the point-phenomenon, viz. $q_j$.

12.2.ii. Nature of the spatial differential operators (grad, div, curl):

The electromagnetic theory in general uses differential operators of grad, div, and curl. Even just in electrostatics, we had to have a differential element at $\vec{r}_j$ to be able to calculate the work done in moving a charge. The $\text{d}r$ is an infinitesimal element of a separation vector, not a zero. Similarly, you need the grad operator for going from the aetherial potential energy $V(q_j)$ field to the aetherial force field $\vec{\mathcal{F}}_{iA}$. Similarly, for the div and the curl (in dynamical theory of EM). In QM, we use the Laplacian too.

All these are spatial differential operators. They refer to two different values of a field variable at two different points, and then take an appropriate ratio in the limit of the vanishing size.

Now the important characteristics to bear in mind here comes from a basic appreciation of calculus. An infinitesimal of something does not denote a definite change. The infinitesimal arises in the context of two definite quantities at two distinctly identifiable points, and then applies a limiting process. So, the proper order of knowledge development is: two distinct points lying at the endpoints of a continuous interval; quantities (variables) defined at the two points; then a limiting process.

This context also gets reflected in the purpose of having infinitesimally defined quantities in the first place. The only purpose of a derivative is to have the ability to integrate in the abstract, and vice versa (via the fundamental theorem of calculus of derivatives and anti-derivatives). Sans ability to integrate, a differential equation serves no purpose; it has no place even in proper mathematics let alone in proper physics. Ontology of calculus demands so.

If you want to apply grad to the $V(q_j)$ field, for instance, and if you insist on a formulation of an empty space and a point-particle for an EC Object (a point-particle itself being an abstraction from a finite particle, but let’s leave that context-drop aside), then you have to assume that the charge-attribute operates not just at $q_j$ but also at infinitesimally small distances away from it. Naturally, at least an infinitesimally small distance around the $\vec{r}_j$ point cannot be bereft of everything; there must be something to it—something which is not covered in the abstraction of the EC Object itself. What is it? Blank out.

An infinitesimal is not a definite quantity; its only purpose is to be able to formulate integral expressions which, when concretized between definite limits, yield calculations of the unknown variables. So, an infinitesimal can be specified only in such a context where definite end-points can be specified.

If so, what is the definite point in the physical space up to which the infinitesimal abstraction, say of $\nabla V(q_j)$ remains valid? How do you pick out such a point? Blank out, again!

The straight-forward solution is to say that the infinitesimal expresses a continuous variation over all finite portions of space away from $\vec{r}_j$, and that this finite portion is indefinitely large. The reason is, the governing law itself is such that the abstraction of infinitely large separations are required before you can at all reach a meaningful datum of $V(q_j)_f$ approaching $V(q_j)_i$.

Naturally, the nature of the mathematics, as used in the usual EM physics, itself demands that whatever it is which is present in the infinitesimal neighbourhood of $q_j$ must also be present everywhere.

That’s your mathematically based argument for an all pervading aether, in other words.

12.3. Polemics: No, “nature abhors vacuum” is not the reason we have put forth:

The line that physicists opposed to the idea of the aether love to quote most, is: “nature abhors vacuum.” As if the physical nature had some intrinsic, mystical, purpose; a consciousness in some guise that also had some refined tastes developed with it.

They choose to reproduce this line because it is the weakest expression of the original formulation (which, in the Western philosophy, is due to Parmenidus).

To obfuscate the issue is to ignore his presence in the context altogether; pretend that Aristotle’s estimated three quarters of extinct material might have had something important or better to offer—strictly in reference to the one quarter extant material (these estimates being merely estimates); to ignore every other formulation of the simple underlying truth that the nothing cannot have any valid epistemological status; and try to win the argument via the method of intimidation.

Trying to play turf-battles by using ideas from mysticism, and thence mystefying everything.

Idiots. And, anti-physics zealots.

Anyway, to end this section on a better, something-encapsulating way:

12.4 Polemics: Aether is not an invisible medium; it is not the thing that does nothing:

It’s wrong to think that the EC Object is what you do see and the aether what you don’t. If you have that idea of the aether, know that you are wrong: you have reduced the elementary EC Object to a non-elementary NM-ontological one, and the aether to the Newtonian absolute space. The fact of the matter is: it is both the EC Objects and the aether which together act to let any such a thing as seeing possible.

In EM, they both lie at the most fundamental level. Hence, they both serve as the basis for every physical mechanism, including that involved in perception (light and its absorption in the eye, the neural signals, etc.) Drop any one of them and you won’t be able to see at all—whether the NM Objects or the “empty” space. In short, the aether is not a philosophical slap on just to make Aritstotle happy!

Both the aether and the massive point-particle of EC Objects are abstractions from some underlying physical reality; they both are at the same ontological standing in the EM (and QM) theory. You can’t have just one without the other. Any attempt to do so is bound to fail. Even if you accept Lorentz’ idea (I only vaguely know it via a recent skimming) that all there is just one single object of the aether, to build any meaningful quantitative laws, you still have to make a reference to the singularity conditions in the aetherial fields. That’s nothing but your plain EC Object, now coming in via another back-door entry. Why not keep them both?

We can always take mass of the spring and dump it with the point-mass of a ball, and take the stresses in the finite ball and dump them in the spring. That’s how we get the mass-spring system. Funny, people never try to have only the the spring or only the mass. But they mostly insist on having only the EC Objects but not the aether. Or, otherwise, advocate the position that everything is the aether, its modern “avataar” being: “everything is fields, dude!”

Oh.

OK. Just earmark to keep them both. Matter cannot act where it is not.

13. A preview of the things to come:

The next time, we will take a look at some of the essence of the derivation of Schrodinger’s equation. The assigned reading for the next post is David Morin’s chapter on quantum mechanics, here (PDF, 495 kB) [^].

Also, refreshen up a bit about the simple harmonic motion and all, preferably, from Resnick and Halliday.

No, contrary to a certain gossip among physicists, physics is not a study of the pendulum in various disguises. It is: the study of the mass-spring system!

A pendulum displays only the dynamical behaviour, not static (and it does best only for the second-order relations). On the other hand, a spring-mass system shows also the fact that forces can be defined via a zeroth-order relation: $\vec{f} = - k (\vec{x} - \vec{x}_0)$. So, it works for purely static equilibria too. That is, apart from reproducing your usual pendulum behaviour anyway. It in fact has a very neat visual separation of the place where the potential energy is stored and the place that does the two and fro. So, it’s even easier to see the potential and kinetic energies trying to best each other. And it all rests on a restoring force that cannot be defined via $\vec{f} = \dfrac{\text{d}\vec{p}}{\text{d}t}$. The entire continuum mechanics begins with the spring-mass system. If I asked you to explain me the pendulum of the static equilibrium, could you? So there. Just develop a habit of ignoring what physicists say, as always. After all, you want to study physics, don’t you?

Alright, go through the assigned reading, come prepared the next time (may be after a week), take care in the meanwhile, and bye for now.

A song I like:

(Hindi) “humsafar mere humsafar, pankh tum…”
Singers: Lata Mangeshkar and Mukesh
Lyrics: Gulzaar
Music: Kalyanji-Anandji