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_is 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 Vs 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

 

Ontologies in physics—7: To understand QM, you have to first solve yet another problem with the EM

In the last post, I mentioned the difficulty introduced by (i) the higher-dimensional nature of \Psi, and (ii) the definition of the electrostatic V on the separation vectors rather than position vectors.

Turns out that while writing the next post to address the issue, I spotted yet another issue. Its maths is straight-forward (the X–XII standard one). But its ontology is not at all so easy to figure out. So, let me mention it. (This upsets the entire planning I had in mind for QM, and needs small but extensive revisions all over the earlier published EM ontology as well. Anyway, read on.)


In QM, we do use a classical EM quantity, namely, the electrostatic potential energy field. It turns out that the understanding of EM which we have so painfully developed over some 4 very long posts, still may not be adequate enough.

To repeat, the overall maths remains the same. But it’s the physics—rather, the detailed ontological description—which has to be changed. And with it, some small mathematical details would change as well.

I will mention the problem here in this post, but not the solution I have in mind; else this post will become huuuuuge. (Explaining the problem itself is going to take a bit.)


1. Potential energy function for hydrogen atom:

Consider a hydrogen atom in an otherwise “empty” infinite space, as our quantum system.

The proton and the electron interact with each other. To spare me too much typing, let’s approximate the proton as being fixed in space, say at the origin, and let’s also assume a 1D atom.

The Coulomb force by the proton (particle 1) on the electron (particle 2) is given by \vec{f}_{12} = \dfrac{(e)(-e)}{r^2} \hat{r}_{12}. (I have rescaled the equations to make the constants “disappear” from the equation, though physically they are there, through the multiplication by 1 in appropriate units.) The potential energy of the electron due to the proton is given by: V_{12} = \dfrac{(e)(-e)}{r}. (There is no prefactor of 1/2 because the proton is fixed.)

The potential energy profile here is in the form of a well, vaguely looking like the letter `V’;  it goes infinitely deep at the origin (at the proton’s position), and its wings asymptotically approach zero at \pm \infty.

If you draw a graph, the electron will occupy a point-position at one and only one point on the r-axis at any instant; it won’t go all over the space. Remember, the graph is for V, which is expressed using the classical law of Coulomb’s.


2. QM requires the entire V function at every instant:

In QM, the measured position of the electron could be anywhere; it is given using Born’s rule on the wavefunction \Psi(r,t).

So, we have two notions of positions for the supposedly same existent: the electron.

One notion refers to the classical point-position. We use this notion even in QM calculations, else we could not get to the V function. In the classical view, the electronic position can be variable; it can go over the entire infinite domain; but it must refer to one and only one point at any given instant.

The measured position of the electron refers to the \Psi, which is a function of all infinite space. The Schrodinger evolution occurs at all points of space at any instant. So, the electron’s measured position could be found at any location—anywhere in the infinite space. Once measured, the position “comes down” to a single point. But before measurement, \Psi sure is (nonuniformly) spread all over the infinite domain.

Schrodinger’s solution for the hydrogen atom uses \Psi as the unknown variable, and V as a known variable. Given the form of this equation, you have no choice but to consider the entire graph of the potential energy (V) function into account at every instant in the Schrodinger evolution. Any eigenvalue problem of any operator requires the entire function of V; a single value at a time won’t do.

Just a point-value of V at the instantaneous position of the classical electron simply won’t do—you couldn’t solve Schrodinger’s equation then.

If we have to bring the \Psi from its Platonic “heaven” to our 1D space, we have to treat the entire graph of V as a physically existing (infinitely spread) field. Only then could we possibly say that \Psi too is a 1D field. (Even if you don’t have this motivation, read on, anyway. You are bound to find something interesting.)

Now an issue arises.


3. In Coulomb’s law, there is only one value for V at any instant:

The proton is fixed. So, the electron must be movable—else, despite being a point-particle, it is hard to think of a mechanism which can generate the whole V graph for its local potential energies.

But if the electron is movable, there is a certain trouble regarding what kind of a kinematics we might ascribe to the electron so that it generates the whole V field required by the Schrodinger equation. Remember, V is the potential energy of the electron, not of proton.

By classical EM, V at any instant must be a point-property, not a field. But Schrodinger’s equation requires a field for V.

So, the only imaginable solutions are weird: an infinitely fast electron running all over the domain but lawfully (i.e. following the laws at every definite point). Or something similarly weird.

So, the problem (how to explain how the V function, used in Schrodinger’s equation) still remains.


4. Textbook treatment of EM has fields, but no physics for multiplication by signs:

In the textbook treatment of EM (and I said EM, not QM), the proton does create its own force-field, which remains fixed in space (for a spatially fixed proton). The proton’s \vec{E} field is spread all over the infinite space, at any instant. So, why not exploit this fact? Why not try to get the electron’s V from the proton’s \vec{E}?

The potential field (in volt) of a proton is denoted as V in EM texts. So, to avoid confusion with the potential energy function (in joule) of the electron, let’s denote the proton’s potential (in volt) using the symbol P.

The potential field P does remain fixed and spread all over the space at any instant, as desired. But the trouble is this:

It is also positive everywhere. Its graph is not a well, it is a peak—infinitely tall peak at the proton’s position, asymptotically approaching zero at \pm \infty, and positive (above the zero-line) everywhere.

Therefore, you have to multiply this P field by the negative charge of electron e, so that P turns into the required V field of the electron.

But nature does no multiplications—not unless there is a definite physical mechanism to “convert” the quantities appropriately.

For multiplications with signed quantities, a mechanism like the mechanical lever could be handy. One small side goes down; the other big side goes  up but to a different extent; etc. Unfortunately, there is no place for a lever in the EM ontology—it’s all point charges and the “empty” space, which we now call the aether.

Now, if multiplication of constant magnitudes alone were to be a problem, we could have always taken care of it by suitably redefining P.

But the trouble caused by the differing sign still remains!

And that’s where the real trouble is. Let me show you how.

If a proton has to have its own P field, then its role has to stay the same regardless of the interactions that the proton enters into. Whether a given proton interacts with an electron (negatively charged), or with another proton (positively charged), the given proton’s own field still has to stay the same at all times, in any system—else it will not be its own field but one of interactions. It also has to remain positive by sign—even if P is rescaled to avoid multiplications.

But if V has to be negative when an electron interacts with it, and if V also has to be positive when another proton interacts with it, then a multiplication by the signs (by \pm 1 must occur. You just can’t avoid multiplications.

But there is no mechanism for the multiplications mandated by the sign conversions.

How do we resolve this issue?


5. The weakness of a proposed solution:

Here is one way out that we might think of.

We say that a proton’s P field stays just the same (positive, fixed) at all times. However, when the second particle is positively charged, then it moves away from the proton; when the second particle is negatively charged, then it moves towards the proton. Since V does require a dot product of a force with a displacement vector, and since the displacement vector does change signs in this procedure, the problem seems to have been solved.

So, the proposed solution becomes: the direction of the motion of the forced particle is not determined only by the field (which is always positive here), but also by the polarity of that particle itself. And, it’s a simple change, you might argue. There is some unknown physics to the very abstraction of the point charge itself, you could say, which propels it this way instead of that way, depending on its own sign.

Thus, charges of opposing polarities go in opposite directions while interacting with the same proton. That’s just how charges interact with fields. By definition. You could say that.

What could possibly be wrong with that view?

Well, the wrong thing is this:

If you imagine a classical point-particle of an electron as going towards the proton at a point, then a funny situation ensues while using it in QM.

The arrows depicting the force-field of the proton always point away from it—except for the one distinguished position, viz., that of the electron, where a single arrow would be found pointing towards the proton (following the above suggestion).

So, the action of the point-particle of the electron introduces an infinitely sharp discontinuity in the force-field of the proton, which then must also seep into its V field.

But a discontinuity like that is not basically compatible with Schrodinger’s equation. It will therefore lead to one of the following two consequences:

It might make the solution impossible or ill-defined. I don’t know enough about maths to tell if this could be true.  But what I can tell is this: Even if a solution is possible (including solutions that possibly may be asymptotic, or are approximate but good enough) then the presence of the discontinuity will sure have an impact on the nature of the solution. The calculated \Psi wouldn’t be the same as that for a V without the discontinuity. That’s inevitable.

But why can’t we ignore the classical point-position of the electron? Well, the answer is that in a more general theory which keeps both particles movable, then we have to calculate the proton’s potential energy too. To do that, we have to take the electric potential (in volts) P of the electron, and multiply it by the charge of the proton. The trouble is: The electric potential field of the electron has singularity at its classical position. So, classical positions cannot be dropped out of the calculations. The classical position of a given particle is necessary for calculating the V field of the other particle, and, vice-versa.

In short, to ensure consistency in the two ways of the interaction, we must regard the singularities as still being present where they are.

And with that consideration, essentially, we have once again come back to a repercussion of the idea that the classical electron has a point position, but its potential energy field in the electrostatic interaction with the proton is spread everywhere.

To fulfill our desire of having a 3D field for \Psi, we have to have a certain kind of a field for V. But V should not change its value in just one isolated place, just in order to allow multiplication by -1, because doing so introduces a very bad discontinuity. It should remain the same smooth V that we have always seen in the textbooks on QM.


6. The problem statement, in a nutshell:

So, here is the problem statement:

To find a physically realizable way such that: even if we use the classical EM properties of the electron while calculating V, and even if the electron is classically a point-particle, its V function (in joules) should still turn out to be negative everywhere—even if the proton has its own potential field (P, in volts) that is positive everywhere in the classical EM.

In short, we have to change the way we look at the physics of the EM fields, and then also make the required changes to any maths, as necessary. Without disturbing the routine calculations either in EM or in QM.

Can it be done? Well, I think the answer is “yes.”


7. A personal note:

While I’ve been having some vague sense of there being some issue “to be looked into later on” for quite some time (months, at least), it was only over the last week, especially over the last couple of days (since the publication of the last post), that this problem became really acute. I always used to skip over this ontology/physics issue and go directly over to using the EM maths involved in the QM. I used to think that the ontology of such EM as it is used in the QM, would be pretty easy to explain—at least as compared to the ontology of QM. Looks like despite spending thousands of words (some 4–5 posts with a total of may be 15–20 K words) there still wasn’t enough of a clarity—about EM.

Not if we adopt the principle, which I discovered on the fly, right while in the middle of writing this series, that nature does no multiplications without there being a physical mechanism for it.

Fortunately, the problem did become clear. Clear enough that, I think, I also found a satisfactory enough solution to it too. Right today (on 2019.10.15 evening IST).

Would you like to give it a try? (I anyway need a break. So, take about a week’s time or so, if you wish.)


Bye for now, take care, and see you the next time.


A song I like:

(Hindi) “jaani o jaani”
Singer: Kishore Kumar
Music: Laxmikant-Pyarelal
Lyrics: Anand Bakshi


History:

— Originally published: 2019.10.16 01:21 IST
— One or two typos corrected, section names added, and a few explanatory sentences added inline: 2019.10.17 22:09 IST. Let’s leave this post right in this form.

 

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
Singer: Lata Mangeshkar, Mohammad Rafi
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.