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

1. Prologue:

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

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

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

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

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

OK. Let’s get going.

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

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

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

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

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

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

3. The mainstream view of the wavefunction:

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

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

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

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

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

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

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

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

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

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

Got the argument?

Alright. What about the term $V$?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

5. Wrapping up the problem statement:

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

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

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

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

So, our immediate problem statement becomes this:

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

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

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

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

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

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

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

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

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

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

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

A song I like:

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

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

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

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Ontologies in physics—3: EM fields in terms of forces; space; and related ontological issues

0. Before we begin:

I wish I could have undertaken some fresh re-readings on the history of science before setting out to write this series of posts. I have most of the material, and over the years, I have gone through a lot of books on history / conceptual aspects of physics and maths. (Off-hand: Kline, Kolmogorov (ed.), Dugas, Truesdell, Hesse, Coopersmith, Encycl. Brittanica, and many, many others, not to mention the Wiki). Further, I’ve just came across a new source: “Energy, Force and Matter,” by Harman. Wish I could’ve gone through at least this one. But I just don’t have the time. …

I would like to finish this series as fast as I can, so that I can then go over to my new theorization regarding QM. So, overall, what I write here turns out to be in a rather abstract way and without specific references to the original writings. I rely too much on my memory, and on the view of these matters I have come to develop. There’s a risk here that in writing without fresh referencing, I might be mixing my own views with the original ideations. This is not at all to my liking. But I have to make do with precisely that for now, purely out of a lack of time.

A great resource for this post (and the last) is the online notes: “A visual tour of classical electrodynamics,” written for an undergraduate course at MIT [^]. There are many great books on history (some being listed above). Apart from the books, here is a relevant paper: Ernan McMullin, (2002) “The origins of the field concept in physics,” Phys. perspect, vol. 4, pp. 13–39 [(PDF) ^]. Also see a brief note at Prof. Philip C. E. Stamp’s Web pages: “The concept of the field in physics.” [(PDF) ^]. Both these resources were mentioned in a tweet I made on 04 September 2019.

1. The background object and spaces in our EM ontology:

1.1 The background object of our EM ontology:

We mentioned the last time that in our EM ontology, there are only two types of objects: (i) EC objects, and (ii) the background object.

In our EM ontology, the background object is a physically existing but a non-NM kind of an object. Our view that the background object is not an NM kind of an object makes it sharply different from Faraday, Maxwell and Newton’s view.

[As an aside, even my c. 2005 paper on QM (covering my old, PhD-time approach) had mentioned the “aether” as a physically existing but non-material object. I guess this idea has been with me for a very long time.]

1.2 The physical space, and mathematical spaces:

When you point your finger to some place in between two EC objects, i.e., if you point out the “empty space” in between them, what you are actually pointing to is an invisible background object—not space—which is present at that place. We can justify this position, though its justification will progress slowly over this and the next two posts.

So, what exists in between two EC objects is not the “empty space.” Not even in the physical sense of the term “space”. (And there is something as the physical space.)

In our view, physically, the concept of space denotes the fact that physical objects (both the EC objects and the background object) have spatial attributes or characteristics like extension, location, and also other spatial attributes like the topological ones. The concept also includes the physically existing spatial relations between all objects. The physical space is the sum total of all the spatial attributes or characteristics of all the physically existing objects, with all the interrelations between them. Mathematically, the concept of space denotes a quantitative system of measuring the sizes (or magnitudes) of those spatial attributes with which objects actually exist. [For a very detailed, in fact very long-windingly written series of posts on the philosophical ideas behind the concept of space, see my earlier posts here [^].]

Notice the logical flow:

What ultimately exist are objects—that is the most fundamental fact. In fact, it is the primary fact assumed by all of physics. A primary fact is one which cannot be analyzed as implied by or arising from other facts. So, objects exist, full-stop. Every object exists with all the attributes that it has; each object has a certain identity.

Attributes exist (as part of identity of an object) only in some specific quantities or sizes. There can be no size-less attribute. As a simple example, a pen has the spatial attributes of length, diameter, shape, etc. Each pen exists with a specific quantity or measure of length and other attributes. Thus: sizes do have a physical existence; sizes do exist in the concrete physical reality out there. However, sizes don’t exist as apart from the objects whose sizes (in different respects) they are.

Mathematics then comes into picture. Mathematics is the science that develops the methods using which physical sizes of comparable objects (i.e. objects having the same attributes but to different measures) can be quantitatively related to each other. Mathematical concepts refer to mathematical objects, not physical, even though these concepts are reached only after observing the size-wise relations among physical objects. Mathematical objects are a result of objectifying the methods invented by us for measuring the existing sizes of the physical objects. The same set of physically existing objects (or their attributes, characteristics, properties, etc.) can give rise to an indefinite number of mathematical concepts.

Coming back to the concept of space: There can be many mathematical spaces, but they all refer to the same physical space. A mathematical space is an abstract quantitative system of measuring extensions and locations of physical objects. It is a mathematical concept which is developed in reference to the facts subsumed by the concept of the physical space.

The referents of the concept of the physical space are perceived directly, but not the physical space itself. The physical space is a concept, not a directly perceivable concrete.

What you directly see, the sense you get in your direct perception that there is some “solidity” or “volume-ness” to the physical world, is not the physical space itself, but the evidential basis for that concept.

1.3 The background object

The background object is a physical substance, but having a “non-material” nature, speaking vaguely—the exact statement here is that it is not an NM object.

The physical space is just one way to characterize the background object. It is possible to spatially characterize the background object because all its parts do possess spatial attributes (like extension, location, etc.).

However, therefore identifying the background object with the physical space (or worse, with a mathematical space) is a very basic error.

Just the fact that the background object possesses spatial attributes, does not make it the same as the physical space. …Would you call an NM-object (like a ball or a table) “space” just because it has spatial attributes?

Realize, the background object, though it is invisible (in fact it is as good as non-existent for the passage of NM-object through it), is physically an active existent at all times. Force-conditions are being sustained in it at all times, which it makes it active in the same sense that a foundation stone of a building is: no overall motion, but still a transmission of forces through it, at all times.

Neglecting gravity, the background object does not interact in any way with the NM-objects. It is for this reason that no inertia or mass can be ascribed to it. This is one of the easy reasons why it can’t be regarded as an NM-object. However, as we shall see later, the background object does possess something like a state of a “stress” within itself. (The nature of these stress-strains is different from what you have in case of the purely NM-objects.) It is worth noting here that even in the NM-ontology (i.e., in solid- and fluid-mechanics), the equations defining the stress/strain fields do not have mass appearing in them. So, the point here is that even if the background object were to have mass, it still wouldn’t matter because it doesn’t enter into dynamical equations involving it.

2. A few words on the difference between the 19th century view of space and ours:

2.1 What space meant to the 19th century physicists:

The 19th century physicists (like Maxwell) took space in the same sense as Newton did—in the NM-ontological sense of the term.

Space, to them, is a directly given (i.e. directly perceptible) absolute, having an independent physical existent apart from any NM-objects there may be. They would regard objects as filling or occupying some parts (some regions) of this already given absolute space.

In our opinion, this view is somewhat Platonic in nature. Faraday and Maxwell received this view of a physical, “absolute” space from Newton.

2.2 Our view: Space as a concept derived from the spatiality shown by physical objects:

We take the view that only physical objects are what is directly given in perceptions. (Perception here also includes observations of EC objects made in controlled EM experiments.) It is true that we directly perceive something space-like—the solidity or volume-ness of objects. It is also true that we also directly perceive the particulars of configurations of objects, including their directly evident locations. But that’s about all. We don’t perceive space as such; only the volume-ness or extended-ness of objects, their spatiality.

This point is somewhat similar to saying that what we directly perceive are blue objects (of different hues). We do directly see that blue objects show blueness. But the concept “blue” is only an abstraction from these objects; it is a product of a concept formation. The blue is not out there; only blueness is, and that too, only as a characteristic or attribute of those actual objects which do possess it.

People call the “empty” region “space” simply because they are unable to think of an invisible (or untouchable etc.) object that can exist in between two visible (or perceptible) NM objects. But realize that this “emptiness” also is a part of what we directly perceive. It is not as if our perceptual fields comes to have an in-principle hole (or complete absence) in the places where there is this absolute “nothing”ness of the “empty space.” Perceptions don’t have “holes” because there is something of what we call “space.” Our perceptual field cannot contain any evidence for the existence of a literal nothing in the world out there—a nothing that supposedly is at par with the things that do exist in the world out there.

So, what we directly perceive are only objects having spatiality, but never “space” itself. Space is something which we conceive of, based on these percepts.

2.3 Filling space with objects—what it means in our view:

It is only at this point in development—after we already have the concept of space—that we are able to trace the concrete-to-abstract relationship in reverse, and say that objects can be seen as coming to occupy some region of space that was initially empty (of similar, NM-type of, objects).

In our view, this statement (that objects fill space) involves an application of a concept, a backward motion in conceptual linkages. Objects can fill the physical space only because there are other NM-objects that help you isolate the “empty” region, which can then be filled with some other NM-objects.

The reverse-tracing of conceptual linkages becomes possible only if you have it in the first place, if you start from an abstract level. Perceptual level does not permit such reverse-tracing.

2.4 Space couldn’t possibly survive literal annihilation of all objects:

To Newton, Faraday and Maxwell, if all physical objects were to be annihilated, then the absolute space would still be left behind. To us, this is an impossibility. There cannot be spatial attributes without there being objects that have spatial attributes, and there cannot be the concept of space without our grasping a certain aspects like extended-ness or volume-ness that they have, and then abstracting a concept out of such physical features.

3. Causality and interactions:

3.1 Our view of causality

A word about causality. We follow the Objectivist view of causality, put forth by Ayn Rand [^]. Thus, at the most basic level, the idea of causality has nothing to do with an orderly progression in time. Properly speaking, the nature (identity) of the objects that act is the cause, and the nature of the actions they show or undergo are the effects.

It must be understood that the concept of causality is in principle applicable to single objects as well as to interactions between two or more objects.

3.2 Characterizing all the causes and effects operative in interactions:

Ontologically, in any interaction between any two objects of any kind, both the objects participating in the interaction must be seen, simultaneously, as being agents of causal actions.

When certain actions of a given object leads to certain other actions by some other object, we say that they are interacting. Suppose one billiard ball hits a second ball that was initially stationary. We often loosely say that the motion of the first ball is the cause and the motion of the second ball is the effect. Actually, the respective natures of the balls themselves are causes—both of them; and the natures of their individual actions are the effects—both theirs. At the instant when the two balls are in contact, both their natures taken together are such that they determine both their subsequent actions (here, motions). Thus, there are two causes and two effects.

If in order to determine an effect (an action by some object) you have to consider the nature of some other object(s) too, then we say that they all are interacting with each other. Thus, in collision of two balls, the ball that is initially moving (the one that hits the other) is not the only cause. Both the objects are causes (and they both hit each other). They both produce effects, even if you typically focus on only the second ball for characterizing the effect (of that interaction).

3.3 Causality as a concept far more basic than an orderly progression in time:

To repeat, it is wrong to characterize the idea of causality in reference to an orderly progression in time. When two NM-objects remain in static equilibrium for a long period of time, they still are obeying (and exhibiting) causality even if nothing about their dynamical states is ever undergoing any changes with the passage of time. Both are interacting at all times, both are causes (they exert forces like weight and reactions), and both their stationarity (“motionless-ness”) is an effect.

4. The electric field ($\vec{E}$) as a mathematical generalization from Coulomb’s law of force:

4.1 Coulomb’s law as operative at only two distinct points, and nowhere else:

It is the electric charge which gives rise to the Coulomb forces between two EC objects.

Consider an EC object of a charge $q_1$ fixed at a position $\vec{r}_1$. Consider a “test charge” of magnitude $q_T$ at some arbitrarily chosen but fixed position $\vec{r}_T$.

The idea behind calling the second charge a “test” charge is that it can be placed at any number of locations. However, the position of the “given” charge $q_1$ always stays fixed.

The force exerted by $q_1$ on the test charge $q_T$ is given by Coulomb’s law:
$\vec{F}_{1T} = \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_1\,q_T}{r^2}\;\hat{r}_{1T} \qquad\qquad r = |\vec{r}_T - \vec{r}_1|$.

The equation of the law tells us that the force $\vec{F}_{1T}$ is a function of both $q_T$ and $\vec{r}_T$. If we could somehow take out the effects of these two factors (both being related to the test charge), then the field will come to depend on the first charge $q_1$ alone. In short, we want to isolate the action of the first (given) charge from that of the test charge.

4.2 How to isolate the actions of the first charge from that of the test charge?: The electric vector ($\vec{E}$)

If the position of the test charge is different, we will obtain another force vector acting at that position. We can imagine having the test charge placed at different locations, successively, in different system configurations. Each $\vec{r}_{T} \leftrightarrow \vec{F}_{1T}$ pair is unique. If we collect together all these (infinity of) unique force vectors, they form a vector field—a function that gives a vector once you plug in some specific $(x,y,z)$ coordinates. Since the idea of the field includes all possible force vectors for all possible positions of the test charge, we don’t have to separately state a specific position—no matter what be the position, it’s already there somewhere in that infinite collection. So, by giving the entire field, we make it redundant to have to specify a single specific position for the test charge. We remove the necessity for specifying any specific position.

To factor out the effect of the magnitude $q_{T}$ of the test charge, we now propose a new vector quantity called the electric vector, denoted as $\vec{E}$. It is defined as:
$\vec{E}_{1} = \dfrac{\vec{F}_{1T}}{q_T}$
What we have effectively done here, by dividing the force at $\vec{r}_T$ by $q_T$, is to suppose that $q_T$ always remains a unit charge. Since neither position nor charge-magnitude of the test-charge have to be specified, we have dropped the $T$ subscript.

Whereas the electrostatic force field depends on the magnitude of the second charge, the electric vector field does not. Thus, we have found an object that captures the effect of the first charge alone. Hence the subscript $_{1}$ to $E$ still remains.

Actually, the second charge ($T$) is physically still there, but its role has been pushed back into an implicit background, using our procedure—via generalization to all space, and normalization to the unit charge, respectively.

4.3 The electric vector’s field as a mathematical device of calculations:

A field is just a simple function of the spatial coordinates. You plug in any specific position into the field-function, and you get the electric field vector ($\vec{E}$) that would be produced at that point—if a unit test charge were to be actually present there. Once you know the electric field vector which would be present at a point, then you can always find the Coulomb force which would be exerted by the first charge (which generates the field) on any arbitrary second charge, if it were to be actually situated at that point: you just multiply the electric field vector at that point by the magnitude of the second charge.

This is the meaning which people (the continental mathematical physicists) had in mind when they first put forth the mathematical idea of such functions. These functions later came to be regarded as fields.

5. Some comments on the MIT notes:

Since I gave a reference to the notes at MIT, a comment is in order. The MIT note defines the electric field via a limit: $\vec{E} = \lim\limits_{q_0 \rightarrow 0} \dfrac{\vec{F}}{q_0}$. Taking this limit is not at all necessary. In fact I fail to see even its relevance. If at all a limit has to be conducted, then it could be for a vanishing size (diameter etc.) of the test charge, so that the point of definition of $\vec{E}$ becomes unambiguous. But taking the charge to vanishingly small charge simply does not seem to bring in anything of relevance. … For others’ comments, see, for instance, here: [^]. The answer given at a JEE-preparation site also is somewhat misleading [^].

The correct way to think about it is to think of a static situation (at least in the vicinity of the test charge). A static situation can be had either (i) by considering just one instant of time in the motion of a movable configuration of charges (EC Objects), or (ii) by introducing some imaginary support forces which keep all the charges fixed at their respective positions at all times.

We differ from the MIT notes in one more respect. They first define the electric field as a limit of the force at a point on a test charge, but without any reference to the other source of that force. Its only after thus defining the electric field that they come to relating it with the Coulomb force exerted by the first charge. Thus, their definition is, strictly speaking, half-arbitrary: it misses one of two crucial objects that are present in the empirical observations.

We regard the idea of an arbitrary field as existing at a higher level abstraction, but insist on noting that no matter how arbitrary an electric field (its pattern or distribution) might get, it still cannot come into existence without some or the EC object(s) producing it. That’s our viewpoint. We emphasize the role of the field-producing charge.

Indeed, when it comes to QM ontology, we do away completely with the idea of arbitrary fields and even arbitrary continuum charge distributions (which they demand). We *restrict* the generation of all permissible electric fields only to point charges because elementary charges are point-particles.

6. An ontological breakthrough: The entire electric vector-field seen as existing physically:

6.1 A possible reason why the continental physicists didn’t  go for a physically existing field:

Coulomb’s law states a relation for what happens at two specific points in space. The law is completely silent on what happens at any other points of space. In contrast, the electric field is mathematically defined for all points of space.

As mentioned in the last post, the continental physicists did work with the mathematical notion of fields for a long time (I off-hand suppose, for at least 70 years) but without thereby necessarily implying its physical existence.

It should not be too difficult to understand their perspective. A field would be just a mathematical device for them; using this mathematical object meshed well with their energetics program, that’s all.

Their development program primarily relied on the underlying idea of energies, and so they would certainly require the potential energy, not forces. Actually, the concept of energy was yet to be isolated as such—they simply used the mathematical functions that we today regard as defining the quantities of energy.

Strictly speaking, having a field of potential energy was not necessary for formulation of laws of physics in their program. Their laws could have been formulated with just a single number for the potential energy of the entire system. (The number would vary with positions of discrete bodies, but it would nevertheless always be a single number.) That is, as far as their laws were concerned.

However, the field idea would be convenient when it came to applying their formulation. Problems having continuous boundary conditions naturally got simplified with the working idea of a function of all possible spatial coordinates. Thus, a field came be supposed for the potential energy. They would see forces as gradients of potential function; forces were secondary or derived quantity for them. Indeed, the problems they worked on, during the development of the potential field concept, came exclusively from gravity.

Thus, all in all, the field was primarily a mathematical idea for them—just a device of calculations, and that too, only for gravity, even if electromagnetic laws also were being discovered during the same period.

6.2 Faraday makes the ontological breakthrough:

It was Faraday who vigorously advocated the idea that the force-field is not just a mathematical idea but also physically exists in the real world out there. He characterized it in terms lines of force. He believed that the space was not empty but filled with a fluid (a mechanical or NM-object like air, water, oil, etc.). The lines of force were imagined by him to be tubes formed by fluid flow. Maxwell then mathematically refined the idea.

It may be perhaps be noted here that the pattern of the magnetic field which is observed when you sprinkle some iron filings on a magnet, does not actually form enough of an evidence to prove the existence of fields. It merely suggests and supports the idea of a field.

But strictly speaking, you can always argue that a field does not therefore exist; only point-wise forces (action at a distance) do. In the context of iron-filings and a magnet, you can argue that magnetic forces are present only at the points where the iron-filings are—not in the empty spaces in between them. The picture of the field pattern produced by the iron-filings, by itself, is thus not sufficient.

The fact is that the field idea can only be introduced as based on a more general thought; it can be introduced only as a postulate, to ensure consistency in theory. We will touch on this issue later. For the time being, we will simply assume that the continuum field does indeed physically exist.

7. Our EM ontology: The electric vector field as an attribute of the background object:

Let’s now note a subtle ontological point.

7.1 An EC object is the cause of the $\vec{E}$ field:

As the position of a given charge (the “first charge” or $q_1$) itself changes, the entire force-field shifts in space too. This is a direct consequence of Coulomb’s law.

The preceding sentence says “changes” and “shifts”. However, note that we don’t thereby mean an actual motion here. We are merely describing the differences which are present between two fields when they are actually produced in two different system configurations. Within each system description, everything still remains static. Taking a difference between two system descriptions does not always have to involve a continuous motion connecting them. That is what we mean here. Thus, the “shifts” here are of the variational calculus kind.

The fact that the field generated by an EC object shifts with its position further establishes the idea that the electric force field is a causal effect of that object.

7.2 The $\vec{E}$ field is not an attribute of an EC object:

However, an important point to note here is that this fact still does not make the field an attribute of that particular EC object.

Mathematically, the field due to a point-charge is a function that is defined at all points other than its own position. Physically, therefore, the field must exist only at those spatial locations where the field-generating EC object itself is not present.

In the EM ontology, EC objects and the background object are the only two categories of object. If the field is not even present at the location of a point-charge, it cannot be an attribute of that charge. Therefore, it must be an attribute of the background object. Matter cannot act where it is not.

7.3 The existence of $\vec{E}$ implies an interaction between two physically existing objects:

If an object is causally responsible for producing an effect, but if that effect is an attribute of some other object, clearly we have two different objects interacting here.

There is a difference in the hierarchical levels of EC objects and electric fields. EC objects are primary existents. In contrast, electric charge and mass exist only as attributes of EC objects. Similarly, the background object is a primary existent. But the electric fields are secondary—they exist only as attributes of the background object.

In calculations, we often get so much used to associating the electric field generated by a point-charge with that charge itself, that we tend to forget that the generated field is not an attribute of that charge itself. Though produced by a charge, the associated field is actually an attribute of the *other* object—the background object.

The electrostatic (and in general, all electromagnetic) phenomena can be explained in terms of the direct contact forces which arise between an EC object and the background object, or between two control volumes (of arbitrarily small size) within the background object.

8. An electric vector field as an effect produced by an EC object acting in isolation of others:

Coulomb’s law has exactly two electrostatic forces. The force exerted by any one charge acts at only one point: at the distant location of the other charge. (The support forces are taken as mechanical in nature here, not electrostatic.) Coulomb’s law thus speaks in terms of a pair of forces. It is physically impossible to have a situation in which only one of the two charges is active (exerts a force on the other).

In abstracting the idea of the electric vector field, we had to factor out the effects due to the test charge. As a result, attributes of test charge do not determine the distribution pattern of the electric field. Interpreted physically, a single charge can be taken to generate the field associated with it; a second charge (say a test-charge) is not at all necessary.

In other words, a complete electric vector field (spread over the entire infinite space) can come into being with the existence of just one charge. In contrast, Coulomb’s law requires and simultaneously relates two different charges. Ontologically, this is a significant difference.

The electric field due to several charges is simply an algebraic sum of the fields produced by each charge acting separately.

Imagine an infinitely large universe that has nothing but just a single charge. (You can’t determine its location.)

If the electric force field exists physically, and if such a field can be produced by each single charge acting singly, then actual forces will come to exist everywhere even in this universe. The field will be spherically symmetric. Faraday’s lines of forces will be straight lines that emanate from the point-charge—they will look like symmetrically distributed “spokes.”

When a second charge is added at a finite distance from the first, then the field condition actually existing in the universe is obtained simply by a linear superposition of the two fields. The effective lines of force will look distorted from their initial symmetrical shape.

A distortion of Faraday’s lines of force is implied by (and implies) the existence of the second charge. However, their very existence does not depend on the existence of the second charge.

9. A preview of the things to come:

The description in this post was mostly in terms of forces and quantities derived from them. In the next post, we will look into an alternative description, one that is couched in terms of energies and quantities derived from them. While both the approaches are physics-wise equivalent, the energy-based approach helps simplify calculations. In fact, most physicists get so thoroughly used to the energy-based approach in their PG years, that they even come to forget that ever was a force-based approach (which is quite unlike engineers—we engineers never come to forget forces, including reaction forces at supports).

We have to look at some of the basics of the energy-based approach to physics, simply because the Schrodinger equation is inductively derived using only the energies. However, since we are covering this material purely from the viewpoint of the Schrodinger picture of QM, we will try to keep the variational calculus ideas as much to the background as possible.

Thus, in the next post, we will go over the following sequence of ideas: the potential energy number of an electrostatic system; the potential energy field in the spatial region of a system; the potential energy field of a single elementary point-charge; the potential of an elementary point charge. We will also try to look into the issue of how the background object comes to support forces within itself. Finally, there also will be an issue of justifying the inclusion of the background object in an ontology—its necessity. We will try to cover it right in the next post. If not, we will do that in a subsequent post, and then we will be done with our EM ontology.

See you the next time, bye for now, and take care….

An important update on 2019.09.17 12:26 IST:

I think that the sections 7. and 8. above have come out as being somewhat misleading.

The present write-up suggests that what physically exists is the $\vec{E}$ field. However, if the magnitude of the second charge $q_2$ or $q_T$ is not $1$, then it is not clear precisely what physical process occurs at $\vec{r}_2$ such that it results in a multiplication of $\vec{E}$ by $q_2$ to produce the force $\vec{F}_{12}$ at that point. This force, after all, must physically exist. But it is not clear what physics is there for the multiplication by $q_2$.

So the proper conclusion to draw seems to be that the actually existing field is always that of the force $\vec{F}_{1T}$ and not of $\vec{E}_{1}$; that the latter is only a mathematical device, a convenience in formulating Maxwell’s system of equations. But what will be the implications for an isolated system having only a single charge? for bringing a second charge into it from infinity? … Need to think through.

… I will think about it for a while and then, if necessary, I will come back and update the above description appropriately.  … In the meanwhile, any thoughts / suggestions?

A song I like:

(Marathi) “hirve, pivale, ture unhaache, khovilesa kesaata ugaa kaa??”
Lyrics: Indira Sant
Music and Singer: Vasant Aajgaonkar

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Ontologies in physics—2: Electromagnetic fields as understood by Faraday and Maxwell

In this post, we shall begin going through the ontology assumed in—or rather, demanded by—the physical phenomena which are covered by the classical (i.e. Maxwellian) electrodynamics. We call this ontology EM for short.

We will mostly be talking in reference to electro-statics. However, note, this ontology just as well applies also to electro-dynamics.

1. A list of the ontological objects used in the physics theory of electromagnetism (EM):

The EM objects basically are of only two types: (1) massive and electrically charged point-particles (of one of two polarities: positive or negative), and (ii) a background object (sometimes loosely identified with the fields induced in it, by calling it “field”; other times identified with an aether of a certain kind by Maxwell; more, below).

Notably, there are no separate magnetically active objects in this ontology even though magnetism has been known as a force for more than at least one millenium. Magnetism is an effect produced by the electrical charges.

2. Electrically Charged (EC) objects:

The ontology of NM (seen in the last part in this series) is basically that of the uncharged bodies. What the EM ontology now does is to further endow these same objects with an additional attribute of the electric charge. This extra attribute considerably modifies the entire dynamical behaviour of these objects, hereafter called the EC objects for short.

For the elementary charged objects (basically, here, only the electrons and protons), the phenomenon of the induced charge/polarity does not come into picture—the charge of each elementary EC object always remains with it and its quantity too remains completely unaffected by anything or any action in the universe.

The electrical charge is just as inseparable an attribute of an EC object as its mass is. As an EC object moves in space, so does its charge too. An electrical charge cannot exist at any spatial location other than that of the massive EC object which possesses it.

Just as in NM, the EC objects too can be abstractly seen as if all their mass, and now charge too, were to be concentrated at a single point. We call such EC objects the point-charges.

3. Electrostatic forces between point-charges—Coulomb’s law:

Coulomb’s law is an empirically derived quantitative relationship. There is no theoretical basis beyond the fact that such behaviour was actually observed to occur in carefully conducted experiments. The statement of the law, however, is mathematically sufficiently refined that it would be easy to suspect whether it was not derived from some other a priori basis. As a matter of fact, it was not. What does Coulomb’s law describe?

Consider an isolated system of two point-charges fixed in space at some finite distance apart. [Help yourself by drawing a free-body diagram, complete with the structural support symbols for each of the charged bodies too. No, these supports, though they look like the electrical ground, are actually mechanics symbols; they don’t discharge the charges by grounding.] It is experimentally found that both the charged bodies experience forces of certain magnitudes and directions as given by Coulomb’s law.

Let the electrical charges of the two point-charges (EC massive point-particles) be $q_1$ and $q_2$, and let their positions be $\vec{r}_1$ and $\vec{r}_2$.

Let the separation vector going from the first point-charge to the second be given by $\vec{r}_{12} = \vec{r}_2 - \vec{r}_1$. (There is no typo in the last equation.) Similarly, let the separation vector going from the second point-charge to the first be given as $\vec{r}_{21} = \vec{r}_1 - \vec{r}_2$. (Again, no typo.)

[Note, a separation vector measures the difference in the two vector positions of two different bodies at the same time, whereas a displacement vector measures the difference in the two vector positions of the same body at two different times. … There is another related idea: The variation in position is a vector that measures the difference in the two vector positions of the same body, without reference to motion (and hence time), but as imagined in two possible and different configurations within two description of the same system. We will not need it here.]

Let $\vec{F}_{12}$ be the force that $q_1$ exerts on $q_2$ at $\vec{r}_2$ (and nowhere else). Similarly, let $\vec{F}_{21}$ be the force that $q_2$ exerts on $q_1$ at $\vec{r}_1$ (and nowhere else).

Coulomb’s law now states that:

$\vec{F}_{12} = \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_1\,q_2}{r^2}\;\hat{r} = - \vec{F}_{21}$,
where
$r = |\vec{r}_{12}| = |\vec{r}_2 - \vec{r}_1| = |\sqrt{\vec{r}_2 \cdot \vec{r}_1}| = |\vec{r}_1 - \vec{r}_2| = |\vec{r}_{21}|$, and $\hat{r} = \vec{r}_{12}/r = - \vec{r}_{21}/r$.

A few points are especially noteworthy:

1. The two electrical forces so produced are always equal in magnitude, regardless of how big or small $q_1$ and $q_2$ may be with respect to each other. Even if $q_1 \ll q_2$, the first object still ends up exerting just as much of an electrical force on the second object as the electrically much stronger $q_2$ exerts on the first.
2. The forces appear on the two EC objects even though they are not in a direct physical contact—exactly as in Newtonian gravity. They also obey an inverse-square law, once again exactly as in Newtonian gravity.
3. The forces don’t at all depend on the respective masses of the bodies; they are only for the electrostatic interaction.In reality, the two point-charges in the fixed configuration would also experience the force of gravity, not to mention the imaginary forces exerted on them by the mechanical supports. However, we ignore gravity in this entire description, and we exclude the support forces from our system—they are regarded as at all times belonging to the environment. (If a charge moves, so does its support, and along with it, the system–environment boundary. In fact a motion of a boundary is necessary for any system to exchange energy with its environment in the form of the mechanical work. We will need to take into account the mechanical work when it comes to defining the electrical potential energy of the system.)

4. Why the big ontological issue of Action-at-a-Distance remained unresolved before EM came on the scene:

The issue slipped under the carpet the last time (while discussing NM) now once again raises its (ugly or beautiful) head: How come two charges are able to exert forces on each other even if they are separated by nothing but the empty space, i.e., when there is no direct contact of the NM-kind between them?

People had thought a lot on this question right since the time that the issue came up in the context of gravity. In fact even Newton himself had once speculated whether an invisible string might not extend between two gravitating bodies like the earth and the Moon. However, Newton was only too well aware of the limitations of the available experimental evidence. The observational data such as that by Kepler indicated very high speeds for transmission of gravitational forces; the data were not refined enough to capture any effects of a finite but high speed for the forces. So, Newton refrained from adopting any definitive position concerning either a mechanism for the transmission. (Unlike Faraday, the poor fellow could not sprinke asteroids between the earth and the moon, and thereby grow confident enough about ideas like tubes of force or space-filling fields.)

The people from the other side—from the energetics approach to formulating mechanics—were not as deeply interested in the ontological matters anyway. In the market-place of ideas, their main market-differentiator was not a superior or more refined physics but a superior method of calculating solutions when the boundary conditions became complex, e.g., too numerous, as with continuous and curved constraints. (The physics implied by the mathematics of both the approaches—Newton’s momentum-based approach and the energy-based approach—was exactly identical. What differed between the two was mathematics—the methods of calculations.)

So, there arose a feedback circle of sorts: people who didn’t care about ontology and foundations of physics, but wanted to do maths, got attracted to the energetics program; people who already were in the program directed their energy in pursuing their strong point further. So, what they kept on developing was maths. Given this feedback circle, any ontological problem concerning the action-at-a-distance couldn’t have benefitted from them anyway. In actuality, it didn’t.

That’s why even if a term for the gravitational potential energy $V$ had appeared as early as ~1773 in Lagrange’s writing (which was devoid of not just ontology but also of even a single geometrical diagram), and even if luminaries like Laplace (~1799), Gauss (about the same time but published in 1840) and Green (1828) happily developed the potential theory for gravitation, they all were perfectly happy working with just “a mathematical function” of “coordinates” for $V$—not with an actually existing physical field. It was Green who described $V$ as the “potential function”.

Of course, these physicists couldn’t have used the term “potential energy” for $V$. The discovery of the first law of thermodynamics and the conceptual clarity on what precisely the concept of energy itself meant, was still only in latency; the explicit identification was at least 20–30 years away.

In any case, the mathematically oriented physicists on the continent didn’t pursue the issue of the physical meaning of potential energy a lot. The quizzical end-result was this: There wasn’t just an equation for the gravitational (and later electrostatic) potential energy of a system, there also was this distinctly further development of an equation for the gravitational potential of a single gravitating body. Thus, the Continental physicists had succeeded in mathematically isolating the interacting system of two charges into components specific to single charges: the potential field due to each, taken in isolation. The potential energy of the system could be found by taking the potential of either charge and multiplying it with the other charge. They were really advanced in maths. They successfully manipulated equations and predicted results. But they didn’t know (or much care) about what ontology their concepts or procedures suggested.

That’s why the ontological issue concerning the action-at-a-distance remained unresolved.

5. Faraday’s lines of force and Maxwell’s fields:

Historically, the crucial step in developing the idea of the field was taken by Faraday. I will not go into the details simply because they are so well known: sprinkling of iron filings near magnets, tubes of force (called “lines” of force by Faraday himself), the broad laws governing their behaviour (attractive force goes with tension, repulsive with sideways pressure), etc.

What is important is to note that Faraday did explicitly advocate the idea that the lines of force physically existed; they weren’t just a device of calculations the way the gravitational potential was to the Continentals. At the same time, he also believed that even the apparently empty space in between two lines of force was also filled with more such lines, that they filled the entire 3D space.

Maxwell got impressed by Faraday’s idea. He even met Faraday, and then, set out to translate the idea into more precise mathematical terms. He arrived at his mathematical description of the electromagnetic phenomena by imagining not just the tubes of force but even many further mechanical mechanisms, invented by him, in order to give a mechanical explanation of the known EM laws (Coulomb, Biot-Savart, Ampere, Faraday, and others.) Eventually, the system of explanations of EM phenomena using mechanical means (essentially, the NM-objects) became too complex. So, in his final synthesis, he simply dropped these complex details, but retained only the differential equations.

However, importantly, he continued to keep the abstraction of the fields in his final synthesis too.

The idea of the field solved the problem of action at a distance. Maxwell put forth the idea that all of space (except for the regions where charged objects are present) is filled with a physically existing field. Charged bodies are in continuous contact with the field, and therefore, are able to induce a condition of force in space—which is the field of force. Maxwell imagined that the force field consists of non-uniform mechanical stresses and strains. (He used the term “displacements” for these mechanical strains; hence the term “displacement current,” which is still in use.) When the same field comes in contact with some other charge, it experiences a net force due to the presence of these mechanical stresses at its boundary. That’s how the forces get transmitted.

Maxwell was smart. What he started out (or invented as he went along) were rather complicated physical mechanisms (all of which were made from NM-objects). But he had a definite sense of which point-quantities to abstract away, using what kind of limiting arguments, and how. Thus showing a refined and mathematically informed judgment, he arrived at an essentialized description of all the electromagnetic phenomena in terms of point-properties of a continuum.

6. The ontological view implied by Maxwell’s ideas:

In Maxwell’s view, the entire universe could be analyzed in purely mechanical terms. This means that all the objects he employed in his synthesis were essentially only NM-objects. These came in two types: (i) point-charges, and (ii) a mechanical continuum for the field.

The two interacted (exchanged forces) using the only mode that NM-objects were allowed: using direct physical contact. The interaction proceeded both ways: from charges to the continuum and from continuum to charges.

The point-charges pressed forces on the continuum of the field at their common boundary, which resulted in there being a mechanical field of stresses and strains inside it. These stress-strain states extended everywhere in the continuum, “up to” infinity.

The continuum, in term, generated forces on the surfaces of the boundary between itself and any charged object embedded in it.

The specific stress-strain field generated by a charge was spherically symmetric around that charge. Hence, it didn’t result into any net force acting on the same charge. However, due to the inverse-square nature of Coulomb’s law, and the fact that charges were point-particles, the field they generated was necessarily non-uniform at all other points. Hence, the continuum did generate a net force on the other charges.

Ontologically, there was little difference between Faraday meant (or strived to indicate), and what Maxwell directly put forth, using mathematical concepts.

7. The basic weakness of what precisely Faraday, and also Maxwell, meant by a field—its ontology:

Maxwell’s proposal of fields had a very great virtue, and a very great conceptual (ontological) weakness.

The virtue was an advocacy of a physically existing condition in what earlier was regarded as completely empty space. This condition was identified with the mathematically defined fields. The idea of fields was not just satisfactory from the viewpoint of broad philosophic principles (we will touch on them in a short while), it also gave a reasonable-sounding solution to the physics problem of action-at-a-distance.

The weakness was that the field, even if defined very carefully (in direct reference to empirically observed electromagnetic laws, and with rigorous mathematical abstraction), still was characterized, explained, and defended as a specifically mechanical kind of a physical existent. To describe the ontology of such fields using our scheme and notation, these EM fields were “pure” NM-objects.[Professional physicists often don’t have very good ontological clarity, but they are referring to the same underlying physical fact when they say that a mechanical aether provides a reference frame that obeys the Galilean-invariance but not the Lorentz-invariance.]

This particular weakness immediately led to conceptual challenges for Maxwell. Eventually, it also led to a lasting confusion for all, a confusion that persists till date (at least in the discussion of EM and aether).

The difficulties posed by the weakness were actually insurmountable. Here is one example. If the field had to be mechanical in nature as Maxwell said, then it would have to possess seemingly impossible combinations of physical properties. It would have to be an infinitely rigid object, and yet allow other massive objects (of the NM-object kind) to pass through them without hindrance. Et cetera. For an interesting history of how creative solutions were sought, and even were supplied see [TBD]. (A candidate explanation: The field acts like a metal ball placed on a block of snow: The ball passes through the block’s thickness even while keeping the block solid everywhere else, but the ball gets reflected when thrown with a sufficiently high speed. Another example I can think of: a jet-plane (say in a tail-pin) that hits the ocean surface. When the speed is great, it first rebounds as if it had hit a solid surface, rather than sinking in the water as it eventually does when it loses speed.) However, such explanations did not bear out—no mechanism would if it produces a mechanical aether in the end.

If Maxwell’s theory were to be less successful, this weakness would be less consequential. However, this was not the case. Maxwell’s was one the most fundamental and most successful theories of his time. (It was what QM and Relativity are to us.) Since the weakness rode on a very strong theory, it irretrivably put people in the mindset that all fields, including the luminiferous aether (which transmits Hertzian waves through it), had to be necessarily mechanical in nature.

8. The consequences of the conceptual weakness of Maxwell’s idea of the field—its ontology:

The weakness had to come out sooner or later, and in the 19th century, it came out rather sooner.

In particular, if the aether was to be space-filling, then the earth would have to move through it during its elliptical motion around the Sun. If this aether was mechanical in nature, then the interaction of the earth with the aether would have to be mechanical in nature. It would be analogous to the motion of a finger through the tub-water. The aether would exert a drag force on the earth’s motion, which could be detected via light interference experiments.

However, experiments such as those by Michaelson-Morely showed that such a drag was not actually present. This momentous experimental finding led to the following situation.

The physicists and mathematicians of the energetics program (mostly on the continental Europe) had labored for almost 70 years to develop the mathematics of fields, but without regarding it as a physical entity. But they had produced excellent mathematics which greatly clarified presentation of physics and simplified calculations. Even Maxwell’s theory had its mathematics developed on the basis of these developments.

The aether-induced drag was only a deductive inference made from Maxwell’s theory. Maxwell’s theory itself was founded on very well established experimental findings. Another deductive inference, namely that light was an EM wave with the speed of $c$, had been borne out by experiment too. So, all these aspects had to be kept intact.

But the drag implied by Maxwell’s aether was not to be found in the experiments. So, this part of Maxwell’s theory had to be corrected for. The simplest way to do that was to drop the whole idea of the aether from the theory!

After all, in his own development, Maxwell himself had started out with a laundry list of different kinds of physical mechanisms for different aspects of electromagnetics. However, eventually, he himself had come to drop all these mechanical features, because it was hard to get all these mechanisms to work together in a simple manner. So, he had instead decided to abstract out just his mechanical field from them. This field was then identified with the luminiferous aether.

But the aether-wind gave problem. So, why not take just another step of abstraction, and entirely do away with the very idea of the aether itself? Why not regard it as just a mathematical entity? Why can’t space once again be completely empty of any physical being, just the way it had been right since Newton’s times—and even during the entire development of the potential theory and all?

If the situation is to be framed as above, then there can be only one logical way out of it. The physicists came to choose precisely that. Without challenging the specifically mechanical nature of the aether (because no one could think of any other kind of a nature for a physical aether, since none could figure out any good philosophical arguments for having a non-mechanical aether), physicists in the late 19th- and early 20-th century simply decided to remove this whole idea from physics.

In the meanwhile, Einstein was advocating a denial of the absolute space and absolute time anyway. If the space itself was not absolute but depended on the relative motion of the observer, i.e., if space itself depended on motion, then was there any point in filling it with anything?

For the rest, pick up virtually any of the hundreds of the pop-sci books on the relativity theory and/or on Einstein’s (IMO at least in part undue) glorification. … If Maxwell to be less successful as a physicist, his wrong ontological views would be much less consequential. Ditto, for Einstein.

9. A preview of the things to be covered (concerning the EM ontology):

The essential error, to repeat, was to conclude that since Maxwell was successful, and since his field was mechanical, therefore every field has to be taken as being mechanical—i.e., as if it had to follow the NM ontology. Wrong.

Next time, we will cover the correct ontological view to be taken of the Maxwellian fields. We will also look into a few issues about ideas like the electrostatic potential, the electrostatic potential energy, and their fields. We look into the details of these concepts only because they are relevant from an ontological point of view. [And yes, there is a difference between just potential and potential energy—just in case you had forgotten it.] This discussion will also help us prepare for the correct ontological view which is to be adopted when it comes to the quantum mechanical phenomena. Hopefully, the whole portion would be over in two more posts, at most three.

See you next time. Bye for now, and take care…

A song I like:

(Hindi) “o sanam, tere ho gaye hum…”
Singers: Lata Mangeshkar and Mohmmed Rafi
Music: Shankar-Jaikishen
Lyrics: Shailendra

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Ontologies in physics—1: Newtonian mechanics

0. Before we begin:

The mechanics described in the last post, namely that of the molecular dynamics (MD) technique, had three salient features: (i) a potential energy which is anchored into the pair-wise separations of neighbouring discrete atomic nuclei (loosely called “atoms”), with its negative gradient forming a force field, (ii) the local force-field accelerating the atoms thereby causing a modification in the latter’s motions (velocities), and (iii) the resulting modifications in the atomic positions leading to a change in the potential energy, thereby forming a feedback loop. Hence, an essentially nonlinear dynamics.

We also saw the ramifications of such a chaotic dynamics, for instance, the obvious stability of phases over wide ranges of the important parameter, viz. temperature (i.e. average kinetic energy i.e. velocities). We also noted that MD is very close to QM, and that in my approach, the equations of QM and MD show a remarkable similarity.

However, the ontologies of QM and MD differ in that QM is not a classical theory. Further, ontology of even purely classical concepts like potentials, used even at the MD level, are not always clearly spelt out in the literature.

Therefore, before we are able to go to my tweets on my new approach to QM, it is now further necessary to clearly understand certain basic facts of life physics—pertaining to various ontologies followed in it over a period of time. We will do that beginning with this post.

1. An ontology as the proper starting point of physics:

The starting point of a physics theory is not a mathematical equation, not even the kind of configurations there are to a system. The proper starting point is: the kind of objects that are presumed to exist in the real world before the exercise of building a theoretical system involving them can even begin. Thus, the proper starting point of any and every physical theory is an implicit or explicit ontology.

Depending on the ontology followed, we may classify the physics theories (up to nonrelativistic QM) into these types:

• Newton’s original mechanics (here called the Newtonian Mechanics or NM),
• Classical Electrodynamics (EM), including:
• the ontological analogy it suggested for the Newtonian gravitational field (NG)
• The non-relativistic quantum mechanics, as in Schrodinger’s formalism (QM).

I have blogged about these ontologies before. Go through a previous blog post [^] if you wish, but also note that my overall understanding of physics has undergone substantial revision since then. Indeed, if necessary, I might further split the ontologies as I go writing about the above three/four.

The reason we must undertake this exercise of identifying a fairly precise description of these ontologies right now is that in the Outline document (on my new approach to QM [^]), in the section on ontology, I speak of some of the QM objects as being “classical.” However, there are certain important nuances to the meaning to even word “classical,” especially when it comes to the NM vs EM distinction. Hence the necessity to state the exact ontological views.

I would have loved to follow the historical order of the development in the ontological views followed in physics. However, I don’t have time for that right now. So, the development will be only very broadly in the historical order.

2. The ontology followed in the original Newtonian mechanics (NM):

2.1 Objects:

The world consists of spatially discrete objects that are spatially separated from each other. They are of finite sizes—neither zero nor infinite. (Ignore all mathematicians and even mathematical physicists who argue otherwise.) Take a piece of paper and draw some blobs for some objects, say for the earth, the sun or the moon. Or, for some neat solid objects like billiard balls. These blobs represent the primary objects of NM.

The objects are perceptually observed to be spatially extended (their opposite ends don’t coincide), and it is perceptually evident that any one object lies in a specific spatial relationship with the other objects, that it has its own location.

2.2 Absolute space:

The objects of NM exist in an absolute space.

Take an imaginary ruler and an imaginary sharp object. Mark some imaginary, straight-line scratches on the empty space, so as to leave an infinite grid of locations on it.

Yes, this is doable. Just make sure to undertake this exercise while being firmly seated in your armchair on the earth, without ever moving. (Don’t worry about some other grid that some other guy sitting in some other arm-chair makes. In the dynamical equations, they don’t conflict with each other.) You just have to realize that in NM, the world is very stable and simple.

The walls of your room, e.g., don’t move or deform. They form a rigid body, and the surfaces of any such a rigid body can be marked with a neat system of lines, like your school-time graph paper. You can also imagine strings being tied tautly, to form straight lines between opposite walls of the room. A system of such strings, when taken to infinitely small size and imagine to offer no resistance to motions of any objects (seen above), easily provides a means to measure locations within the room. A similar kind of straight lines extended in all directions and infinitely, yields a system of measurement.

But we need to make a distinction between a system of measurements and the thing that is being measured. (We are into ontology.) Here we suppose that the volume inside an empty room is not completely empty. It is filled with a background object. It is a physical object but of a special kind—it offers no resistance to any motion of anything through it.

The grid marked by you never moves because the background object that is the empty space also does not move. They both remain fixed in all respects at all times and forever.

However, objects of the first kind (solid ones like moon, Sun, etc.) are often seen as moving through the aforementioned, unmovable, undeformable background object—called the absolute space—in a lawful manner.

The concepts of position and distance are abstracted from those of locations and extensions of objects.

The concept of space has two meanings: (i) as the physically existing background object, and (ii) as a mathematically devised system of establishing quantitative measures like positions, distances, and relationships between them.

2.3 Configurations and changes in them:

Objects taken together with their (absolute) positions are said to form a configuration.

It is physically observed that configurations of objects are continuously changing from one state to another. There are an infinite number of states in between any two states, and they come to occur in some specific (observed) order. The order being followed in going through all such states (and all the attributes of the stated orderliness) is lawful—it cannot be changed arbitrarily. The individual states are described in reference to the positions of objects against the absolute space. The orderly progression in them occurs because the configuration of the universe is always changing (whether the one you see around your armchair does or not).

2.4 Absolute time:

The immutability of the order in the universal progression of changes in configurations implies a certain measure called time.

With time, you compare and contrast the perceived speed with which a progression in the states of a system undergoes changes: the faster the perceived changes, the smaller the changes in the elapsed time.

Perceiving differences in the speeds of changes of configurations is easiest when the phenomena are of perceptually reproducible speeds and hence durations, which most saliently (though not exclusively) is the case when they are periodic. For instance, pendulum comes back to a certain position (in a single cycle of oscillations) much faster; the sand in a sand-clock gets exhausts much slower; the Sun rises again at a pace that is even slower.

The perception of the speeds in the changes of physical configurations is at the basis of the concept of time.

Time is a high-level concept. It not at all the most fundamental one. (Both Kant and Einstein were summarily wrong here.) It certainly is not as fundamental as the concept of space is. Let me repeat the logic:

Objects come first. Then come the perceived extensions and locations of objects. Then comes the concept of space as a physical object. Then the concept of mathematically defined absolute space, and then of configurations. Then the orderly and continuous changes in configurations. Then we arrive at the idea of a defining a certain kind of a measure for such changes by comparing two continuous changes with each other on the basis of their perceived rapidity. It’s only at this point in the logical development that we can even think of time, or refine this concept by ascribing to it a mathematical quantity that continuously increases. Space and time are not on the same footing—neither in physical terms nor in the complexity of reasoning underlying their mathematical definitions.

This attribute of the perceived speediness of changes (i.e. the attribute of time) is common to all the changes occurring to all the objects in the universe—not just to their motions. Hence, any change whatsoever can be measured using time.

Thus, the physical universe itself has this attribute called time. Time physically exists—via the inverse relation of relative speediness, which is directly observed.

Since time is common to all changes all points of the absolute space in the universe, it can be put to use when it comes to quantitatively characterizing the changes associated with motions of objects.

In NM, the measures of time also are uniform at all locations in the absolute space.

Many of these considerations remain exactly intact even in the relativity theory. What changes in the relativity theory are only the mathematically defined systems for space and time measurements. But neither the fact that they physically exist, nor the fact that they are physically entirely different in origins and at uneven levels in the knowledge hierarchy. Any one who suggests they don’t is stupid—be it a Kant, a Poincare, an Einstein, or your next rising star on the pop-sci horizon.

Now, given the absolute space and the absolute time, it is “time” to study motions (of objects).

2.5 Mass:

Objects have mass. Mass is a dynamically defined measure that happens to match exceedingly well with the notion of amount of matter (“stuff”) possessed by objects. In NM, mass is measured (as in practice it still is) by measuring weight—i.e. the strength of an object’s response to the earth’s gravitational field (which is in common to all the objects being weighed—in fact is quantitatively constant for all of them).

Mass is an attribute of individual objects. Hence, when a given object moves and thereby changes its location, so does its mass. Thus, mass has no location other than that of the object whose attribute it is. Obvious, no? (In the NM ontology, it is.)

2.6 Point-particles:

Objects can be abstractly regarded as point-particles via the idea of the center of mass (CoM). The CoM is the distinguished point which, when entered into dynamical equations, correctly reproduces the observed motions of the actual objects, especially those with spherical symmetry (so that angular momentum etc. are not involved).

The view of objects as point-particles is an abstraction. What metaphysically exist are only spatially finite objects. However, via abstraction, objects can be taken as massive point-particles (i.e., particles having no extension).

Some of the salient features associated with the motions of point-particles are: (i) their trajectories (the continuous and mathematically simple paths that they trace in the absolute space over absolute time), (ii) their displacements, (iii) their speeds and directions (velocities), (iv) the changes in their motions i.e. their accelerations, etc.

2.7 The direct contact as the only means of interactions between objects:

Objects can be made to change (some or more of the measures of) their motions due to the actions of other objects.

In NM, physical objects cannot be made to change their motions through mental action alone. They change motions only after interaction with other physical objects.

In NM, the only mechanism through which two physical objects can come to change their motions is: via a direct physical contact between them.

The contact may last for very short durations (as happens in the collisions of billiard balls), which can be abstractly described as an instantaneous change. The contact may last, continuously, for a long time (as happens with motions of billiard balls on a table with friction; or the idealized, frictionless motion of a ball through air; or of an ideal bead sliding without friction on an ideal wire, etc.).

2.8 Momentum and force:

The dynamically most relevant measure of motion (in Newton’s words, its “quantity”) is: the momentum of an object. It at once captures the effects of both mass and velocity on an object’s dynamical behavior.

The physical mechanism of how two objects affect each other’s motions is: the direct physical contact. The (mathematically devised) quantitative measure of how much an object’s motion has been affected is the force, defined as time-rate of change of its momentum.

Thus, in NM, forces arise only by direct contact between two bodies, and only for the duration that they are in contact.

Since in NM, mass of a given object always remains constant, force and acceleration amount to be just two different terms to describe the essentially same quantitative measures of the same physical facts. Any acceleration of a point-particle necessarily implies a force acting on it; any (net non-zero) force applied to a point-particle necessarily accelerates it. There also is no delay in the action of a force and the acceleration produced in reality—or vice versa. (Deceleration of one object while in contact with second object is a production of a force by the first on the second.)

The universe obeys the law of conservation of momentum.

2.9 An interaction, but without direct contact—gravity:

In the ontology of NM, the only exception to the rule of interaction via direct contact is: gravity.

No one knows how can it be that one object affects—forces—another object at a distance, with literally nothing in between them. Let’s call it an instantaneous action at a distance (IAD).

This issue of the presence of IAD in gravity is a riddle for NM because physical contact is the only mechanism allowed in it by which forces can ever come to arise, i.e., the direct contact is the only mechanism available for one object to affect another object.

[The legal system till date recognizes this principle. To show that a moving knife involved in a murder was not wielded by you, you only have to show that there was no direct physical contact between you and that knife, at that time.]

Coming back to the ontological riddle, no one knows how to resolve it within the context of the NM ontology. Not even Newton. Therefore, the dynamical equation that is Newton’s law of universal gravitation is an incomplete description. Even though it works perfectly in explaining all the observed data concerning the celestial motions (such as those by Kepler).

2.10 The energetics program and the potential energy:

The same physics as is given by Newton’s laws can also be described using a different ontological term: energy.

An object in motion has an attribute called the kinetic energy (whose quantitative measure is defined as $1/2 mv^2$). Objects in a perfectly elastic collision conserve their total kinetic energy. This is a direct parallel to Newton’s original analysis via the conservation of total momentum.

In the energetics program (pursued by Leibnitz, Euler, Lagrange, and others), two objects interacting at a distance with each other via gravity, say a massive ball and the earth, have an additional energy associated with them. This energy is associated not with their motions, but with their common configuration. This energy is called the potential energy.

Consider a ball held in hand at some height, which is about to be released. So long as the ball is not released, the configuration of the ball and the earth stays the same over any lapse of time. Though both the objects have zero kinetic energy, their configuration still is considered to have this second form of energy called potential energy. For an unreleased ball, since the configuration of ball–earth system stays the same in time, the potential energy of this configuration also stays the same.

The potential energy measures the unrealized capacity of a configuration to undergo change, if the physical constraints restricting the possible motion, such as the support for the ball, are removed.

When the support is removed, the ball falls down. It accelerates towards the ground.

In the energetic analysis, the ball acquires a kinetic energy (of motion). If initial KE is zero, and if total KE is conserved, then where does this KE of the falling ball come from? It comes about because the ball–earth system is supposed as simultaneously losing its potential energy. When the ball undergoes free fall the system configuration is continuously changing. So, the energy associated with the configuration (relative positions) also is continuously changing. For conservation law to work, the system has to lose PE so that it can gain KE. Gaining of a KE is regarded as a process of realization of a potential. The realized potential is subtracted from the initial potential energy.

Just before the ball comes to rest at the ground, its speed is the highest. That’s because almost all of its initial potential energy has been realized; the realization consists of this particular instantaneous state of motion (of the highest speed).

Thus, the potential energy of the ball (its capacity to undergo motion) is higher at a height, and it is zero at the ground. (After all, once it’s on the ground, it can’t move any further down.) Mathematically, the potential energy of a system is given as $mgh$.

When action-at-a-distance forces like gravity are part of a system description, the total energy of a system at any instant is the sum, at that instant, of the kinetic energies of all its separate constituent objects taken individually, and the potential energy associated with all their positions taken at once—i.e. their configuration.

Thus, notice, the potential energy belongs to the configuration—to the entire system—and not to any one object. That’s in contrast to the kinetic energy. Each object has its own kinetic energy (when it’s in motion). But a single isolated object does not have any potential energy, be it stationary or in motion. Only two or more objects taken together (as a system) possess PE.

For this reason, in NM, the KE has a point-position: it is always located where that object is, during motion. In contrast, the PE does not have any spatial position. It is an attribute of the relative positions of two or more objects taken at once. That’s why, in NM, there is no spatially distinguished point where the PE of a falling ball exists—there is no PE of a ball in the first place!

The conservation of law for the universe is: KE + PE = constant.

2.11 A recap of the NM ontology:

In short, the ontology of NM is this: The objects that NM studies are massive (like solid balls), and isolated from each other in the absolute space. They can move and affect each other’s motions primarily through direct contact. In an extended description, two objects can also act via gravity, though mechanism for such action at a distance is not known in the NM ontology. (In a tentative substitute for the ontology, gravity is taken to act as if it were through an invisible string that connected two spatially separated objects.) In NM, the motions and interactions of objects can be described with reference to the passage of a common universal time. Point-particles don’t physically exists, but form a useful abstraction.

Notice, specific ideas like Newton’s laws, or the law of conservation of momentum or energy, though mentioned above, are not a part of NM ontology as such—they form a part only of its physics, not of ontology.

2.12 In NM, potentials don’t form fields, and so, are attributes of configurations, not of individual objects:

Notice also that while potential energy has entered the physics analysis using NM, it is still not being regarded as a field. Neither gravity nor potential is still being regarded as a field. An object like a field is missing from the NM ontology.

In principle, for visualization of what the world is like using Newton’s own approach, you can draw isolated dots in space representing massive point-particles; indicate (or show in animation) their velocities/momenta; and also indicate the forces which arise between them—which can happen only during a direct contact.

Forces arise and act at the point of direct contact but nowhere else. Therefore, forces arise only at the point-positions of particles when they are in direct contact—and it is for this reason that forces are able to affect the particles’ motions. You can use Newton’s laws (or conservation of the sum of PE and KE) and calculate the motions of such particles. If objects of finite sizes have to be dealt with as such, they are to be seen as collections of infinitely many particles each of which is infinitely small. It is the particles that are basic to the NM ontology.

In using the Leibniz/Euler/Lagrange’s energetics program, you still draw only isolated dots for particles. However, you now implicitly suppose that they form a system.

“System” actually is a much later-date concept. Using modern ideas, we can draw an imaginary box around the particles which are being considered for a dynamical description. We can then imagine as if a meter is attached to this imaginary box. This meter displays a number, and calculations involving it enter into analysis. The reading on the meter gives the potential energy for the overall system—for all the particles put together, in the configuration in which they are found together. Thus, this number is not associated with any one particle in the system, but with the overall system taken as a whole (or, the system taken as an abstract object of sorts).

Thus, to repeat, the potential energy “of a ball” is a rather loose expression, if you follow the NM ontology. The PE is not an attribute of a single object. Hence, PE is not something which moves in space along with it. PE remains a global property of a system with unspecified spatial properties (like position) for it.

The idea of a potential as something that is an attribute of an individual object itself (regardless of the system it is in), though so familiar to us today, actually forms a part of a distinct development in ontology. This development is best illustrated with Maxwell’s electrodynamics. I will come to it after a few days.

… In the meanwhile, GaNapati festival greetings, take care, and bye for now…

A song I like:

(Marathi) “too sukhakartaa too du:khahartaa…”
Singer: Ashalata Wabgaonkar
Lyrics and Music: Vijay Sonalkar

History: Originally published (~2,700 words) on 2019/09/02 15:48 IST. Considerably extended (but without changing the sub-paragraphs structure or altering the basic points—~3,900 words) on 2019/09/03 15:04 IST. … Now am leaving it in whatever shape it is in.

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Relating the One with the Many

0. Review and Context: This post is the last one in this mini-series on the subject of the one vs. many (as understood in the context of physics). The earlier posts in this series have been, in the chronological and logical order, these:

1. Introducing a very foundational issue of physics (and of maths) [^]
2. The One vs. the Many [^]
3. Some of the implications of the “Many Objects” idea… [^]
4. Some of the implications of the “One Object” idea… [^]

In the second post in this series, we had seen how a single object can be split up into many objects (or the many objects seen as parts of a single object). Now, in this post, we note some more observations about relating the One with the Many.

The description below begins with a discussion of how the One Object may be separated into Many Objects. However, note that the maths involved here is perfectly symmetrical, and therefore, the ensuing discussion for the separation of the one object into many objects also just as well applies for putting many objects together into one object, i.e., integration.

In the second and third posts, we handled the perceived multiplicity of objects via a spatial separation according to the varying measures of the same property. A few remarks on the process of separation (or, symmetrically, on the process of integration) are now in order.

1. The extents of spatial separation depends on what property you choose on the basis of which to effect the separation:

To begin with, note that the exact extents of any spatial separations would vary depending on what property you choose for measuring them.

To take a very “layman-like” example, suppose you take a cotton-seed, i.e. the one with a soft ball of fine cotton fibres emanating from a hard center, as shown here [^]. Suppose if you use the property of reflectivity (or, the ability to be seen in a bright light against a darker background), then for the cotton-seed, the width of the overall seed might come out to be, say, 5 cm. That is to say, the spatial extent ascribable to this object would be 5 cm. However, if you choose some other physical property, then the same object may end up registering quite a different size. For instance, if you use the property: “ability to be lifted using prongs” as the true measure for the width for the seed, then its size may very well come out as just about 1–2 cm, because the soft ball of the fibres would have got crushed to a smaller volume in the act of lifting.

In short: Different properties can easily imply different extensions for the same distinguished (or separated)“object,” i.e., for the same distinguished part of the physical universe.

2. The One Object may be separated into Many Objects on a basis other than that of the spatial separation:

Spatial attributes are fundamental, but they don’t always provide the best principle to organize a theory of physics.

The separation of the single universe-object into many of its parts need not proceed on the basis of only the “physical” space.

It would be possible to separate the universe on the basis of certain basis-functions which are defined over every spatial part of the universe. For instance, the Fourier analysis gives rise to a separation of a property-function into many complex-valued frequencies (viz. pairs of spatial undulations).

If the separation is done on the basis of such abstract functions, and not on the basis of the spatial extents, then the problem of the empty regions vaporizes away immediately. There always is some or the other “frequency”, with some or the other amplitude and phase, present at literally every point in the physical universe—including in the regions of the so-called “empty” space.

However, do note that the Fourier separation is a mathematical principle. Its correspondence to the physical universe must pass through the usual, required, epistemological hoops. … Here is one instance:

Question: If infinity cannot metaphysically exist (simply because it is a mathematical concept and no mathematical concept physically exists), then how is it that an infinite series may potentially be required for splitting up the given function (viz. the one which specifies the variations the given property of the physical universe)?

Answer: An infinite Fourier series cannot indeed be used by way of a direct physical description; however, a truncated (finite) Fourier series may be.

Here, we are basically relying on the same trick as we saw earlier in this mini-series of posts: We can claim that what the truncated Fourier series represents is the actual reality, and that that function which requires an infinite series is merely a depiction, an idealization, an abstraction.

3. When to use which description—the One Object or the Many Objects:

Despite the enormous advantages of the second approach (of the One Object idea) in the fundamental theoretical physics, in classical physics as well as in our “day-to-day” life, we often speak of the physical reality using the cruder first approach (the one involving the Many Objects idea). This we do—and it’s perfectly OK to do so—mainly because of the involved context.

The Many Objects description of physics is closer to the perceptual level. Hence, its more direct, even simpler, in a way. Now, note a very important consideration:

The precision to used in a description (or a theory) is determined by its purpose.

The purpose for a description may be lofty, such as achieving fullest possible consistency of conceptual interrelations. Or it may be mundane, referring to what needs to be understood in order to get the practical things done in the day-to-day life. The range of integrations to be performed for the day-to-day usage is limited, very limited in fact. A cruder description could do for this purpose. The Many Objects idea is conceptually more economical to use here. [As a polemical remark on the side, observe that while Ayn Rand highlighted the value of purpose, neither Occam nor the later philosophers/physicists following him ever even thought of that idea: purpose.]

However, as the scope of the physical knowledge increases, the requirements of the long-range consistency mandate that it is the second approach (the one involving the One Object idea) which we must adopt as being a better representative of the actual reality, as being more fundamental.

Where does the switch-over occur?

I think that it occurs at a level of those physics problems in which the energetics program (initiated by Leibnitz), i.e., the Lagrangian approach, makes it easier to solve them, compared to the earlier, Newtonian approach. This answer basically says that any time you use the ideas such as fields, and energy, you must make the switch-over, because in the very act of using such ideas, implicitly, you are using the One Object idea anyway. Which means, EM theory, and yes, also thermodynamics.

And of course, by the time you begin tackling QM, the second approach becomes simply indispensable.

A personal side remark: I should have known better. I should have adopted the second approach earlier in my life. It would have spared me a lot of agonizing over the riddles of quantum physics, a lot of running in loops over the same territory (like a dog chasing his own tail). … But it’s OK. I am glad that at least by now, I know better. (And, engineers anyway don’t get taught the Lagrangian mechanics to the extent physicists do.)

A few days ago, Roger Schlafly had written a nice and brief post at his blog saying that there is a place for non-locality in physics. He had touched on that issue more from a common-sense and “practical” viewpoint of covering these two physics approaches [^].

Now, given the above write-up, you know that a stronger statement, in fact, can be made:

As soon as you enter the realm of the EM fields and the further development, the non-local (or the global or the One Object) theories are the only way to go.

A Song I Like:

[When I was a school-boy, I used to very much like this song. I would hum [no, can’t call it singing] with my friends. I don’t know why. OK. At least, don’t ask me why. Not any more, anyway 😉 .]

(Hindi) “thokar main hai meri saaraa zamaanaa”
Singer: Kishore Kumar
Music: R. D. Burman
Lyrics: Rajinder Krishan

OK. I am glad I have brought to a completion a series of posts that I initiated. Happened for the first time!

I have not been able to find time to actually write anything on my promised position paper on QM. … Have been thinking about how to present certain ideas better, but not making much progress… If you must ask: these involve entangled vs. product states—and why both must be possible, etc.

So, I don’t think I am going to be able to hold the mid-2017 deadline that I myself had set for me. It will take longer.

For the same reasons, may be I will be blogging less… Or, who knows, may be I will write very short general notings here and there…

Bye for now and take care…