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):
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).
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.)
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 ). 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 .
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.