Both Classical and Quantum Mechanics seem to be based on a mathematical model that views physical space as a set of locations or points. This brings about the inconsistency that a continuous space is being defined by discrete points.
This is the first part of paper that looks at the above inconsistency. It starts by taking a conceptual approach to the subject of inertia.
Inertia
Here are some definitions from “Newton’s Principia for the Common Reader”:
Definition 1: The quantity of matter is the measure of the same, arising from its density and bulk conjointly.
Newton refers to the quantity of matter as body or mass. It is proportional to weight.
Definition 2: The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly.
Quantity of motion is momentum (mass x velocity) in our present terminology.
Definition 3: The vis insita, or innate state of matter, is a power of resisting, by which every body, as much as in it lies, continues in its present state, whether it be of rest, or of moving uniformly forwards in a right line.
Newton says, “This force is always proportional to the body whose force it is and differs nothing from the inactivity of the mass, but in our manner of conceiving it. A body, from the inert nature of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita may, by a most significant name, be called inertia (vis inertiae) or force of inactivity. But a body only exerts this force when another force, impressed upon it, endeavors to change its condition; and the exercise of this force may be considered as both resistance and impulse, it is resistance so far as the body, for maintaining its present state, opposes the force impressed; it is impulse so far as the body, by not easily giving way to the impressed force of another, endeavors to change the state of that other. Resistance is usually ascribed to bodies at rest and impulse to those in motion; but motion and rest, as commonly conceived are only relatively distinguished; nor are those bodies always truly at rest, which commonly are taken to be so.”
Definition 4: An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of uniform motion in a right line.
Newton says, “This force consists in the action only, and remains no longer in the body when the action is over. For a body maintains every new state it acquires, by its inertia only. But impressed forces are of different origins, as from percussion, from pressure, from centripetal force.”
Law 1: Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.
Conceptually, inertia is the inherent tendency of a state of motion to maintain its status quo.
Law 2: The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
This is stated as, “Force = mass x acceleration.” Force is proportional to acceleration, and the proportionality factor is the mass (quantity of matter). Mass is equivalent to the force that is required to produce unit acceleration.
Law 3: To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.
The internal resistance encountered due to inertia equals the force applied. Thus, inertia may be accounted for in terms of mass.
The above discussion may be summarized in the following concepts.
- Inertia is the inherent tendency of a state of motion to maintain its status quo.
- Inertia comes into play when a force is impressed from outside, and acceleration is generated.
- Inertia is the property of quantity of matter, or mass.
- Mass depends on the density and volume of the body.
- Inertia increases or decreases as mass increases or decreases.
- Mass may be measured in terms of force required to produce unit acceleration.
Center of Mass
Here are some corollaries from “Newton’s Principia for the Common Reader”:
Corollary 4: The common center of gravity of two or more bodies does not alter its state of motion or rest by the actions of the bodies among themselves; and therefore the common center of gravity of all bodies acting upon each other (excluding external actions and impediments) is either at rest, or moves uniformly in a right line.
Newton says, “It is manifest that the common center of all never suffers any change in the state of its motion or rest from the actions of any two bodies between themselves… And therefore the same law takes place in a system consisting of many bodies as in one single body, with regard to their preserving in their state of motion or rest. For the progressive motion, whether of one single body, or of a whole system of bodies, is always to be estimated from the motion of the center of gravity.”
Proposition 24, Cor. 7: And hence appears a method both of comparing bodies one with another, as to the quantity of matter in each; and of comparing the weights of the same body in different places, to know the variation of its gravity. And by experiments made with the greatest accuracy, I have always found the quantity of matter in bodies to be proportional to their weight.
This conclusion is in support of Newton’s Law 2.
From Maxwell’s paper on Matter and motion cited in “Newton’s Principia for the Common Reader”:
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Center of mass of two particles: C is a point such that if the masses of A and B were concentrated at C, their mass-vector from any origin O would be the same as when A and B are in their actual positions. The point C is called the Center of Mass of A and B.
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Center of Mass of a system: The center of mass is therefore a definite point in the diagram of the configuration of the system. By assigning to the different points in the diagrams of displacement, velocity, total acceleration, and rate of acceleration, the masses of the bodies to which they correspond, we may find in each of these diagrams a point that corresponds to the center of mass, and indicates the displacement, velocity, total acceleration, or rate of acceleration of the center of mass.
The above discussion may be summarized in the following concepts.
- A body responds to motion as if its weight is concentrated at a point.
- Mass is proportional to weight.
- A body responds to motion as if its mass is concentrated at a point.
Physical Location
Classical mechanics treats physical objects as if all their mass is concentrated at a center. A dimensionless point is used to represent this location. The whole object is not dimensionless, but its center of mass comes closest to pinpointing its physical location, under the laws of classical mechanics.
This location appears in reference to the object itself. It depends on the mass of the object and how that mass is distributed within that object. Thus, the universe also has a location with respect to itself. This location is invariable if the total mass remains constant.
As the mass of an object increases it is harder to move. We may say that its physical location is becoming more centered. From this point of view, the most centered objects in this universe are Black Holes.
As the mass of an object decreases, its physical location becomes less centered even when it is looked upon as “in motion”. The location may still be approximated by a geometrical point, but it is increasingly unstable. It may be altered easily with respect to more stable locations. When there is no mass then there is no physical location that can be defined as a point in space. A massless photon shall have no physical point location even when it is looked upon as moving.
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The above discussion may be summarized in the following concepts.
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A physical point location in space depends on the distribution of mass around it.
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If there is no mass in a section of space then there is no physical point location within that context.
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Such physical point locations are discrete and they seem to be distributed in space according to the law of gravity of classical mechanics.
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A mathematical point is different. It is discrete and it may be assumed anywhere in a continuous space even when there is no mass.
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A physical location may be represented by a mathematical point. But a mathematical point in space does not necessarily represent a physical location.
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The physical location of a massless photon may not be represented by a point or points as conceived in mathematics.
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This paper essentially points out the difference between a physical location and a mathematical point.
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Comments
I see matter as a state of motion. Newton assumed that matter is continuously divisible down to point particles. He didn’t envision motion within a particle. Atomic physics came much later than Newton and showed that there is motion within a particle (atom). Is there a motionless particle within an atom? I doubt it as there is no evidence for it. This tells me that we do not fully understand the subject of motion.
Classical concept of motion cannot be applied to atomic and quantum physics.
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Concept of point to denote physical location cannot be applied in space where no mass exists.
Even when one uses a density function along with the idea of point to demark physical location, it would be inapplicable when density is zero, unless it can also address the “density” (consistency) of massless photon.
The notion of physical location should be consistent through the spectrum of inertia of which mass is just a part.
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There is a fallacy of “appeal to authority” regardless of whether that authority is worthy or unworthy. Newton was a worthy authority but Einstein still discovered limitations of his assumptions. I think that one should NOT be discouraged from questioning authorities.
If authorities are right then they would not fall down, but if they were limited by their assumptions, there is a possibility of discovering new knowledge.
My path of investigation is that a MATHEMATICAL POINT in space cannot always be equated with a PHYSICAL LOCATION, I would like to examine how the assumption of equating these two things has influenced the conclusions of many authorities.
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Does this mean you are getting ready to consider the non-physical space defined by the mathematical points resulting from the “root of -1”?
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Square root of -1 still represents a point that cannot morph into continuity by whim.
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It can’t be dismissed by whim but it can certainly be used to define a continuous “space”.
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How do you propose that it does? As I see square root of -1 is still a finite unit even when it is on a different dimension.
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V: “As I see square root of -1 is still a finite unit even when it is on a different dimension.”
Perhaps I don’t understand you here.
“Root -1” is the equivalent of a scalar. It is the “identity element” of the “imaginary” number space.
Any Real number could be multiplied by “root -1” making the Real an “Imaginary” number.
The point is that you have more than the single number “root -1”, you have a continuous field of numbers from “imaginary” minus-infinity to “imaginary” plus-infinity, and this number space can support continuous functions (like sine).
William Rowan Hamilton worked out the way to create a mathematical 6-space (3 real dimensions x,y,z and 3 imaginary dimensions i,j,k). This math is what it takes to accurately predict astronomical and planetary motion. This idea of dimensions and spaces is the subject of quaternions, if you wish to look it up.
The “imaginary” operator also is critically important in creating resonant structures.
Quantum structures (like photons) appear to be resonant structures (what you call “motion”).
Schrodinger’s equation shows the wave function being operated on by the “imaginary” operator.
If you took a physical ball and multiplied each of the physical coordinates of the ball by the imaginary operators i,j,k, you would have a ball as defined in the Hamilton quaternion space. You couldn’t physically see it but the ball would be fully defined as that “imaginary” space is fully continuous.
So particles are resonant structures and resonance takes them between the “real” state and the “imaginary” state.
We know what we have in the “real” state of the particle: we have the seemingly “solid” particle. What then of the condition of the particle in the “imaginary” state? Well, why would it be different than the “imaginary” ball? It becomes a mapped coordinate set, of sorts: it becomes a “real” coordinate set mapped to an “imaginary” coordinate set.
This mapping (transformation from real to imaginary) is the complete equivalent of an image or picture of the “real” thing. I call that image the space-picture because it is an image-copy of the space of the particle in the “real” state.
These space-pictures of particle state would be lasting “memories” of past physical locations of the particle.
This memory phenomena would fully account for the probability distribution of particle location.
This is because if the particle has a memory image of every place it has been, then during the condensation phase of the resonance of the particle, an “old” memory has a probability of being “recalled” and that old memory would then define the location that the particle would condense to.
This effect has been recorded with a single electron where the electron will appear to be in multiple locations at once.
The apparency is that the electron is in multiple locations at once (superposition) but the more likely event is that the electron changes position every Planck time-unit (5 x 10^-44 sec), because a different memory (space-picture) is active at the time of condensation simply giving it an apparency of superposition by any slower-speed observing mechanism (camera).
The reason these different “memories” can be active has to do with the well understood phenomena that cause changes of quantum states.
So that is a reasoning that is consistent both with mathematics and with observed particle behavior as to how the space defined by the “root -1” would operate.
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SCALAR = representable by position on a scale or line; having only magnitude: “a scalar variable.”
“Root of +1” is the unit representation on a real number line. A point on the real number line is a rational multiple of “Root of +1.”
“Root of -1” is the unit representation on an imaginary number line. A point on the imaginary number line is a rational multiple of “Root of -1.”
A rational number, whether real, imaginary or complex is a discrete entity. A discrete entity has a boundary of value. A set of discrete entities is a collection of such values with boundaries.
Space is a continuous entity. A continuous entity has continuously varying values. There is no bounday that defines an exact value as in the case of discrete entities.
I simply do not see how a continuously varying value without boundary can be defined as a collection of discrete values with boundaries.
The moment you refer to a “number” you have placed a boundary around a value by identifying it. A continuous scale is a continuous scale. A lot of numbers may fit on it, but these numbers remain separate from the scale. A continuous scale is not the same thing as a set of discrete numbers. When you say, “There is a continuous field of numbers,” all you are saying is that there is a continuous field that may be filled with numbers. That continuous field is still there when it is not filled with numbers. The numbers DO NOT MAKE the continuous field.
A continuous function like SINE is by itself. It is not made up of numbers, though infinity of numbers may fit into any segment of it. The bottom line is,
“INFINITY OF DISCRETE ENTITIES DO NOT TOGETHER MAKE A SINGLE CONTINUOUS ENTITY.”
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V: “Space is a continuous entity.”
Continuous and boundless can be different. Functions can be continuous within a domain but discontinuous at a boundary. The tangent function is just one of many examples.
Space may not even be continuous. The boundaries created by black holes may be just one example. Some physigists suggest that space is more like bubbles that form between separating masses (though I wouldn’t necessarily subscribe to that myself).
V: ““INFINITY OF DISCRETE ENTITIES DO NOT TOGETHER MAKE A SINGLE CONTINUOUS ENTITY.””
They certainly can if the space is continuously differentiable. That is the key element.
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We are talking about continuous and discrete entities, and that infinity of discrete entities do not make a continuous entity. What I am presenting is the viewpoint consistent with Physics. In Mathematics you may assume anything that is consistent within the domain of mathematics, but not all mathematics is necessarily consistent with physics.
A continuous entity is continuous throughout. When it acquires a boundary it becomes discrete. It may have a fabric that is continuous inside, but it is the boundary, which imparts the characteristic of discreteness.
There are astrophysicist who are assuming fantastic things, but these fantastic things are mainly derived from fantastic mathematics that is out of touch with physics.
A derivative, such as, dy/dx, provides a ratio between two discrete quantities that are infinitesimal and their ratio reaches a limit. Such derivatives are possible only when a space can be filled with discrete elements that are uniform in their characteristics. This is not possible in space that exists at atomic dimensions.
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V: “A continuous entity is continuous throughout.”
V: “Such derivatives are possible only when a space can be filled with discrete elements that are uniform in their characteristics. This is not possible in space that exists at atomic dimensions.”
You’re contradicting yourself.
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At atomic dimensions, inertia as particle-mass and inertia as wave-frequency exists side by side. There is transition of inertia happening between discreteness and spreadingness. The characteristics of “inertial elements” that are filling the space is not uniform. That is why the use of probability comes in.
Due to this non-uniformity of “inertial elements” it seems difficult to carry out calculus operations in this region with any certainty.
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V: “There are astrophysicist who are assuming fantastic things, but these fantastic things are mainly derived from fantastic mathematics that is out of touch with physics.”
Be a bit more specific here, please. Are you aiming this comment at Hamilton or some generic physicist who has made some statement about the formation of space?
I would like to hear what you think of the Hamiltonian use of complex coordinates as this goes to my argument that the imaginary component of complex number cannot be dismissed. It requires you to understand it. One cannot get away with poo-pooing it.
It is not possible to get a consistent or meaningful and complete explanation of inertia – especially at the quantum level – without a grasp of this concept and the operation of the cross product. If you are familiar with precession, you could use that motion and mechanism to understand the vector torque created by a transition from real 3-space to complex ijk space and back.
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My comment is aimed only at fantastic claims that are based entirely on mathematical speculations of new dimensions as in String Theory.
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V: “The characteristics of “inertial elements” that are filling the space is not uniform. That is why the use of probability comes in.
Due to this non-uniformity of “inertial elements” it seems difficult to carry out calculus operations in this region with any certainty.”
Applying calculus in a complex system of many particles would certainly be difficult but probably not totally impossible. This is the sort of problem considered by mathematicians looking at the motion patterns in a large flock of birds or a school of small fish.
What is interesting in complex atomic systems is how there are some similarities to the motion and clustering of macro systems like birds and fish.
For instance, many liquids can go through a an apparent crystalline phase (basis of the LCD) and form ordered arrays that are mathematically predictable in structure and function. The liquid is still comprised of uniquely atomic structures (molecules) and the crystalline structure comes not from a change of molecular structure but from a mass orientational structure.
With some liquids, a multiple of crystalline structures can appear depending on the external energies (or quantum fields) present. This has even been shown with water.
What this tendency of coalescence to crystalline structure shows is that there can be bands of probabilities of observable motion and momenta.
The effect of being in one of those bands would be like having someone think of a particular subject. If you said “Think about ice cream” then you could predict that if you took a random moment and said “what were you thinking about?” the answer might come back as “Chocolate ice cream”. You couldn’t necessarily predict the reply if the person has had many experiences with different flavors of ice cream, but you could predict that the reply would be more likely to be about ice cream than about cigars.
Now if the person had only ever had one experience with ice cream, and that was chocolate ice cream, then you could make a much more accurate prediction of reply. Of course the person could also go off on a tangent of chocolate flavors and end up anywhere from chocolate bars to cookies, and that is why all replies are probabilistic.
The point here is that the history of the person is relevant to the positional outcome (like recalled flavor), and so too is the history of an electron relevant to where one is likely to find it condensing. This is now being studied in Quantum Mechanics and has been described by the term “causal sets”. These are essentially the paths that an electron has taken. The “causal set” is effectively the memory of the electron.
This memory is, of course, vast beyond comprehension, giving any particular electron a probability that it could condense at any particular point in the universe where it has formerly been (condensed).
However, the likelihood (probability) of that location change on a random basis is fairly low, similar to the probability of someone concentrating on ice cream suddenly recalling battery acid. If the probability wasn’t low then particles would be very unstable and the universe much more chaotic.
All of these behaviors are similar to inertia at the classical level – once you get something going in a direction (like thinking about ice cream) then it will continue along that pattern until an external condition comes along that is forcible enough to precipitate a change.
So while the momentum and location of individual, discrete, quantum elements may cause local perturbations of the “space” that the particle is in – making its path, momentum and location seemingly discontinuous and non-differentiable, the same need not be said of macro structures.
However, for classically oriented purposes (like path determination), the path of some macro structure – such as a projectile or a baseball – is adequately differentiable to give extremely good prediction (adequate prediction, high probability) of where it is going to located on a moment to moment basis.
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2ndxmr, thank you for your participation. You are indeed a man deeply interested in science.
Right now my main problem is,
(1) How to clearly explain the inertial form of wave-frequency? The frequency does generate a state of motion that maintains its status quo.
(2) How to explain the transition of intertia between its wave-frequency form and its particle-mass form?
(3) Are there simpler mathematical concepts that can encapsulate the above explanations? How can such concepts be presented?
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V:”(1) How to clearly explain the inertial form of wave-frequency? The frequency does generate a state of motion that maintains its status quo.”
That is a property of resonance. We are used to resonances that decay, like the sound from a crystal glass after it has been pinged. The lack of decay of resonances set up at the quantum level indicates that the medium of the resonance (crystal glass is an example of a medium) is perfectly elastic and lossless. In a lossless medium there is no decay so the resonance could go on infinity long. That is how it maintains its status quo.
What this comes back to is that “space” is the most likely explanation for this medium.
Water is a medium for aquatic life. It is H2O plus pollutents, but let us just think H2O and look at the general properties:
– it has density
– it has conductivity and permeability
– sound waves have a definable propagation velocity
Space has properties that correlate directly. This indicates that space could probably be modeled as a fluid.
Particles moving through a fluid are influenced by the fluid:
– A boat moves through water easier when its long axis is aligned with the direction of motion.
– The boat experiences a resistance to the initiation of motion.
– Once in motion a boat will continue to move with relatively low applied force (a force which is only compensatng for drag resistance).
– On removal of force there is an initial tendency of the water to push at the back end of the boat, continuing its motion.
Mass-endowed particles moving through space see similar effects.
The shape and orientation of a boat in water (geometrical aspects) determine interactions with the water which affect its velocity of motion.
It is extremely likely that geometrical aspects – such as long-axis orientation – determine the velocity characteristic of particles in space.
These considerations are what incline me to view the inertia problem as a sort of fluid-dynamics problem which becomes easy to understand and predict once you can model the fluid.
I expect that space, as a fluid, will model with geometries representing the four forces: electro-magnetic, strong-and-weak nuclear, and gravity.
We have been well schooled in the existence of these forces.
Now it is time to consider the geometries that could account for them.
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V: “(2) How to explain the transition of intertia between its wave-frequency form and its particle-mass form?”
This is a different type of inertia than the inertia to motion.
This is the realm of what I have termed quantum inertia: the “To be or not to be” question that a particle appears to make in response to conditions of its environment – like double slits and observers.
We are told that an electron can behave as a particle or wave and I expect that behavior will be seen to be a simple matter of geometric orientation of the electron.
This geometric orientation phenomenon is really no different than how you solve the geometrical concerns of inserting a key into a lock – there are limited options and a few rules based on key type.
Or, another way to view the geometrical orientation is to consider the difference between looking at a sheet of paper face-on or edge-on. It looks different depending on the direction of viewing, but it is still a piece of paper.
The same applies to the electron: it appears to preserve its basic properties of mass, charge and spin independent of its wave or particle form. This behavior is what makes me think it is responding with a geometrical change, an edge-on to face-on transformation.
I expect that things like double-slits act as memory restimulator for the geometrical orientation the electron should take. It is much like walking up to a door and having to remember which key on the key-ring is the appropriate one for the door. That becomes a memory-based choice.
Adding an observer to watch the slit(s) elicits another behavior, but one which can still be shown to derive from the memory model.
Quantum inertia and Newtonian inertia are alike in that they are the tendency to continue on a given path (Newtonian) or orientation (Quantum) until an external force or condition causes a change.
They are also alike in that Newtonian inertia deals with the effect of a geometrical orientation phenomenon and Quantum inertia sets and maintains the geometrical orientation.
Sort of a dumb and dumber duo.
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I am questioning assumptions that are being exposed in the light of atomic phenomena. There is no longer any evidence to keep backing up those assumptions. Can classical concept of motion be applied to the motion within an atom? Can a physical location be always represented by a point? It is not that I am asserting a new theory. I see problem with the existing theory.
Quantum mechanics has gone so deep into mathematics that it has lost touch with physical concepts. The concept underlying the position operator still assumes that a point always means a physical location in space, and so it gets complicated by the task of figuring out probabilities.
The basic inconsistency that I see is that discrete points are being used to define a continuous space. Something discrete cannot be continuous. That is a hidden assumption to think that it can be.
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If the mathematical concepts cannot be translated to simple physical concepts then they cannot be applied to physics.
They remain in the area of metaphysics.
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I do understand that it is very hard when one’s deep set beliefs are questioned.
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The following is the response on Quora to the question, “Is a point in physical space the same thing as a physical location?”, which quoted the above paper.
http://www.quora.com/Is-a-point-in-physical-space-the-same-thing-as-a-physical-location
The answer is, ‘yes, they are the same’. What else, after all, would “point in physical space” and “physical location” mean?
Your blog is not convincing. Though it is good that you start out with quotes from Newton himself, I do not know how you got the idea that “Inertia is the inherent tendency of a state of motion to maintain its status quo” out of his words. It simply does not follow. You seem to have skipped his words in Definition 3: “innate state of matter”.
Furthermore, you compare in your opening, classical and quantum mechanics, without any consideration for the revelation in relativity that is is not space, but spacetime that must be considered as the fundamental manifold on which we build our physical description of the world.
Finally, you seem to lose track of a fundamental feature of science: the use of models. Newtonian mechanics and dynamics use one model, that described by Newton and elgantly refined and reformulated by others such as Lagrange, Laplace Jacobi and Hamilton; quantum mechanics and relativity make careful changes to this model to make new and different models.
The reason I bring this up here and now is that the use of ‘points’ you consider contradictory is NOT contradictory in classical mechanics, where space IS a continuum of points (the math of this continuum is not contradictory as long as we find no contradiction in the axioms of arithmetic). Resolving the apparent contradiction in quantum mechnaics is harder, since Hilbert’s Sixth Problem is still not solved. But this could just be an indication that the theory is not yet ripe for completion, that we are still using the wrong abstractions. Even the Nobel Prize Laureates who worked out QED have expressed such concerns.
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Here is further clarification of the original paper.
(1) A discrete object acts as if its mass is concentrated at a point in space. I called such a point as a distinct physical location. But let’s call it a Center of Mass (COM) to avoid confusion.
(2) A discrete physical body has only one COM. Other points on that body cannot be treated as COMs.
(3) Similarly, a system of physical bodies, which are interacting mechanically, has only one COM. Other points in that system cannot be treated as COMs.
(4) Thus, within a specified context, a distinction may be made between a point that is a COM and another point that is not a COM.
(5) Other points in space are then mathematical extrapolations with respect to COMs.
(6) Thus, a COM has the additional characteristic of acting as a reference point in space. This characteristic comes from the mass associated with it.
(7) A physical body, or a system of bodies, is made up of points in space, but all these points are referenced from its COM.
(8) Two COMs interact with each other per Newton’s Law of Universal Gravitation. The distance between them may be determined as, r = SQRT (G m1 m2 / F)
The distances in space, therefore, do not behave as multiples of some absolute unit of distance. They depend on the masses associated with COMs and the observed force between them.
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The next part of this paper is going to show that spacetime is not independent or absolute. It rather depends on INERTIA.
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On its face the above paper seems silly but it arises from the inconsistency of a continuous space being defined as a set of discrete points. Mathematics fails to define the relationship between point and space properly. It seems that there is a better explanation underlying this inconsistency.
Science has progressed to a point where it is looking at the process of observation itself. It is difficult to come up with some absolute reference point, from which to look at the rest of existence. God seems to be a placeholder for what we do not understand. The only possible avenue to take is to take one inconsistency at a time and resolve it as best as one can, keeping in mind that one may have to come back to it later to resolve it further.
The universe is obviously finite in terms of what one is actually able to observe. We may postulate the universe to be infinite but that does not add anything more to our actual awareness or understanding. The idea of location is limited to the mechanical universe that we actually observe, independent of what one might postulate mathematically or otherwise.
We are now confronted with the idea of motion that goes beyond the classical concept of a physical body moving in space and time. A physical body consists of motion within itself at atomic levels. We cannot say if there is a point within the atom that is absolutely still. The stillness may be computed only in relative terms as difficulty involved in shifting a point. This is where the concept of inertia springs from. Thus, the idea of a location in space is intimately tied with the idea of inertia. This is what I meant by the concept of “centeredness” of CoM. I do not know if there is any limit to ideal centeredness. That is subject to observation.
An electromagnetic wave does not have its momentum concentrated at a point; rather it is evenly spread throughout. So, it has a physical presence that cannot be described as located at a point, as it can be done with mass. A photon may be described as having a physical presence as a wave packet, the size of which depends on its frequency. The higher is the frequency, the smaller is the region that describes its presence. One may say the location of a photon is spread-out in space, as opposed to being concentrated at a point as in case of mass. There is continuity that seems to exist within that “spread-out location”.
As frequency of the photon decreases, its location seems to “spread-out” further as proportional to its wavelength. This is looking at location in terms of inertia. The inertia lessens as frequency decreases. This spread may assume infinite proportions as frequency approaches zero. Here we achieve an approximation of space as a primitive concept that does not depend on the concept of point. In fact, space and point seem to appear at the opposite ends of a scale of inertia. Here is a picture that I get.
So, an absence of “point location” does not mean “absence of location”. It means a location spread out in a manner of continuity that is very different from a “set of points.”
The statement, “A physical point location in space depends on the distribution of mass around it” means that there is the continuity of space corresponding to “zero inertia” that acts as the background. In this background exist point locations as “concentrated inertia.” I shall explain this as further questions arise.
Basically, I believe that physical space and location can be described more coherently in terms of inertia. This description bypasses the contradiction posed by a mathematical description of space as a set of points.
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Here is Einstein’s point of view with which the above paper agrees:
From Wikipedia:
“In addition, Einstein used the text to defend the utility of field theories amid the advances of quantum mechanics. The best way to do that was to view particles not as independent objects but as a special manifestation of the field itself: “Could we not reject the concept of matter and build a pure field physics? We could regard matter as the regions in space where the field is extremely strong. A thrown stone is, from this point of view, a changing field in which the states of the greatest field intensity travel through space with the velocity of the stone.”
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Here are some comments as a realist and not as a mathematician.
DISCRETE per dictionary means, “apart or detached from others; separate; distinct” as in “six discrete parts”; or “consisting of or characterized by distinct or individual parts; discontinuous.” I do not see discrete points becoming continuous, regardless of how you scale them, because these points are dimensionless. Points may be looked upon as fractals. The concept of infinity does not change their basic characteristic. They do not coalesce into each other.
The continuous space is a primitive notion on its own. Just becuase it may be filled with points, does not make its definition dependent on the primitive notion of point. “dimension of continuous space” and “dimensionlesss and discrete point” are two primitive notions that are very different from each other. A line is a “single-dimension space” that is continuous. It may be filled with infinity of discrete points, but there are still going to be gaps and breaks between those points. The very fact that a line has no gaps tells you that it is not made up of points. However, a single point may be stretched to form a continuous line. That would be consistent, but that is not allowed in classical mathematics.
There is nobody making the claim that physical spacetime behaves like the mathematical model. I simply see the mathematics failing to adequately model the motion in subatomic space with its “point-space” conceptualization. That is making Quantum Mechanics so math-heavy that it seems to have lost touch with reality, and it is simply floundering around. It is about time to move away from conceptualizing space as a set of points.
I have looked at Newton (& Leibniz) in this regard. Infinity of points may approximate space when one is dealing with inertia in that space either in the form of particle-mass or in the form of wave-frequency, but not as both. This condition is violated at subatomic regions.
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This post is inspired by 2ndxmr’s post here:
https://vinaire.me/2015/01/31/a-conceptual-model-of-inertia-mass-location/#comment-59667
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An electromagnetic wave is produced when there is an oscillating charge. The frequency of oscillation determines the frequency of the electromagnetic wave. The electromagnetic wave maintains that same frequency throughout its propagation. That is a good argument for inertia at wave level.
I see space as an “electromagnetic disturbance of zero frequency and infinite wavelength.” There is complete absence of inertia in space because there is no frequency. Space is pure extendedness. Space is not made up of discrete dimensionless points, because points have no extendedness. Space is extendedness that may be filled with dimensionless points.
Electromagnetic wave is made up of rotating electromagnetic field, as shown in a Wikipedia graphic below. I wonder what makes the electromagnetic wave move forward in this corkscrew fashion.
Electromagnetic Wave Animation
An electromagnetic wave introduces inertia in space. I would love to understand the relationship of the electromagnetic field with space. It is like an electromagnetic disturbance is trying to fill the space with inertia. It probably thins out as it spreads, but it stays there for ever. This is mind boggling.
It almost seems as if space breaks into electrical and magnetic fields when disturbed.
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