The resolution limits of space and time

ABSTRACT: A discussion on the problems associated with the point assumption of classical physics leads to the postulate that there must exist resolution limits in classical space and time. It is shown that this postulate is superior to the existing point assumption because it allows both classical and quantum physics to be described by a single mathematical framework instead of two separate and conflicting frameworks as they are today.

A. THE POINT ASSUMPTION OF PHYSICS AND ITS PROBLEM

  Physics deals with what I would call “physical measurables”. The concept of “measurable” is extremely important, because even if philosophically I think that this whole universe is an illusion, I would have no problem agreeing with you, for example, that (in this illusory universe) this page measures 8 ½ inches across; and that is enough for you and me to work together.

  There is one very important assumption in physics that cannot be justified even as a mathematical concept. This is the assumption that space and time are infinitely divisible. I will call this the “zero-point assumption” or simply the “point assumption”. I will not go into the reason why the point assumption is a troubling mathematical concept, but I will discuss briefly why its application to physics creates many serious problems.

  I will use the example of a line segment of length L₁. According to mathematics, this segment consists of an infinite number of points. Since I have an infinite number of points, each of size zero, the mathematical relation for the length L₁ is:

  ∞ × 0 = L₁                                                                           (1)

  Which, interestingly, is allowed in mathematics, as the product ∞ × 0 could be anything, and may indeed take the value L₁. But since physics and mathematics are not the same thing, each application of mathematics to physics must be qualified to make sure that it describes physics correctly. An a priori assumption of physics is that, if none of the parts are interacting with one another, they must add up to the whole. Now if the point assumption is applicable to physics, the (mathematical) points must be the parts, and the line segment the whole. For another line segment of length L₂, where L₂≠L₁, the formula would read:

  ∞ × 0 = L₂                                                                           (2)

  Since the left side of the two equations are exactly the same while L₁ and L₂ are different from each other, the point assumption leads to an ambiguous description of physics. While this ambiguity has not caused any problem in classical physics (e.g., Newtonian mechanics), it is not unreasonable to speculate that the failure of classical physics in the quantum realm is a consequence of the point assumption.

  To evaluate this speculation I will seek an alternative to the point assumption. I will evaluate my alternative assumption in 2 steps:

  Step 1: I will verify that my alternative assumption leads to a consistent description of classical physics so that I’m justified in going to the next step.

  Step 2: I will verify that the same alternative assumption leads to a consistent description of quantum physics.

  If my attempt is successful, I can claim that my alternative assumption is superior to the point assumption, because the point assumption requires two separate and opposing paradigms to describe the two branches of physics whereas my alternative assumption requires only one.

  I will speculate that the point assumption is only an approximation of how Nature operates. An approximation will have to break down at some point. As the next logical step, I postulate the possible existence of a classical length scale L₀ and a classical time T₀ that could not be reduced any further. I will call this the “resolution limit assumption”. The resolution limit assumption is promising because it eliminates the ambiguity associated with the point assumption.

  I proposed the resolution limit assumption because, from my 20 years experience in the high technology industry, I know that every measuring tool has a resolution limit. My favorite example is my own set of eyes; which qualifies as a measuring tool. I’m near sighted. Without my eyeglasses I cannot see these words clearly from 2 feet. I would say that without my eyeglasses my ability to resolve images is much worse than any person with a normal set of eyes. But even when I put my eyeglasses on I still cannot see the bacteria floating around me. I would say because they are too small for human eyes to resolve.

  My eyes also have limits in resolving time. A movie appears continuous to me only because the time lapse between frames is smaller than the resolution ability of my eyes. A card trickster can cheat me by dealing the second card (instead of the top card) right before my eyes because his action is too fast for my eyes to resolve; etc.

  Can I generalize this idea of space and time resolution to all physics phenomena? I think I can because, strictly speaking, the word “phenomena” means “phenomena as observed”, and all observations involved measurements. Although most measurement tools are superior to my eyes, I don’t see why any of them can have perfect resolution.

  The reader will note that I have not made any statements regarding the nature of L₀ and T₀. All I have said is that their existence is necessary. Thus, I have allowed the possibility that one or both of these resolutions are process dependent.

  At this point I will rename the resolution limit assumption as the resolution limit postulate, simply because the word “postulate” sounds better than the word “assumption” in physics.

  Before going on, I must make an extremely important point. If this point sounds redundant to you, please ignore it. If it sounds too philosophical to you, I apologize; but this point has to be made. We must differentiate space and time as man made concepts and space and time as they mean to physics. As man made concepts, space and time could be either continuous or discontinuous, depending on personal perception and/or inter-personal agreement. But in physics by "space" we mean "space as measured" (e.g., with a 'standard unit length', a 'standard unit area', a 'standard unit volume'), by "time" we mean "time as measured" (e.g., with a watch, a clock). It is only in the sense associated with physics that the resolution limit postulate is intended.

  Why is this point so important? Just imagine a universe which is an infinite void with absolutely no measurable existence, i.e., no photon, no electron, no proton, no neutron, no neutrino, no Starbucks café, etc. I think you will agree with me that, while it is still possible to conceptualize space and time philosophically in such a universe, there will be no way to define space and time in the sense of physics. Thus, in the sense of physics space and time mean the space and time related to the various forms of existence. In other words, space and time in physics are measures of existence. So when I say there are spatial and temporal resolution limits, I don't mean space and time are inherently non-uniform, I only mean that the space and time components of one form of existence (the measured) can never be perfectly resolved by another form (the measurer).

  Return to the question at hand, I acknowledge that I am not the first one who has the idea that space and/or time may have resolution limits. However, I hope to be the first one who has a sound argument for the resolution limit postulate.

Newtonian mechanics (17^th century)

  Newtonian mechanics centers around the mathematical idea of “derivative”, which later developed into what we now call differential Calculus or simply Calculus. For example, the derivative of position x with respect to time t is the velocity v, the derivative of v with respect to t is the acceleration a. For convenience I will use the Leibnitz notation (e.g., dx/dt for the derivative of x with respect to t). The definition for v = dx/dt is:

  v = dx/dt ≡ limit of Δx/Δt when Δt→0           (3)

  In this definition I see the ghost of the point assumption because in order for Δt to approach zero, the idea of a point of zero measure in the time dimension must be permitted. This is one reason why the point assumption seems necessary for Newtonian mechanics.

  My contention is that the point assumption is not really necessary for physics. With the resolution limit postulate (3) becomes:

  v = dx/dt ≡ limit of Δx/Δt when Δt→T₀          (4)

  I will now compare the performance of (3) and (4).

  1. For a generally smooth function of x, (4) is an approximation of (3). The math purists may not like (4), but it still works in calculations, and is definitely a more practical definition than (3) as far as physics is concerned.

  2. For a kinked function of x, definition (4) is superior over (3) because it removes the conceptual difficulty at the corners (that velocity and acceleration do not exist, which we know to be untrue for actual motions).

  Up to now the differences between the point assumption and the resolution limit postulate only have academic meaning, and I agree that the point assumption is more mathematically convenient in solving most problems in Newtonian mechanics. However, the point I want to make is that conceptually the resolution limit postulate is a more accurate description of Newtonian mechanics. The point assumption works in Newtonian mechanics not because it is fundamentally correct, but because in the realm of Newtonian mechanics it is a good approximation of the resolution limit postulate!!!

  It is in Maxwellian mechanics that the resolution limit postulate starts to show its practical superiority over the point assumption. I will discuss this point next.

Maxwellian mechanics

  By Maxwellian mechanics I mean the part of physics that deals with electromagnetic phenomena. Maxwellian mechanics forces the idea of “waves”, contrasting the idea of “particles” of Newtonian mechanics. However, since the mathematical method of Newtonian mechanics still worked well, Maxwellian mechanics was readily accepted as a new addition to classical physics.

  All electromagnetic waves move at the speed of light c, but curiously each wave consists of repeating units called “wavelengths”. Wavelengths were simply taken as properties of waves because there was nothing in classical physics that could relate to them.

  However, with the resolution limit postulate, I’m forced to conclude that wavelengths are the resolution limits of space! It naturally follows that the time required to complete a wavelength is the resolution of time.

  Let λ stand for wavelength. The time required to complete a wavelength is called the "period", which can be expressed as λ/c.

  In conclusion, for electromagnetic phenomena:

  L₀ = wavelength λ            (5)

  T₀ = period λ/c              (6)

  At this point I seem to be confronted with an insurmountable problem. I know that the wavelength λ theoretically could be, say, 10 trillion miles long, and the period could be, say, 10 billion light years. If I analyze a length smaller than the wavelength of an electromagnetic wave, which --according to the resolution limit postulate-- is beyond the grasp of classical physics, I consistently obtain very precise information regarding the wave. Does this mean the resolution limit has led me to nonsense in Maxwellian mechanics?

  By re-examining the matter more carefully, I find the answer! In a loose sense I will call the spatial and temporal resolution limits the unit length and unit time respectively. In Newtonian mechanics the length scale and time scale of interest to me are much much larger than the unit length and the unit time. For this reason, the events that take place inside a unit length or a unit time are overwhelmed by the Newtonian events, and therefore appear to me as if they don't exist. But in Maxwellian mechanics, what happens within a unit lenght or/and unit time is exactly the subject of my analysis. Thus, it is conceptually wrong for me to try to describe Maxwellian mechanics the same way as Newtonian mechanics. Strictly speaking then, Maxwellian mechanics should not be classified as a branch of classical physics like it is today.

  My point is that Maxwellian mechanics is a part of quantum physics. Thus, the validity of the resolution limit postulate in Maxwellian mechanics must be established in the framework of quantum physics. While this may sound like a delay tactic on my part, I think when the evidence is in the reader will agree with me.

Performance of the resolution limit postulate in classical physics

  From the above discussion, the resolution limit postulate creates no fundamental problem in the description of Newtonian mechanics. In the case of Maxwellian mechanics, it gave me interesting information regarding the nature of wavelength and period, which I would not be able to deduce with the point assumption. This information forces me to realize that Maxwellian mechanics has nothing in common with Newtonian mechanics, and should have been considered as the first theory of quantum physics. I don’t need to touch Einsteinian mechanics as it is a combination of Newtonian and Maxwellian mechanics.

  Next, I will conduct the more crucial test of quantum physics on the consistency of the resolution limit postulate.

The logic of quantization

  With the resolution limit postulate, all lengths and time periods are integer multiples of L₀ and T₀ respectively. The unit velocity v₀ is naturally defined as:

  v₀ ≡L₀/T₀                               (7)

  It is clear that all velocities have to be integer multiples of v₀:

  v = κv₀                                (8)

  Where κ is an integer.My immediate interest is in the atomic realm, where the quantum velocity is much lower than the speed of light. In this environment, the momentum p can be defined in the conventional way; that is the product of the mass m and the velocity v:

  p ≡ mv = m(κv₀) = m(κL₀/T₀)= κ{m(L₀/T₀)}            (9)

  Since L₀ and T₀ are irreducible units, it follows from (9) that momentum is quantized. This fits the observations in the quantum realm, and serves as the first justification of the resolution limit postulate.

  Next I consider an electron moving in an idealized circular orbit in the Hydrogen atom. Since L₀ is the resolution limit of space, the following must hold true:

  2πr = ηL₀                               (10)

  Where η is also an integer. Work out the mathematics for angular momentum pr, I find:

  pr = p{(ηL₀)/(2π)} = η(pL₀)/(2π)                  (11)

  Thus angular momentum is quantized. This fits observations in the quantum realm and serves as the second justification for the resolution limit postulate.

De Broglie's wavelength as spatial resolution limit

  Comparing (11) with the well known relationship for the Bohr’s Hydrogen atom, namely:

  pr = ηh/(2π)                             (12)

  Where h is the Planck constant. I get the relationship:

  L₀ = h/p                               (13)

  Comparing (13) with the famous de Broglie’s hypothesis introduced in 1924, where λ_d is the de Broglie’s wavelength:

  λ_d = h/p                               (14)

  I find:

  L₀ = λ_d                               (15)

  Since in the case of light the de Broglie wavelength λ_d is the wavelength λ, by comparing (15) with (5) I conclude that the resolution limit postulate has given me a consistent description of quantum physics, with Maxwellian mechanics being a part of it.

 &esmp;At the time of this writing, the point assumption is still the status quo of all branches of physics, including quantum mechanics. Specifically, many quantum physicists hold on to the point assumption because they have run into difficulties with many models along the same line as the resolution limit postulate that I proposed in this paper. For example, the probability numbers would not add up to 100% in the theory of quantum electrodynamics (QED), as pointed out by Richard Feynman in his classic book "QED - The Strange Theory of Light and Matter".

  Then why did I propose the resolution limit postulate? My answer is that quantum mechanics should not have resorted to probability in the first place! As long as probability is still in its foundation, quantum mechanics will continue to be a collection of many brilliant patches with no overall consistency; but once probability is removed, the resolution limit will work wonders, and all “quantum weirdness” will fade away like morning fog on a summer day. I know that this is a big claim, therefore I had to write the book "The End of Probability and the New Meaning of Quantum Physics" to present my case in detail. You can get a feel for why probability has no place in quantum physics (despite what the experts claim), by reading my other article “Logic for the End of Probability”, also at this website.

  The de Broglie’s wavelength was first postulated as the wavelength of a physical “quantum wave” then later modified to mean the wavelength of a “probability wave”. The “probability wave” postulate was a very big scientific controversy when it was first introduced in 1926 and remains so today.

  The resolution limit postulate has two advantages over the probability wave postulate:

  1. There is no need to assume that a quantum wave exists.

  2. There is no need to assume that quantum phenomena are inherently probabilistic. The apparent probabilistic nature of quantum phenomena can be explained as a manifestation of the resolution limits of space and time.

  This point was demonstrated in my treatment of the Heisenberg’s uncertainty principle, which is also posted on this website.

  I hope you will at least agree with me that, the phenomenon of quantization is a natural consequence of the resolution limit postulate, whereas it can only be accounted for after-the-fact in an ad hoc manner with the point assumption. Thus, the resolution limit postulate may qualify as a fundamental ground of all branches physics, while the point assumption most likely will lead physics to the dead end of incompleteness.

  This point alone makes a very strong case for the resolution limit postulate.