Dimensionality in Physics by Armin Nikkhah Shirazi

Dear Marcel,

Thank you for your comment. My definition could have been formulated differently. For example, I could have defined the absolute dimensionality of an object in terms of the number of basis vectors of the vector space imposed on the space occupied by the object. But this overly technical way of defining things is in my view unnecessary when most people have a good grasp of what dimensionality means.

Thanks again,

Armin

Dear Juoko,

Thank you for your comments. I did not claim that "all can be derived from "lengths"". Rather, I claimed that dimensional abatement is a more fundamental way of thinking about relativistic length contraction.

I had a look at your paper, and I will shortly write a comment. To prepare you, it will contain tough but honest criticism of your work. Of course, the format of this essay contest discourages this kind of thing, but for this contest, my main objective is not to win but to get as many professional physicists, mathematicians and philosophers as possible to read my two papers and seriously consider the ideas presented therein.

All the best,

Armin

Dear Joe,

Thanks for your comment.

Armin

Dear Edwin,

Thank you for your kind words. I will directly address some of your specific comments below:

"I very much like your explanation that Einstein's invariance of the laws of physics first focused on "the independence of the speed of light from the speed of its source" but this was then recast "in terms of the invariance of the speed of light." Thank you! These are not the same and you are the first I have observed to contrast them!"

You may find the discussion in the article by Baierlein which I cited of interest.

"In my essay The Fundamental Nature of Time I (as have others) propose local gravity as the medium through which light propagates, i.e., "the ether". In this case the independence from the speed of its source is preserved, but the invariance of the speed of light (in all frames) is not!"

Well, from my point of view, this is a rather delicate issue. As I mentioned in my paper, one cannot have Minkowski spacetime without at least some object with a non-zero proper time and that implies at least some object with non-zero mass, which, by the equivalence principle implies, non-zero gravity field. So I am sympathetic to the idea that gravity is lurking in the background even in Minkowski spacetime, and I have some of my own ideas of how I would make that explicit, but this is right now a backburner project for me.

"You then reconceptualize the Lorentz contraction in terms of dimensional abatement. As I understand this you are assuming that Lorentz contraction is physically real."

As real as anything in a 3-dimensional slice of spacetime, which, I mentioned in my paper, is the arena of our reality.

"In the literature the Lorentz transformation is always derived between two different inertial frames, each of which has its own 'universal time' dimension. I have in An Energy-Based Derivation of Lorentz Transformation in One Inertial Frame derived the Lorentz transformation in one inertial frame, and shown that any length contraction is only apparent. The corresponding 'time dilation' that is implied by space-time symmetry is re-interpreted in terms of energy-time conjugation, and the result is that "the relativity of simultaneity" vanishes and time regains its meaning as universal simultaneity."

Well, I would have to look at how you implement this, e.g. whether you are claiming that relativity of simultaneity is not "real" in some sense but still necessary in our mathematical description of spacetime events. But let me just say that eliminating simultaneity altogether is in conflict with the very geometric structure of spacetime.

"In short, our assumptions differ, but we both treat the Lorentz-based relativity in novel ways. I hope you enjoy my essay as much as I have enjoyed yours, and I welcome any comments you might have."

I will be happy to look at your paper.

"You consider a body moving with v = c. But is this limiting case really possible?"

Well, certainly it is not possible for any spacetime observer to attain that speed in space in any frame.

"If I understand you correctly, you then attribute "redundant dimensionality" to such a hypothetical frame. You trace this to the assignment of a 4D coordinate frame."

The third bullet point claimed that in a speed of light frame it is *spacetime* which has redundant dimensionality. I do not trace this to the assignment of a 4D coordinate system but to the fact that in such a frame both the timelike and the spacelike direction of motion become lightlike, and therefore linearly dependent.

The mention of "assignment of a 4D coordinate frame" comes in when I examine how this conundrum can be easily solved: You cannot assign a 4D frame to such objects, but that does not mean you cannot assign a coordinate frame whatsoever (as it is commonly believed today)! The what seems to me in retrospect obvious solution is that you assign a 3D frame, which in spherical coordinates wholly contains a lightcone without any matter in it.

"My Lorentz derivation in one frame supports only one time dimension. The 4D Minkowski rotates time into space, but if only one time dimension exists, it only projects time onto itself, leaving 3D objects to rotate and translate in 3D."

The orthodox explanation for the existence of two distinct kinds of time parameters is that this simply reflects the fact that observers in relative motion "slice up" spacetime differently. When we get right down to it, according to special relativity there are no objects in space for which "time passes". There are only extremely long and narrow and often branching 4-dimensional tubes. However, I can understand that this may not be such a satisfying answer, and that worries that special relativity implies an infinite number of time dimensions may persist.

In part 2, the companion paper to this one, I address this issue in a novel way: Proper time is reconceptualized in terms of duration of existence in spacetime. In that way, time dilation becomes reframed in terms of a comparison of the observed durations of existence of objects during an interval of the observer's own duration of existence. In this way, time dilation becomes an entirely about a comparison of different local phenomena, as opposed to different global phenomena: There is only one time dimension, but different objects may differ on their relative durations of existence in spacetime. I know these ideas are very unfamiliar, hopefully my paper will be finished soon and, if you like, you can read the details there.

"You link the dimensional reduction to invariance of the speed of light. Does reduction occur in ether - either gravity or 'quantum vacuum'?"

Well, this is another delicate subject which I cannot adequately answer at the moment because the distinctions which are necessary to give a satisfactory answer have not been defined yet. However, the companion paper will address this issue: The second part shifts from special relativity to quantum mechanics.

"When discussing photons, it's fascinating to note that the Maxwell-Hertz equations are Galilean invariant. I only recently learned this, and I believe it is significant."

Perhaps...I will have to look at it.

I've not yet understood how this relates to your demonstration that "magnetic fields are line integrals of dimensionally reduced versions of electric fields". I will try to study this until I understand it. Nor have I understood whether this depends upon the Lorentz transformation. It does not seem to, at first glance, but this may conflict with your appendix.

Well, since I am not familiar with the Maxwell-Hertz equations, I cannot tell, either (yet?). But there is a clear relationship with the Lorentz transformations which I did not have enough space in my paper to point out. In some textbooks, magnetic fields are explained as a "relativistic effect" by imagining the following scenario: a point charge near a wire in which the uniformly drifting electrons are exactly canceled by the stationary positive charges in the cable material "observes" a net zero electric field. Now, another point charge moving parallel to the wire will observe the wire to be contracted, which means that the charge density increases, but because the relative motion of the positive charges is different from the relative motion of the negative ones, the charge densities increase by different amounts and therefore no longer cancel, so that the moving charge "sees" a net electric field, which exerts a force that, when transformed back to the original charge has exactly the form we attribute to the magnetic force.

What this shows is that Lorentz contraction plays a key role in generating magnetic fields, but the usual ways of conceptualizing it are indirect, involving a transformation to a moving frame and then transforming back. My analysis is direct: Lorentz contraction can be described as dimensional abatement, and if it causes some phenomenon, then that phenomenon must also have a description in terms dimensional abatement. Fields are infinitely extended objects, so dimensional abatement there cannot occur by means of length contraction since an infinitely long object contracted by any finite factor is still infinitely long. Rather, it occurs by superimposing E and B fields (notice how it makes no sense to say that an E field is "length contracted", we can only talk about the field strength of components of the field at a point in space).

Incidentally, I feel that this point should just be visually very obvious, but the feedback I have gotten so far makes it seem as if it may not be? So let me ask you: Imagine a Coulomb field; all the force arrows are aligned radially in 3 dimensions. Now imagine the magnetic field of an infinitely long straight wire. All the force arrows are aligned radially in 2 dimensions, and to get a 3D description you have to integrate over the length of the wire. Does this not strike you as a visually obvious example of my claim that B-fields are line integrals of dimensionally reduced analogs of E-fields?

Of course, there are more complicated examples, like magnetic dipole fields etc. but these complications do not negate the underlying simple structural relationship between E and B fields. As I mentioned in my paper, this is mathematically implied by the fact that to transform from a frame with a pure E-field to a frame with a pure B-field requires v=c, which indicates dimensional reduction.

"You conclude that dimensional abatement is a more fundamental concept of nature than Lorentz contraction. In my essay Lorentz transformation is interpreted as an energy-time interpretation of reality, not a space-time reality. I wonder if you will see if our papers make sense together, or are sent down different paths by being based on different assumptions."

I should have a better idea once I read your paper.

Thank you again for reading my paper and for your extensive essay comments.

All the best,

Armin

Dear Peter,

Good to hear from you. I will reply to some of your comments below:

"Having spent a good part of my career as an instrumentation specialist watching the highly relativistic beams in RHIC, designing pickups to measure their various properties, and spending a little time thinking about just what i was actually looking at, i find your unconventional approach to the implications of SR hard to mix with my pretty much congealed worldview of how things behave relativistically."

First, I am glad that you have an awareness that your views reflect a particular worldview. Since my worldview is probably very different, it is not so easy for me to discern which of the things I mentioned you find "hard to mix" with yours:

a. Dimensional diminution is mathematically consistent with Lorentz contraction at speeds less than c, and dimensional reduction is consistent with the complete Lorentz contraction of a body characterized by v=c.

b. I am unaware that anybody had ever pointed out the invariance of absolute dimensionality or the homodimensionality of space, but I suspect that is just because it was "too obvious".

c. Probably the most novel idea in my paper is that one can assign coordinate frames to speed-of-light objects as long as they are 3D, not 4D, and I made a case that this is not only consistent with SR, but hinted at by other parts of the theory, such as the fact that null-vectors have only 3 independent components. I think the resistance to the idea does not come from SR but because this realization is genuinely foreign to the contemporary worldview; It is like a piece of a jigsaw puzzle that doesn't fit anywhere. My reaction is simply that after over a 100 years, special relativity still holds some surprises (and that is nothing compared to the analogous claim in the second paper, ha!)

d. The claim about magnetic fields being line integrals of dimensionally reduced versions of electric fields seemed visually so obvious to me that I did not bother to design diagrams directly comparing the force field of a Coulomb field and the magnetostatic force field of a current, but I am coming to regret that now because, to my shock and bafflement, apparently it does not seem obvious to others. Let me ask you as well: When you compare the direction of the forces of a Coulomb field and a magnetostatic field of a current, does this relationship not immediately jump out at you? The force field in one has spherical symmetry, and the force field of the other has circular symmetry, which, when integrated over the current, gives you just the relationship I claimed. More complicated field arrangements make this relationship a lot less obvious but that doesn't change anything because this relationship holds at a differential level. Of course, the mathematical evidence in the appendix requires that it is already recognized that v=c implies dimensional reduction. Maybe that is what you had difficulties with?

"Hard to get a sense of just how the pieces fit together, what happens to the scale invariant properties of various models,..."

I am not sure which models you are referring to, but let me say that I reject scale invariance as a fundamental principle of nature because the ratio of different powers of length changes with scale. Most relevantly to us, the relationship between surface area and volume changes with scale. For example, a ball of radius 1 meter has 100 billion times as much volume per unit surface area than a ball of Bohr radius. When you consider this together with density, this profoundly affects the behavior of objects at different scales. I consider this at bottom the reason why, for example, we don't see large rocks floating in the air or why only planet-sized or larger objects eventually turn into round balls. To paraphrase Philip Anderson, as far as I am concerned, bigger is different.

"Assumption i guess has to be that everything you're doing is consistent with SR, that the equivalence is either proven or provable."

I listed the four major claims of my paper above. Tell me please, so I have input from an outside perspective, which of the four you are skeptical about (more than one choice is of course okay)

"Will part two address the quantum? "

Absolutely! This is the reason my paper is getting delayed. I had originally intended to only touch on some quantum concepts, but then I realized that it is really hard to just try to give a brief glimpse without it being confusing. So I have kept adding details, and as the result the paper is getting longer, and I am still not done.

"Clifford algebra is the language of QM. What is missing in mainstream is the geometric interpretation of the algebra. Geometric product of Geometric Algebra mixes dimensionality. For instance the product of two lines is a point and a plane. I'm not aware of any calculations in the literature showing some sort of smooth deformation (of what? geometry of electric and magnetic fields?) during evolution of the geometric product, showing just how two lines gradually morph into point and plane during dimensional abatement or dimensional 'enhancement' (got a better word for this?)"

Although my knowledge of Clifford Algebra is very little, I tend to think that it might hold the key to some profound insights yet to be discovered. I don't know enough to be able to assess your claim that it is "the language of QM". My first reaction is that we use different mathematical languages of QM depending on our needs. Whether we use Hilbert spaces, Path integrals or Clifford algebra, it seems to me, is dictated by the physics.

As for how the geometric product of two vectors "happens", I suspect that it does not reflect a smooth deformation of the kind you wonder about, but instead reflects a system of symbolic manipulations which exhausts the ways in which two vectors can be usefully combined in a way that conceptually qualifies as a `product'. I take this `exhaustiveness' of Clifford Algebra to be its strength.

"Curious regarding how your ideas might be applied to details of the interaction of two geometric wavefunctions (comprised of point, line, plane, and volume elements)."

Well, one difficulty I see right away is that wave-functions live in configuration space, whereas my ideas apply to real space. If anything, it seems to me that it would have to be the path integral which is affected by them.

Thank you for reading my essay and your thoughtful comments. I will read your essay soon and provide feedback as well.

All the best,

Armin

Dear Silviu,

Thank you for reading my essay and commenting on it. to answer your question, yes "fundamental" is a property that I argue is not intrinsic to anything because it depends on the background worldview, the paradigm in which it is considered, and so it can very well be thought of as an idea or point of view.

I don't quite understand what you mean by saying it "will not be fundamental in the search of what is fundamental", but the most charitable interpretation I can give is in agreement: If something that hints at the next paradigm is supposed to be fundamental, then, once the next paradigm arrives, then it will no longer be fundamental because that term would be reserved for things that point to the paradigm after.

This presupposes a Kuhnian worldview according to which progress in science is not really a progress toward the truth; although I share a lot of views of Kuhn, on this issue I tend to be skeptical of them. I like to think that in some sense we do progress more toward some kind of truth, which perhaps could be expressed in many ways that seem different but turn out to be equivalent. If that is true, and there is such a thing as a final paradigm, then the conception of fundamentality I proposed will fail at that stage. However, I am not worried about that too much.

Thank you again,

Armin

Dear Terry (if I may),

There seems to be a slight mix-up in that you replied to Silviu's response to your comment by name, yet indicated something in your last paragraph which suggests that you may have thought that you were addressing me. Indeed, I am coming a little late to this conversation, but let me thank you right away for taking the time to read my essay and making some thoughtful comments. Also, I am flattered by your appellation, but I am not a professor, just a researcher. I would like to respond to some of your comments in detail.

First, your essay contestant pledge. What a wonderful idea! I have participated in this essay contest every single time except the last, and I have expressed my distaste for the author rating system both to the moderators and even to Max Tegmark himself. I understand why he does it, but unfortunately it comes at great costs, including that ratings are invariably bound to be influenced by factors other than the rater's best judgement of the essay's quality. With your permission, I would like to forward the pledge to Max, and suggest that either it or something similar be made a requirement for participation in future contests. It won't deter the determined colluders and cheaters, but it will help make the whole thing less farcical, I believe.

Second, some thoughts on your feedback on the essay itself:

"In terms of physics, I think your observation that there is a unbreakable dimensionality barrier between sub-light (massive) and light speed (massless) particles is a nice alternative way of saying massive versus massless."

It is actually more subtle than that, as I will try to show In the second part of the 2-part series of which my essay entry was the first part. The second part will, after developing an analogous reconceptualization of time dilation, apply these ideas to quantum mechanics and derive some novel distinctions, including one involving mass.

"I don't see a new invariant in this. I've read your dimensional abatement several times, and every time I get back to the same conclusion: This is still one-dimensional Lorentz contraction, just with more dimensions added."

I understand your point of view and agree that by itself, dimensional abatement does not contribute anything not already contained in Lorentz contraction.. However, in the second paper I will combine it with its analog for time dilation to derive certain insights that could not be derived using just the concepts of length contraction and time dilation and use these as a starting point exploring for my real target: quantum foundations.

"I don't immediately see the advantage of defining the photon as 2D, even though I find that an intriguing idea."

Thank you, at this stage I agree that the value is more philosophical, but I am glad that you are open-minded enough to consider that this really is an implication of special relativity which has gone unrecognized so far.

Photons are... odd, and complex, and can push the limits of quantum theory even now if you dig deep into them. (I have a great little book on that that I seem to have lost, hmm...I)

I agree and I would love to find out which book you have in mind, if you remember.

"For example, if you imagine a photon as a 3D sphere (technically a 2-sphere for purists) with a point orbiting around its equator, you can model all possible polarizations, from linear through elliptical to circular, as nothing more than different orientations of the sphere that is then projected (made 2D) along the axis of propagation. The cohesive value in that model of treating the photon as a 3D sphere before making the final 3D to 2D projective reduction in its dimensionality says to me that one should not simply discard the 3D view as irrelevant even for a particle traveling at c."

I was unaware of this model and it sounds very interesting. As you know, while the photon is according to SR completely length contracted, it does have a wavelength and period in the direction of motion, and this can result in 3D-behavior associated with photons, of which certain polarization states like circular polarization are an excellent example. So I agree with you: My claim that photons are dimensionally reduced was not meant to make any 3D conceptualizations of photons irrelevant. Rather, it was meant to elucidate what I believe to be an overlooked implication of special relativity.

For example, if you imagine a photon as a 3D sphere (technically a 2-sphere for purists) with a point orbiting around its equator, you can model all possible polarizations, from linear through elliptical to circular, as nothing more than different orientations of the sphere that is then projected (made 2D) along the axis of propagation. The cohesive value in that model of treating the photon as a 3D sphere before making the final 3D to 2D projective reduction in its dimensionality says to me that one should not simply discard the 3D view as irrelevant even for a particle traveling at c.

"Here's my biggest problem, and it's one that affects many of the large number of submissions this year: You did not answer the question that FQXi asked, which was to explain what makes a theory "more fundamental". "

My answer to the essay question is at the top of page 2:

"And this gives us a clue for identifying the most fundamental things of a theory within any given paradigm: it has to be those things which point to, or at least hint at, the next paradigm."

I structured my essay so that the theory I presented was an illustration of that answer. Now, if you had criticized my answer by saying that I did not sufficiently elaborate on it, or that I shoe-horned my own theory into the paper, I might have agreed with you, but to me, it seems a little unfair to say that I did not answer it at all. Be it as it may, my objective in this essay contest is not to win but to reach physicists, mathematicians and philosophers of physics who have a strong interest in foundational physics with some ideas which I believe are going to lead us to the next paradigm.

Actually, from your comments on another essay I saw, it appears that you do have an interest in the foundations of quantum theory, and so you are exactly part of the core of the target audience I am trying to reach. My essay entry was just meant as an advertisement for the second paper, which uses the ideas of the essay entry and a multitude of novel ideas to provide an ontology for the textbook interpretation of quantum mechanics. My hope is that my essay entry intrigues people enough so that they are willing to invest spending time and thought on the second essay, which is conceptually far more challenging than the first.

Thank you again for your well-considered comments.

All the best,

Armin

Dear Satyavarapu Naga Parameswara Gupta,

Thank you for your comments,

Armin