Faradave's View on Quantum Gravity

[hyksos]

[Mod Note: This thread has been split off from Faradave's Is Gravity Relative? thread, becasue it veered outside of the Physics section guidelines.]

I'm pretty sure this material is repeated over a dozen times by Faradave, in various parts of this forum. I already told Faradave that he appears to be deriving spherical wave mechanics in 3 dimensions, but replacing the textbook orthodoxy with his own neologisms. If he ever gets off the this forum and goes and studies this topic, he will find that indeed the amplitude of the wave falls off with distance, exactly in the way he keeps repeating his argument about "density of rope".

His hijacking of established physics terms (superposition) is making me itch, but there is nothing 'wrong' with it as far as forum etiquette.

By "model gravitation fundamentally", I assume he means he thinks he is deriving a model of Quantum Gravity.

[Faradave]

...physicists tend to think of 'real' as the result of measurements ...

The term Taylor & Wheeler used instead of "real", for the magnitude of an invariant 4-vector was "well defined", which I find agreeable.

"Despite this arbitrariness in the means that we use to describe the 4-vector AB, the 4-vector itself is well defined. Not only does the interval AB have the same magnitude in all frames of reference. More significantly, the locations of events A an B themselves...are as well...regardless of what coordinates one uses..."

They consider the 4-vector a unifying idea between invariant spacetime intervals (I) and invariant rest mass (m₀). In natural units (c = 1) we see:
ΔI² = ΔX² – ΔT² and Δm₀² = ΔE² – Δp².

I also find this agreeable, though I prefer their Euclidean representation.

...your idea of renormalization is very different from the mainstream meaning.

Understood. I simplistically interpreted, "Renormalization replaces the originally postulated mass and charge with new numbers such that the observed mass and charge matches those originally postulated. ... If the electron is assumed to be a point, ... the mass diverges, because the field is inverse-square."

Though limited, the rope field analogy is much clearer than QED. Renormalizing reveals some places it could be improved.

You accept that "rest mass" of a system includes the energy of its spin, an intuitive way of representing energy at rest. I push that concept in conceiving a fundamental particle as a spinning field element, which embodies mass-energy equivalence. QED sees the field as a far more convoluted swarm of self-interacting virtual particles. This seems to apply to electric charge but it neglects mass charge (thus "quantum electrodynamics"). Either way, an instance of a field (G or EM) corresponds to a particle's future light cone at that moment.


Tidal forces, which confound the equivalence principle, are avoided by Einstein's "sufficiently small region of spacetime". In the limit, as that region goes to zero, we're left with a single, lightlike field element, from which generation of a particle's field may be imagined with "chronaxial" spin.

Getting back to the OP, Einstein was able to elegantly deduce time dilation (duration contraction) with his imagined light clock. Length contraction follows from SR's 2^nd postulate, as the length:duration ratio is preserved by invariant c. I believe "relativistic mass" should be as simply derived. Not finding that, I attempt it below.

Recall the light clock scenario, considering only the half paths needed. Light path t' relates to t by equivalent expressions √(1-v²) and cosθ.

The light clock relates paths t & t' by both Pythagoras and trigonometry.

Now, consider a massive particle, the designated center of a field. It's persistence defines a particle worldline, vertical in the particle's rest frame. Regular instances of that field can be marked along it. An identical particle, seen to be in motion (here 0.71c), has a worldline at 45°, in Euclidean coordinates. A triangle so defined is similar to the light clock’s in that the angle between the vertical and diagonal paths goes to 90° as v goes to c.

Here, worldline angle (θ) indicates speed. A moving particle (m) is seen to accomplish more instances of its native field per unit of rest time of (m₀). Thus, observed mass (m) increases with v toward limit c (90°).

Like a tilted stack of checkers, a motion worldline registers more field instances per vertical unit (proper duration) of an observer's rest frame. This holds, even in the limit as that duration goes to zero. A resting observer will therefore measure a greater gravitational field intensity for the moving particle, which implies greater "relativistic" mass. Similarly, more instances of a particle make it harder to push.

But moving clocks run slow. Why should a moving object have more instances of its field?
A diagonal worldline is the proper time coordinate for the moving particle. Like the tilted stack of checkers, it sees nothing amiss, the same density as always. Further, a moving particle is seen to have the energy to supply the extra instances observed. True, a moving clock is seen to run slow, but consider the slowed motions in this video.


This "slow motion" video requires a higher image rate (5,000 fps) than normal cameras (60 fps). Apparent motion slows asymptotically to a stop with increasing fps (frames or "instances" per second).

[BurtJordaan]

By "model gravitation fundamentally", I assume he means he thinks he is deriving a model of Quantum Gravity.

Yes, in that the field is modeled as arising from personally characterized quantum spin (i.e. chronaxial). But I don't adhere to string theory or extra dimensions and I do unify G & EM simply (as a Personal Theory). So, there are distinctions from what's found in the literature.

This is exactly why these posts do not belong in the Physics section. It becomes extremely time consuming to filter out (and correct) all the false arguments, of which there are quite a few in the last few posts. Until I have the time, this thread is closed, pending moderation.

[BurtJordaan]

They [Taylor & Wheeler] consider the 4-vector a unifying idea between invariant spacetime intervals (I) and invariant rest mass (m0). In natural units (c = 1) we see:
ΔI2 = ΔX2 – ΔT2 and Δm02 = ΔE2 – Δp2.

I also find this agreeable, though I prefer their Euclidean representation.

The problem is that space-time is not Euclidean. Add to that confusing terms like "radiant energy (p)" and "light interval (I)" and you are creating complete confusion. Neither of these terms have the meanings in physics that you attempt to attach to it.

QED sees the field as a far more convoluted swarm of self-interacting virtual particles. This seems to apply to electric charge but it neglects mass charge (thus "quantum electrodynamics"). Either way, an instance of a field (G or EM) corresponds to a particle's future light cone at that moment.

Never heard of "mass charge" - what's that? Please also define "instance of a field (G or EM)" in standard terms (or at least provide a reliable reference).

In the limit, as that region goes to zero, we're left with a single, lightlike field element, from which generation of a particle's field may be imagined with "chronaxial" spin.

What is a "lightlike field element"? And "chronaxial spin" has been refuted by Lincoln, myself and others in the past. Why are you persisting with it?

I believe "relativistic mass" should be as simply derived. Not finding that, I attempt it below.

It has been simplistically derived by Einstein, in its energy form.

I think your next diagrams have the potential to create a lot of confusion. It mixes two reference frame views into one diagram. It is really an Epstein space-propertime diagram, which never caught on due to its confusing mixing of "my space" and "your time".

But moving clocks run slow. Why should a moving object have more instances of its field?

Since I assume you are talking inertial movement here, there are no "moving clocks", only clocks that move relative to reference frames and then the situation is reciprocal - each clock "run's slow" according to the others perspective. Hence moving clocks do not run slow. It is just an observational perspective.

Also, in your scenario, no clock has "more instances of its field". It is only the perception of the "other guy". One of the pitfalls of your diagrams.

The only time you can have an absolute (frame independent) difference between elapsed times is when you consider a two-way situation like in the classical "twin-paradox". We had plenty of threads on that in this forum, so no need to belabor it again. The point is: do not try to build your private theory of quantum gravity on false perceptions.

[Faradave]

false arguments

The numbers work! The interval equation (X² = I² + T²) and energy equation (E² = m² + p²) are analogous. Further, a mass triangle and Einstein's energy triangle must agree, by mass-energy equivalence.

The problem is that space-time is not Euclidean.

I didn't say it was. But Einstein's light clock and energy triangles are Euclidean! That should be sufficient to question naïve acceptance of archaic space and time coordinates to represent the continuum.

Fortunately, we aren't stuck with clumsy spacetime. It has legacy value but it's not taking us anywhere new. As you've informed us, physics also employs light cone coordinates, which entail lightlike and spatial coordinates. It's slowly catching up to my combination of lightlike and temporal coordinates, Times Square (T2).

[Your] confusing terms like "radiant energy (p)" and "light interval (I)" ...are creating complete confusion. Neither ... have the meanings in physics that you attempt to attach

I'm sorry about any confusion. "Radiant energy" is nothing special. Taylor & Wheeler use it for kinetic energy, particularly in the case where a "particle" has zero rest mass. Such radiant energy transmits via lightlike intervals.

We also know radiant energy as the "momentum component" (pc)* of the total energy, even for massive particles. An equivalent term would be"radiant mass".

*just p in natural units, where c=1

Never heard of "mass charge" - what's that?

As I consider energy to be the more fundamental aspect of mass-energy. "Mass charge" is energy in any form. It could also be called "inertial charge" or "gravitational charge" by the equivalence principle. I meant to invoke both latter terms with the former.

Please also define "instance of a field (G or EM)" in standard terms (or at least provide a reliable reference).

An instance of a particle should be understood to be an event along that particle's worldline, i.e. the particle at a particular instant. Each instance of a particle has an associated future light cone, which defines the entirety of its potential causal interaction. That future light cone corresponds to an instance of that particle's field.

A field is a region of potential force pairing (e.g. potential interaction with a test particle.). As QED sees forces mediated by massless force carriers, an instance of a field is the future light cone of a particle. It is the collection of all potential forces, which can originate from an instance of a particle.

What is a "lightlike field element"?

"A cone is formed by a set of line segments, half-lines, or lines connecting a common point, the apex... in the case of half-lines, it extends infinitely far. ... The definition of a cone may be extended to higher dimensions." Let the half lines (i.e. rays) be "radial elements". A lightlike field element is a radial element of a future light cone (in 4D).

A more elegant and insightful definition is that a lightlike field element is an unpaired force. This sees force as an object, not unlike its conventional treatment as a vector. I model a force (i.e. field element) more fundamentally, as a particle-interaction wormhole (pinhole). With chronaxial spin, a particle's field is generated.

And "chronaxial spin" has been refuted by Lincoln, myself and others in the past. Why are you persisting with it?

1. You're too intelligent to continue denying one of four available, independent spin axes.
2. It's the least I can do, considering all I've learned from you.
3. With all due respect, said refutations have been insubstantial. (Remind me, if there was a good one.)
4. Chronaxial* spin is the best and only available explanation of quantum spin. QM is woefully incomplete without it.

*More generally, it occurs about a particle's worldline, which parallels proper time in the particle's rest frame.

["relativistic mass"] has been simplistically derived by Einstein, in its energy form.

Good for him! My feeling is that length contraction, time dilation and relativistic mass-energy all occur together and if the light clock example gives us one, it should give us all three.

I think your next diagrams really an Epstein space-propertime diagram, which never caught on due to its confusing mixing of "my space" and "your time".

Yes! My post included a link to Epstein diagrams. He and Minkowski both suffer with spatial coordinates. The simplicity of Epstein diagrams is retained by replacing spatial with invariant lightlike coordinates, agreeable to all observers. As I showed, the geometry is as elegant as with the light clock.

...you are talking inertial movement here, there are no "moving clocks" ... each clock 'run's slow' according to the others perspective. It is just an observational perspective.

Those are equally legitimate "observational perspectives", wrt physical law. As you know, SR allows no privileged inertial frames. There was nothing about Einstein's light clock example nor my analogous diagram that would deny reciprocal perspectives. In fact, that helps legitimize them.

no clock has "more instances of its field". It is only the perception of the "other guy".

I beg to differ. We agreed that gravity is relative. I now provide a mechanism.

Two stacks of coins oriented at an angle to each other, each think the other stack has more coins per its own vertical measure. Do the same for instances of a gravitational field along angled worldlines and you've got relative gravity.

Parallel lines from either stack measure higher coin counts in the other angled stack. Parallel simultaneities measure mutually higher temporal densities of gravitational fields in different inertial frames.

Length, duration and mass (inertial and gravitational) are indeed frame dependent. I take that as the meaning of "relative" in SR. Additional terms like "reciprocal" and "mutual" apply but are redundant.

[hyksos]

No machinery has ever been invented that “explains” gravity without also predicting some other phenomenon that does not exist.

-- Richard Feynman. (Caltech, 1963)

[hyksos]

http://www.feynmanlectures.caltech.edu/I_07.html

[Faradave]

I love the Feynman lectures, which I own on CDs (several sets) but I especially like his videos.

No machinery has ever been invented that "explains"” gravity without also predicting some other phenomenon that does not exist.

So far, yes. Nevertheless, gravity is here along with all the phenomena which do exist. It is thus, also the case for its explanation (my goal).

[hyksos]

I hope that my words will not come across as combative or inflammatory. Where ever I say something which might be construed as such, I have placed the qualifier (advice) next to it.

(advice) Wikipedia is for reference, not for learning.

The following wikipedia links should act as 'portals' for your own reading, as jumping-off points to deeper research. There are three main 'thrusts'/'directions' in physics regarding how gravity acts at the smallest scales. Roughly in the order they were introduced in history,

• 1.) Canonical quantum gravity.
  https://en.wikipedia.org/wiki/Canonical_quantum_gravity
• 2.) Loop quantum gravity
  https://en.wikipedia.org/wiki/Loop_quantum_gravity
• 3.) Spin foam
  https://en.wikipedia.org/wiki/Spin_foam

I will include two more notable mentions. I separate these from the 'main three' directions, because in some cases they fit underneath them as subheadings. But the wall-of-text will often obscure these topics, and they would be unfortunate to overlook.

• 4.) Semi-classical gravity.
  https://en.wikipedia.org/wiki/Semiclassical_gravity
• 5.) Euclidean quantum gravity
  https://en.wikipedia.org/wiki/Euclidean_quantum_gravity

Your posts on this topic in the forum are closest to semi-classical gravity -- as I can see you making references to the "gravitational field". You refer to the field in the classical sense, rather than say, a quantum field. Euclidean QG (number 5 above) may be of interest to you, particularly the part about Wick rotation. You seem to have put a lot of thought into rotations performed in Minkowski space. But don't let that go to your head -- they are using Wick rotations to simplify a path integral. The mathematics is very erudite.

(advice) Speaking in rough terms, it is unlikely you can make any progress in this topic using college-freshman-level mathematics. Claiming that you have a theory of quantum gravity, and then presenting the theory with triangle diagrams (that look like repeated derivations of the Lorentz factor), this will only lead to pain.

Remember what Feynman said,

"Finally, let us compare gravitation with other theories. In recent years we have discovered that all mass is made of tiny particles and that there are several kinds of interactions, such as nuclear forces, etc. None of these nuclear or electrical forces has yet been found to explain gravitation. The quantum-mechanical aspects of nature have not yet been carried over to gravitation. When the scale is so small that we need the quantum effects, the gravitational effects are so weak that the need for a quantum theory of gravitation has not yet developed. On the other hand, for consistency in our physical theories it would be important to see whether Newton’s law modified to Einstein’s law can be further modified to be consistent with the uncertainty principle. This last modification has not yet been completed."

[BurtJordaan]

false arguments

The numbers work! The interval equation (X² = I² + T²) and energy equation (E² = m² + p²) are analogous.

They may look the same, but they represent totally different concepts. Saying they are analogous is a great way of getting everyone confused. Here's why. The spacetime interval equation embodies the hyperbolic nature of spacetime. Note that takes the place of your here.

The spacetime interval definition

If we solve these equations, we get a spacetime diagram like this:

The hyperbolic nature of spacetime

The energy equation has the rest mass taking the position of in the interval equation, but they are not related to each other. The other two (total energy E and momentum p are self explanatory, but would a diagram like the hyperbolic spacetime depicted make any sense with mass in place of ? Certainly not. Hence, the two concepts are not related.

[Your] confusing terms like "radiant energy (p)" and "light interval (I)" ...are creating complete confusion. Neither ... have the meanings in physics that you attempt to attach

I'm sorry about any confusion. "Radiant energy" is nothing special. Taylor & Wheeler use it for kinetic energy, particularly in the case where a "particle" has zero rest mass. Such radiant energy transmits via lightlike intervals.

The standard meaning of "radiant energy" is given in this Wiki. I do not know how Wheeler and Taylor got away with that, but I have never seen that in any paper - I presume it is because books are not refereed! In any case, to insinuate that one can use the same term for a massive particle's kinetic energy seems quite absurd.

OK, I'll leave you alone with your quantum gravity effort now, and only comment where I think statements are to detriment of this forum. I see Hyksos has given you great advice.

=J

[Faradave]

[The interval equation (X² = T² + I²) and energy equation (E² = m₀² + p²)] ... may look the same, but they represent totally different concepts. ... The energy equation has the rest mass taking the position of [the spacetime interval] in the interval equation, but they are not related to each other.

Let's think about that. At rest, a mass proceeds along a vertical worldline, corresponding to its proper time coordinate. So, corresponding rest components are m₀ and ΔT*. Each equation has a spatial component (E & X) and a motion component (I & p).

Mere coincidence? The geometric representations of space-time-interval separation and total-rest-radiant energy correspond exactly, suggesting the former is the coordinate system upon which the latter occurs.

Each equation also has invariant terms (m₀ & I) while angle (θ) corresponds. Perhaps nature is telling us something.

*For simplicity, I omitted Δs from my interval equation, presuming the values are represented from the origin.

The spacetime interval equation embodies the hyperbolic nature of spacetime.

OK. But it's just as well to say that rearranged, the equation represents the Euclidean nature of Times Square (time & light coordinates). It's not unusual for science to get things a bit sub-optimal the first time. Spacetime coordinates have their use but lack a justification so robust as to preempt all others.

If we solve the [conventional] equations, we get a [hyperbolic] spacetime diagram

And if we solve my single rearranged equation: X² = T² + I², we get a simpler diagram. One in which the otherwise inexplicable future "light cone" becomes a simple "light disk". Speed limit c exists as a consequence of the unidirectionality of time and the mysterious quantum spin becomes clearly evident by its spatial projections, exactly as measured.

...to insinuate that one can use ["radiant energy"] for a massive particle's kinetic energy seems quite absurd.

It needn't seem that way. Consider three related scenarios:
1. An electron with kinetic energy (KE) is seen to have velocity (v). As KE increases, v approaches lightlike limit c, suggesting a lightlike influence of KE.
2. The electron's northeast velocity can be said to have components which are north and east, even though it never travels due north or due east. In the same way, an electron worldline which makes an angle with your proper time can be said to have timelike and lightlike components, though it is not seen at rest or at speed limit c.
3. The electron annihilates with positron but their kinetic energy is not lost. It remains lightlike, recoiling away (with the rest energies of the annihilated particles) as gamma emissions.

[BurtJordaan]

OK. But it's just as well to say that rearranged, the equation represents the Euclidean nature of Times Square (time & light coordinates). It's not unusual for science to get things a bit sub-optimal the first time. Spacetime coordinates have their use but lack a justification so robust as to preempt all others.

FD, I've started along the same route (approx your "Times Square") two decades ago and thought that I found an improvement on standard SR representation. In the end I realized the limitations and the potential confusion that it can create. The reason is that whether you regocnize it or not, it mixes reference frame values in a confusing way.

I found that such efforts lead to no valid insights that Minkowski spacetime does not offer (provided one spend the time to understand them properly). Plus, Minkowski offers a whole bunch of thing that your "space-propertime" does not, like half of Minkowski spacetime is spacelike - a quite important domain!
Eventually, I 'saw the light' and I did not have to look back again.

I will leave you with these closing remarks, because I do not intend to continue spending time on the reading of private theories. There is an immense amount of accepted science that I still have to read...
If and when your ideas become mainstream, I'll surely pick up on it again. ;)

Regards, Jorrie

[Faradave]

Wikipedia links The mathematics is very erudite.

Thanks! They're interesting but I have the impression that particles aren't so good at math. That doesn't mean the experts won't derive a workable model for gravitation. But history suggests, a working model with loads of epicycles may actually be yearning for a simple change of coordinates (from geocentric to heliocentric, in that particular case.)


"Any path—periodic or not, closed or open—can be represented with an infinite number of epicycles. ... This is because epicycles can be represented as a complex Fourier series; so, with a large number of epicycles, very complicated paths can be represented in the complex plane."

The methods of your references are going to be far more complicated than with my Times Square coordinates. Currently, an esteemed group are proposing my spinholes (spatial-interconnecting wormholes) as the basis of gravitation. A simple report is below. Here's a more sophisticated lecture.


.
That's fine but since spinholes have internal span, it's going to be hard to create a field which counters dimensional separation, all the way to a point of contact. It's much simpler using my zero-internal-span pinholes (particle-interaction wormholes) instead.

Claiming that you have a theory of quantum gravity, and then presenting the theory with triangle diagrams (that look like repeated derivations of the Lorentz factor), ...will only lead to pain.

It didn't hurt Einstein! I allow the "quantum gravity" characterization only because my theory identifies the chronaxial spin upon which it depends as instances of quantum spin (specifically, the spin½ of fermions). There IS a lot of relativity, thus a Lorentz factor must be apparent. Einstein would have gotten to GR faster, had he understood fields in terms of instances, as I illustrated above.

FD, ... whether you recognize it or not, [Times Square] mixes reference frame values in a confusing way.

Adopting lightlike coordinates does not mix different coordinate frames, it employs coordinates common to all of them (inertial or otherwise). That's a major advantage of invariant coordinates.

Minkowski offers a whole bunch of things that your "space-propertime" does not, like half of Minkowski spacetime is spacelike - a quite important domain!

Nonsense! First, Times Square (T2) coordinates are interval-proper time (specifically lightlike intervals), not "space-propertime". Space is derivative in T2.

The spacelike domain is not lost in T2, it's merely shifted down, entirely consistent with the SR assertion that faster-than-light travel is backward in time. That's something that Minkowski diagrams really screw up!

Unlike Minkowski spacetime, T2 correctly depicts the geometric independence of non-aging lightlike intervals by orienting them perpendicular to the aging coordinate (time).

I will leave you ... There is an immense amount of accepted science that I still have to read...

That's perfectly understandable. But it should be equally well understood (by Lincoln as well) that being busy is not a substantive refutation. The same goes for unfamiliarity, dislike, not being mainstream, etc. I've defended Phyxed against all substantive criticisms ventured.

If and when your ideas become mainstream, I'll surely pick up on it again. ;)

You'll be more than welcome Jorrie! Thanks for your (and hyksos's) thoughtful participation.

[hyksos]

Faradave,

You have linked a video from the Stanford Institute of Theoretical Physics. https://www.youtube.com/watch?v=WQU9yOtWrQk That lecture is about Maldacena's AdS/CFT correspondence. You claimed you are working on the same thing, (or even that you discovered this before Maldacena found it.)

Are you claiming to this forum that you have been doing mathematical work in the topic of dualities in string theory?

Or are you claiming that you are working on a conformal field theory?

[Faradave]

Are you claiming to this forum that you have been doing mathematical work in the topic of dualities in string theory?
Or are you claiming that you are working on a conformal field theory?

Heck, no! You seem anxious to put words in my mouth. I'm claiming that the esteemed group is taking an unnecessarily difficult path to gravitation. Their approach arises from the realization that entanglement is equivalent to non-traversable, space-like wormholes. [1, 2, 3]

I've previously shown you that I described non-traversible, spacelike wormholes as constituting entanglement in two other public forums (six total postings) three years prior to those authors' first publication. Here's one from PF and one from Amazon. I tried to contact the five authors to offer my term, "spinholes" but got no replies.

Whether I gor spinholes by math or magic is irrelevant! Spinholes were in fact, a logical spin-off of my reasoning toward pinholes.

Note: Penzias and Wilson discovered cosmic microwave background radiation, essentially by accident! (They tried everything they could think of to rid themselves of the signal!) That didn't stop them from getting 1978 Nobel Prize in Physics for their discovery. They got there first.

P.S. T2 is has 3 of 4 coordinates lightlike (invariant). I don't know if that makes fields so represented "conformal".

[hyksos]

Okay I'm reviving this thread -- mostly because I saw Faradave's youtube. I watched almost all the videos in one sitting. I guess this will count as my response to the youtube channel.

First things first, Minkowski spacetime is not a literal description of the world. It is a toy model of Special Relativity where space is squashed to 2 dimensions. I notice in the youtube vids, you keep talking about a "cone". Well there is no "cone" in the actual spacetime we are sitting within now. That spacetime is a 4-dimensional subject , with 3 Ds of space and another D that marks off local duration in a reference frame. The D=4 is not "time" per se, (because vis-a-vis ronjanec, the word "time" is fraught with semantic potholes. )

If Minkowski is a toy, then you might want the full description of the theory in 4 dimensions. In my humble opinion, you are at a stage in your physics where you probably need it badly. Ockham's razor is a guideline and a fine tool of justification. But you cannot use Ockham to simply remove actual dimensions of space that are actually there.

4-vectors

You must learn everything about 4-vectors and how they are multiplied by matrices. The basic gist is that all the transformations reduce to multiplying 4-vectors by a Hermitian matrix. I like this link below because they laboriously draw out the whole matrix in every example so you can see exactly what is happening.

http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/vec4.html

Learn this material like the back of your hand. Really you will need to apply it to actual problems to get the "hang" of the procedure. You cannot bluster over this material, because eventually every textbook and website you read will stop writing them as a matrix, and instead just show multiplications by a matrix given by greek letter Lambda.

Next you will have to visit and skim (or read carefully) the material in these two articles. Again, this is not for learning, because wikipedia is not for learning. This is so you can familiarize yourself with the terms.

1.) Lorentz Group
https://en.wikipedia.org/wiki/Lorentz_group

2.) Representation Theory
https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group

In particular , I would like you to familiarize yourself with the word "orthochronous". You will find it numerous times in the above material. Along with the hyperphysics page, consider this another piece of homework : find out what orthochronous means.

We can get into endless arguments about which person "invented" Rock and roll music. Isaac Newton and Leibniz are usually attributed as the people who "invented" calculus. It is true that they were the first people to apply calculus-inspired thinking to mathematics. However, the way calculus is taught at the university level is quite alien from these original formulations , and the college students are actually seeing formalisms invented in the 19th century.

The same is true of Special Relativity. While Hendrik Lorentz and Albert Einstein were the first to formulate it, the way it is taught at universities is very different now. They teach it with the material given above, as matrices (called 'Lorentz Transformations') applied to 4-vectors.

One of the reasons why physics students are forced to wrangle with Representation Theory is because academia is attempting to prepare them to be able to apply S-R to quantum mechanics and QFT. Einstein invented it, yes, but the theory now stands alone with its own jargon. S-R has grown legs and moves around independently of its founders. It is now possible to learn S-R, write books about S-R and teach S-R without ever making any reference to the quote, "...assumptions which Einstein made". You can wrangle with the theory, as a theory.

It is unfortunate that the internet is going to throw a wall of equations at you, and in this regard, you might fall back to habit of Okham's razor simplifying something.

There is nothing here that needs simplifying.

This is the raw depiction of the theory of S-R in the simplest terms. This process of multiplying matrices by 4-vectors neatly captures every aspect of the entire theory. We do not need separate equations to describe time dilation, relativistic mass, and length contraction, and et cetera. The procedure of Lorentz Transformation of 4-vectors captures all the phenomena in one fell swoop. That is truly in the spirit of Okham's principle!

[Faradave]

I watched almost all [Faradave's] videos in one sitting.

Good for you! Thanks. I appreciate the time you invested and your considerable suggestions above and in general.

Minkowski spacetime is not a literal description of the world. It is a toy model of Special Relativity where space is squashed to 2 dimensions.

I prefer to think of a "Minkowski diagram" as a 2D slice from 4D Minkowski "spacetime". In both cases, space and time coordinates are mapped onto the 4D continuum (or manifold) of our existence.

I like toys! So did Feynman, with spacetime the coordinate choice for his ubiquitous Feynman diagrams. As this toy finds extensive utility in both Relativity and QM, it's worth playing with.

there is no "cone" in the actual spacetime we are sitting within now. That spacetime is a 4-dimensional subject

The literature is replete with references to "light cones" so, I find the model unavoidable. Of course, you and I know that being a structure in 4D, the referenced "cone" is better described as a 3-cone, incorporating 3D rather than the depicted surface. Any rim (3-conic section) is thus understood to be a surface (e.g. a sphere) rather than just a loop.

3 Ds of space and another D that marks off local duration in a reference frame. The D=4 is not "time" per se, (because vis-a-vis ronjanec, the word "time" is fraught with semantic potholes.

Agreed! Dimensions quintessentially provide the potential for separation, three offer bidirectional translational freedom, while one (typically referred to as "time") is unidirectional. This can be modeled from a unique, central, Big Bang event, from which a 4D temporal field emanates and "space" is an enclosing 3-surface.

you cannot use Ockham to simply remove actual dimensions of space that are actually there.

A 2D "slice" ignores but does not remove the other two dimensions. It keeps things simple when motion is restricted to one direction. Clearly though, objects like orbitals require 3D & 4D.

You must learn everything about 4-vectors and how they are multiplied by matrices.

I expect I will but only to rescue those poor souls who have overlooked the benefits of Euclidean coordinates. I'm not saying they get incorrect answers. I'm saying they work too hard and miss a lot that should be obvious. For example, the absence of massless particles (i.e. Though orbital transitions are indeed quantized, there are no photons).

There is nothing here that needs simplifying. This is the raw depiction of the theory of S-R in the simplest terms.

Spacetime is clearly a well-developed legacy with much to offer but that's no reason to neglect the simpler, Euclidean perspective I provide with interval-time coordinates.

With all the emphasis QM places on "locality" vs. "non-locality", it seems especially important to realize the full extent of locality in 4D. Yet, as far as I can tell, I'm the only one. Is it any wonder I'm hesitant to adopt their convoluted methods? It's as if they're trying to be remain ignorant.

"Local" means "contact". While there is no such thing as action-at-a-distance, the majority of contact is spatially remote. This is clearly and undeniably evident with Euclidean coordinates in the limit as ∆s approaches ∆t.

Is it any wonder I'm hesitant to adopt their methods? It's as if they're trying to be remain ignorant.

another piece of homework : find out what orthochronous means.

"Lorentz transformations that preserve the direction of time are called orthochronous."

Thanks for the reference. I'll need to do more than just find a quote. Transformations are different from behaviors such as chronaxial spin (i.e. fermion spin½). I was concerned that I might have overlooked some prior art.

[hyksos]

Spacetime is clearly a well-developed legacy with much to offer but that's no reason to neglect the simpler, Euclidean perspective I provide with interval-time coordinates.

Nobody is neglecting Euclidean coordinates. Nobody is neglecting you. You have posted on this forum for years.

I watched 80% of your youtube videos. You are neglecting the way S-R is taught in the full 4 dimensional versions. If I thought there was absolutely nothing to gain for studying it that way, I would have never mentioned it to you. You will gain the world if you study the full 4D version.

I like toys! So did Feynman, with spacetime the coordinate choice for his ubiquitous Feynman diagrams. As this toy finds extensive utility in both Relativity and QM, it's worth playing with.
==
The literature is replete with references to "light cones" so, I find the model unavoidable. Of course, you and I know that being a structure in 4D, the referenced "cone" is better described as a 3-cone, incorporating 3D rather than the depicted surface. Any rim (3-conic section) is thus understood to be a surface (e.g. a sphere) rather than just a loop.

Yes, I am aware and we are all aware of the "extensive utility" of the chalkboard versions for quick napkin calcs and deep discussions in the professor's office at the white board. Yes. Nobody is arguing these points.

However I am outwardly telling you : you are missing some crucial conceptual points by neglecting the full 4D version. In particular what it says about "time" and "energy" and how they relate to each other.

Agreed! Dimensions quintessentially provide the potential for separation, three offer bidirectional translational freedom, while one (typically referred to as "time") is unidirectional. This can be modeled from a unique, central, Big Bang event, from which a 4D temporal field emanates and "space" is an enclosing 3-surface.

It is this kind of talk that led me to propose a doctor's orders for your remedy. You clearly believe this is what Special Relativity is saying and I'm telling you with complete honesty, mano a mano, that you have it wrong. I am giving you the tools to get it right.

The doctor's prescription is to study the full 4D version with matrices (called Lorentz transformations) and vectors (called 4-vectors).

There are certain geometrical operations that work on vectors in 3D that do not extend or analogize to four dimensions. In short, 4-dimensional space is not something that you get for free by hand-waving. The 4th dimension holds the skeleton key to what S-R is saying about the nature of reality.

Special Relativity is a theory from 1905 -- (and while it clearly is not the end-all-be-all of all physics) it does make assertions about the nature of reality. You will never correct the theory, nor extend upon it, until that time in which you capture its fundamental claims. To abandon a theory , you must first authentically state it.

"time" is very wobbly in S-R. It is far better to begin to conceptualize the 4th dimension as representing energy. Without this conception, you risk rotting away in a cul-de-sac where the theory only makes predictions about a single observer's frame.

But the universe we inhabit is made up of a bunch of different observers. There must be some physical principle that is orchestrating all these allegedly separate observers in their own (allegedly) isolated reference frames. Since the length of durations of time will change from one reference frame to another, the 4th dimension cannot be "time" as normally understood. Energy is what it really is.

S-R strongly suggests that "time" moves in the forward direction because energy starts at 0.0 and increases. In other words, there is no such thing as "negative energy" and so this is why all observers think "time" goes in one direction only. It is not, as you have suggested because we are "emanating away from a Big Bang". (we can argue this point in a different context, but we aren't talking about Quantum Gravity, we are speaking only of physics of 1906 at this juncture).

I can blabber the above points at you and you will read them, then forget them as if I were writing poetry. I can tell you things like "It is an emotional attachment to classical time that must be abandoned." You will think this is a pretty flower to be enjoyed temporarily and moved on.

It is not my calling to use this forum to be your personal physics tutor. This will require elbow grease on your part. If you study the full 4D version, everything I have written above will jump out at you. It will fall in your lap. I cannot make you see it. There is no shortcut. You must see it for yourself.

You have all the tools.

[Braininvat]

I can blabber the above points at you and you will read them, then forget them as if I were writing poetry. I can tell you things like "It is an emotional attachment to classical time that must be abandoned." You will think this is a pretty flower to be enjoyed temporarily and moved on.

Unless you happen to have a precise simulation (down to the quantum level) of Faradave's brain, I find it unlikely you can extrapolate his thoughts or actions regarding your posts. Really, it may be best if you don't offer lesson plans and stick with traditional critique. Anything else risks condescension.

Also, are you sure there is "no such thing as 'negative energy'"? That could use a bit more support, perhaps.

[Faradave]

You will gain the world if you study the full 4D version.

Thanks for your continued time and candor hyksos. I didn't take offence from your post but let's both keep Forum Rules in mind.

I loved Fleisch's little book on Maxwell's equations so much that I'm now finishing his book on wave theory and own his book on tensors to read next. It has some 4D matrix algebra in it and I promise to read it at least twice. I do not however, expect it will change the irrefutable aspects of Phyxed, only the language in which it may be presented.

Nobody is neglecting you.

Certainly, I haven't been neglected at SPCF. However, I am having difficulty getting through to the ALPHA team at CERN, regarding the possibility of documenting photo-annihilation. Similarly, with numerous authors as they're busy and I have the profile of a crank.

Nobody is neglecting Euclidean coordinates. You've been posting ... for years.

.
If they aren't being neglected, why haven't I found or been referred to such a reference in all those "years"? The closest has been Epstein diagrams but he mistakenly attempted to retain spatial coordinates (when he should have switched to interval coordinates).

Folks like you read and reply to my posts but these fall short of either affirming or substantively refuting the fundamentals presented.

You are neglecting the way S-R is taught in the full 4 dimensional versions.

Regardless of the dimension of presentation, vectors are themselves inherently 1D.* That's why they're readily depicted by arrows. An interval occurring in a 2D slice of spacetime (or interval-time) is still a perfectly legitimate 4-vector!** As it turns out, that's all I need to make my point in many cases.

*The degenerate case of a "null vector" may be considered 0D.
**Just add zeros for the two components not presented.

you are missing some crucial conceptual points by neglecting the full 4D version

I'll be overjoyed to discover these. In the meantime, I could say the same about anyone who persists in the notion of a photon.

In particular what [4-vectors say] about "time" and "energy" and how they relate to each other.

You have already conceded that "time is very wobbly". I believe that "proper time" of SR represents time experienced (i.e. aging) by objects sharing the coordinate frame. If I'm missing some crucial point about that, it's one of philosophy rather than physics.

Energy is a different issue. Physics currently lacks a model for energy from first principles. Phyxed provides one that is consistent with the Standard Model. I'm happy to elucidate but my point is that I can't be "missing a crucial point" of physics, which physics does not itself yet have.

Special Relativity …You will never correct the theory, nor extend upon it, until that time in which you capture its fundamental claims. To abandon a theory, you must first authentically state it.

I get what you're saying but one can be a terrific cardiologist without also being a neurosurgeon. It is sufficient to know well, the limited area in which one is contributing. Einstein did a lot with triangles before he understood tensors. So can I.


All physical contact is invariant, interval contact. Remote contact is by far, the most available form of such locality.

Interval-time coordinates are consistent with SR and clearly reveal that for light transmission, no photon is required. Indeed, working models can accommodate photons, but that doesn't make photons necessary.

Respectable textbooks (including yours) define lightlike intervals as representing "zero interval separation". Phyxed simply offers the clarity of calling this direct, physical contact. If you want to play with 4D, you must allow for more than 3D of contact! I'm not asking for tutoring or a prescription but if we’re having a discussion, it's time to affirm or refute.

You clearly believe [a curved-space, radial-time model] is what Special Relativity is saying and I'm telling you … you have it wrong.

One step at a time. Curved-space, radial-time can wait.

"time" is very wobbly in S-R. It is far better to begin to conceptualize the 4th dimension as representing energy. Without this conception, you risk rotting away in a cul-de-sac where the theory only makes predictions about a single observer's frame.

Length contraction, time dilation and relativistic mass-energy are all part of Phyxed and entirely consistent with Relativity, QM and the Standard Model. Phyxed is just simpler, more complete and more explicit.

[hyksos]

Length contraction, time dilation and relativistic mass-energy are all part of Phyxed and entirely consistent with Relativity, QM and the Standard Model. Phyxed is just simpler, more complete and more explicit.

I am skeptical of its applicability. Also, the lack of any expressions for energy is suspicious.

[Faradave]

I am skeptical of its applicability.

Phyxed is as applicable as the Standard Model but...

Simpler:
It has Euclidean geometry.
It requires only unidirectional dimensions fundamentally. (A sphere with unidirectional radii does not restrict bidirectional freedom on its surface.)
It eliminates massless particles.

More Complete:
Speed limit c is no longer an unexplained postulate. Phyxed fully explains and naturally requires a universal speed limit. Note the diagram above gives both magnitude (zero) and direction (orthogonal to the aging coordinate) for any lightlike vector.*
Intrinsic spin, gravity and energy are just a few of the many mysteries solved by Phyxed.

More Explicit:
Too many examples to list (and may seem strange out of context) ...
Spin½ has a real axis to go with its real angular momentum.
"Zero interval separation means "contact"!
Contrary to convention, time (radially) and the Big Bang (central) have a place in the balloon analogy of the expanding cosmos.
Force is fundamentally an object, a lightlike pinhole, thus energy transmission is always achieved by direct contact.
A field (e.g. gravitation or electric) is a force in radial superposition about time.

lack of any expressions for energy is suspicious.

I was trying to be brief. Here's the video.




*Did you miss the part about affirm or refute? Most people (even physicists) do. Yet it's difficult to advance the Phyxed framework without consensus at the foundation.

[hyksos]

I see Phyxed does contain a formulation of energy.

By "applicable" I mean whether Phyxed would harmonize with other theories outside of it. That's really asking too much from the original author. It is a hard road for anyone wanting to rewrite some established theories from 103 years ago.

The rest mass of electrons has self-energy corrections. Electrons have a slightly different mass than expected because you have to add in all the various possible particle interactions from point A to point B. In other words, you have to "Feynman integrate" over all the plausible paths. That this actually happens was not discovered overnight. It took roughly 20 years to figure it out. It is unlikely electron mass could be explained in simpler terms.

To harmonize with theories outside of it, Phyxed would have to produce quantitative numbers about particles. Electron mass is one example. However, Electron mass would be difficult to tackle first. ( it was the only example that popped off the top of my head. )

[Faradave]

In my view, as this thread addresses gravity, the more relevant harmony is that of modeling a gravitational field and energy with the same fundamental mechanism (chronaxial spin of a pinhole).

As valuable as General Relativity has been, the Standard Model still lacks a fundamental explanation of gravity. Mass-energy is associated with spacetime "curvature" but there hasn't been a hint as to how mass-energy does that.

Phyxed replaces "curvature" with a corresponding field of "separational insufficiency". This results from the radial superposition of a pinhole about time. That doesn't just take energy, it IS energy!

Chronaxial spin rate (ω₃) is the fundamental manifestation of energy, which obligingly correlates to gravitational intensity (thus, gravitational mass). ~ Beauty ~

[hyksos]

As valuable as General Relativity has been, the Standard Model still lacks a fundamental explanation of gravity. Mass-energy is associated with spacetime "curvature" but there hasn't been a hint as to how mass-energy does that.

A wise man once said that there is a hidden danger in physics. A person can arrive at the correct answer by using fallacious reasoning.

At first hearing this, it sounds like a warning for college freshman, and that somehow the professors in ivory towers have "grown out of it". These days I think this is premature. It appears to me now that this danger of fallacious reasoning effects the Ph.d in equal intensity to the high school student. This is particularly true if we consider this recent explosive avenue :

https://en.wikipedia.org/wiki/Entropic_gravity

https://en.wikipedia.org/wiki/Erik_Verlinde



I'm using the word "explosive" in the Chinese sense.

I see what you are doing with the stacked cones here, with a cone being a sweep of a light cone around a temporal axis. Phyxed would direly require that this rotation about an axis be written in the full 4D version. The mathematics is well within our grasps, and it is just a matter of time of doing it. A will include a caveat that it cannot be done in an evening.

There is something crucial hiding here. Our universe does not have 2 dimensions of space with a 3rd dimension of time going up. Our universe appears as a 3Ds of space and 1D of time. What I will show here is not a starting point, rather this is a warning about what will happen much later after we have formulated precisely , what a chronaxial rotation is in terms of equations.

This is the dot product in Minkowski space, with a minus sign on the "temporal" component. {dpms, "dot product in minkowski space"} I doubled the minus sign (--) because it is easily overlooked with a glance. This is not the equation of the dot product in 4D. Dpms will haunt us not now, but later on, for reasons we can't yet see at this time.

a * b = a₁b₁ + a₂b₂ + a₃b₃ -- a₄b₄ {dpms}

The first leg of this journey is to find out whether 4D rotations are as simple as they are in the 3D diagrams we have all used up to now. This link below suggests the answer is no.

https://math.stackexchange.com/questions/1402362/rotation-in-4d

So what does a rotation look like in 4D? I found this so far

http://www.euclideanspace.com/maths/algebra/clifford/d4/transforms/index.htm

This gives an overview, but additional googling indicates that I will really need to get a textbook on the subject. What I have gleaned so far is that 4D rotations are 'reminiscent' of rotations performed on the complex plane with complex numbers. In complex rotations you multiply two vectors together and then shrink them by some "eigenvalue" calculated separately. Something similar happens in 4D rotations. You multiple 2 vectors and then you get two extra components hanging off the end called a bivector. (The analogy is that it is a single number in complex plane <== corresponds with ==> a bivector in 4D) The bivector can't be divided out with a pen stroke. You have to do some other kind of translation to make it vanish.

Regarding the "need to get a textbook", I have scratched the surface here. ( Homework assignment : What is an isoclinic rotation ?? )

After all that jazz , and the smoke has cleared, then I would tackle trying to fold in the dpms, with its weird minus sign lurking there.

There are two possibilities after this. Either this plane we are building will crash and burn. Or maybe it will all work out. We might find out some sexy insight : such as "chronaxial rotation entails quantum spin". I merely speculate!

[Faradave]

I like your enthusiasm.

Our universe does not have 2 dimensions of space with a 3rd dimension of time going up. Our universe appears as a 3Ds of space and 1D of time.

Agreed. Though diagrams appear to have only 2D of space, it is to be understood that the labeled "hypersurface" is a 3-plane incorporating 3D of space. So, any "circular" cone rim is actually a sphere. Importantly, a chronaxial rotation is thus, solid-angular, entailing 4 pi steradians rather than 2 pi radians.


The unidirectionality of time is evident in that rotation about a spatial axis is always observed in a classical plane (a 2-plane) of rotation. Time cannot support the required oscillation. But time can serve as an axis for point particles and lightlike field elements.

dot product in Minkowski space, with a minus sign on the "temporal" component.

That's the conventional approach, which seems burdensome. Euclidean, interval-time coordinates obviate the minus sign by rearranging the interval equation to:
∆space² = ∆interval² + ∆time².
Alas, no one's ever heard of it (except at SPCF).

find out whether 4D rotations are as simple as they are in the 3D diagrams we have all used up to now. This link below suggests the answer is no.

Interesting but they seem to be suffering under the impression that rotation can only occur in a classical 2-plane (e.g. XY, YZ, XZ, etc.) with as many pairs as the dimension allows. By contrast, Phyxed is perfectly comfortable with solid-angular rotation in an XYZ 3-plane (about time). That's how fields are generated and it's why "independent" spin components of a single fermion are entangled.

So what does a rotation look like in 4D?

Though I can't claim to understand either, the language of your reference seems very much the same as that for spinors, which exists as a abstract description (but not an explanation) of fermion spin.

That's fine as math. But if its to be physics, they're going to have to recognize chronaxial spin in a 3-plane. It should be as clearly stated as that. Physics is about the physical.

[hyksos]

Update : I talked to some mathematicians in a chat room. It seems this plane is going to fly.

Some initial take-aways , briefly :

In 4D you must stop thinking about rotations "around an axis" and instead think of them as if they are rotations in a plane spanned by axes u and v.

In 3D, an axial rotation moves all points except those in a single line on the axis itself. But in 4D, the rotations must rotate exactly 2 dimensions, and must leave fixed exactly 2 other dimensions. This is a hard law. Call it the "Hyksos Rule" if that helps.

Given a point (x,y,z,t) imagine that we want to rotate by an angle ω I will be using the notation that if a point has an {ω} next to it, then it was changed by the rotation. If it has nothing next to it, it did not change at all; was fixed in other words. Applying the Hyksos Rule, these are some examples of coherent rotations.

• ( x{ω} , y{ω} , z , t )
• ( x{ω} , y , z{ω} , t )
• ( x{ω} , y , z , t{ω} )
• ( x , y{ω} , z , t{ω} )
• ( x , y , z{ω} , t{ω} )
• ( x, y{ω} , z{ω} , t )

This list is not exhaustive, but the rule is followed in each case. There are always exactly 2 modified components, and exactly 2 left fixed. Saying these out loud in english is illuminating.

( x{ω} , y{ω} , z , t ) <-> "Rotate around the plane spanned by <z,t> all <x,y> points will move."

( x , y{ω} , z , t{ω} ) <-> "Rotate around the plane spanned by <x,z>. All <y,t> points will move."

In terms of physics, it might help to consider the following list.

• ( x{ω} , y{ω} , z{ω} , t )
• ( x{ω} , y , z{ω} , t{ω} )
• ( x{ω} , y , z , t )
• ( x , y{ω} , z , t )
• ( x , y , z{ω} , t )
• ( x, y , z , t{ω} ) (*ǂ)

Rotations in the above list break the rule. They are (literally) mathematically and geometrically incoherent. In particular consider the first one.

( x{ω} , y{ω} , z{ω} , t ) <-> "Rotate around the t axis, <x,y,z> points move."

This is impossible. It makes no sense, geometrically speaking.

Stuffy grad students speak their own private language about eigenvectors and eigenvalues. They will tell express the same misgivings as "There cannot exist eigenvalues [1 1 1] or [1 -1 -1]" et cetera.

Double Rotations.

From the (Hyksos Rule) there must always be 2 dimensions fixed. You might ask if we can rotate those independently of the others that are changing. The answer is yes. Given a point (x,y,z,t) imagine that we want to rotate by an angle ω in 2 of the components, and "simultaneously" rotate the other two components by an angle φ. Possible?

While this kind of rotation is likely impossible for the human mind to mechanically understand, it makes perfect geometrical sense -- it is called a Double Rotation. It is well-defined in 4D, but has no analogy in 3-dimensional space.

An example briefly,

( x{ω} , y{ω} , z{φ} , t{φ} ) <-> "Rotate around the plane spanned by <z,t> by angle ω, all <x,y> points will move. At the same time, rotate around the plane spanned by <x,y> by some other angle, φ, causing all <z,t> pairs to move."

This is permitted.

Isoclinic rotations.

Consider the ratio of omega and phi.

k = ω / φ

If k is a rational number (or better yet a whole number) then interesting weird stuff happens in terms of topology. Grad students like to stay up all night writing papers about the stuff. Now consider the case in which ω = φ . In that case k = 1. These are special rotations in 4D, and are referred to as "isoclinic rotations". (grad students call them Clifford displacements, because they work in N dimensions and are stuffy people.)


Where now?
Consider a fermion with rest mass m. Place a 4D axis system with the fermion at the origin, so that coordinates are in the fermion's inertial reference frame. Points in the frame are (x,y,z,t) Suppose the fermion is massless if it were simply coasting in its frame. Now impose a rotation about <x,y> at angular velocity ω. We might assume there is still no mass and so no angular momentum. But suppose the fermion's actual rest mass scales as a function of ω.

m = F( ω )

Now we have a syllogism. If a fermion has mass, it must have rotation about t. More precisely. m = F(0.0) then m = 0.

Applying the geometry rules from earlier, there must be one (and only one) other unique component of points that is changing. (see (*ǂ)) For 4D Minkowski space, all the other axes are spatial. So there is exactly one spatial component that is changing.

( x , y , z{ω} , t{ω} )

Given a free choice to label axes, apply that change to z dimension. For human observers living in spacetime, they would not perceive t{ω} directly, but only indirectly as some attribute of the fermion. The change in z{ω} would be perceived as some kind of regular pulse in the particle that repeats at a fixed interval. It would definitely not appear as a spatial rotation, (those go as (x,y{ω},z{ω}))

I was going to ask if z{ω} is the fermion's quantum spin. But I'm cutting this off.

[hyksos]

I have run into some conceptual issues in the above post. This is what I meant by this exercise taking up more than one evening!

( x , y , z{ω} , t{ω} )

That is not a rotation that fixes the time component and this makes little physical sense now that I think about it. We need something that fixes the time and looks more like this :

( x{ω} , y{ω} , z , t )

Though I can't claim to understand either, the language of your reference seems very much the same as that for spinors, which exists as a abstract description (but not an explanation) of fermion spin.

I agree with this. I'm likely stepping through the same beaten path for spinors.

Interesting but they seem to be suffering under the impression that rotation can only occur in a classical 2-plane (e.g. XY, YZ, XZ, etc.) with as many pairs as the dimension allows. By contrast, Phyxed is perfectly comfortable with solid-angular rotation in an XYZ 3-plane (about time). That's how fields are generated and it's why "independent" spin components of a single fermion are entangled.

Yes. You are a few steps ahead of me. The mathematicians in the chat room seemed to talk endlessly about how Minkowski space is very strange. One of them said that you can have negative distances. That's mostly what they talked about when I was there, really.

It seemed to me something murky was hiding in "solid-angular rotation in an XYZ 3-plane".

[Faradave]

I talked to some mathematicians in a chat room. … In 4D you must stop thinking about rotations "around an axis"

That's an absurd restriction by surprisingly narrow minds, which defies the very notion of rotation. Be careful what you listen to. (You'd be doing them a favor by referring them to: Spin½ "Plane" & Simple: https://youtu.be/oGdXt2gZI1I

"Rotate around the t axis, <x,y,z> points move." This is impossible. It makes no sense, geometrically speaking.

That's the result of their absurd presumption. Rotation about the t-axis is solid angular rotation in the XYZ 3-plane. It's perfectly reasonable, geometrically speaking.

Solid-angular rotation about a 4th dimensional axis naturally occurs in a 3-plane. It so happens that this also entails a 720° rotation (in any 2-plane) of the angle bisector to return to its original state (see video below).

consider the case in which ω = φ

I can't be sure but this reminds me of the Bloch sphere used to represent a qubit:



Note that constraining θ = Φ describes a solid angle.

A conventional description of a solid angle corresponds to a Bloch sphere constrained (i.e. simplified) to equal coordinate angles. Radius is ħ/2 or h/4pi.

It seemed to me something murky was hiding in "solid-angular rotation in an XYZ 3-plane".

Yes. It's called "probability amplitude" but it doesn't have to be murky.



Popular convention regards probability amplitude as the square root of a probability! If they had paid more attention to the entailed half angle, it would have become obvious that "probability amplitude" is itself a probability. It is the projection of 100% self correlation across that half angle, plane and simple!

[hyksos]

I talked to some mathematicians in a chat room. … In 4D you must stop thinking about rotations "around an axis"

That's an absurd restriction by surprisingly narrow minds, which defies the very notion of rotation. Be careful what you listen to. (You'd be doing them a favor by referring them to: Spin½ "Plane" & Simple: https://youtu.be/oGdXt2gZI1I

"Rotate around the t axis, <x,y,z> points move." This is impossible. It makes no sense, geometrically speaking.

That's the result of their absurd presumption. Rotation about the t-axis is solid angular rotation in the XYZ 3-plane. It's perfectly reasonable, geometrically speaking.

XYZ 3-Plane a.png

I'm afraid we will have to part ways here.

The mathematicians can prove that axis rotations are impossible in 4 Dimensions. In fact, some of them start off by saying "our regular way of talking about rotations about an axis is flawed". Like the English is flawed versus the mathematics. In other words, it is more true to the equations to say that a rotation "about the Z axis" should instead be spoken of as a "rotation of the points on the XY plane". In short, "axial rotations" is the wrong way to say this even in 3 dimensions.

(A brief sketch of this proof. They do it using eigenvalues and eigenvectors. First we agree that "rotation around an axis" means there exists an eigenvector in the transformation whose eigenvalue equals exactly 1.0. In other words, the points on that single line do not move. You then show such a vector does not exist under the rotation.)

What happens next, the mathematician will declare that an axis-of-rotation will exist in N-dimensional space , if and only if N is odd. So you regain axial rotation in 5 dimensions.