Consider three physicists: Feynman, Weinberg, and Hawking.

All are so far better in depth and breadth of their understanding of general relativity and quantum field theory than you are I (in particular) will ever be. They devoted much effort toward this goal but have yet to resolve the gap.

Here are five possibilities for the chasm that remains despite diligent work on the subject:

1. GR and QFT are right. A new bridge must be constructed between the two.

2. GR is wrong or incomplete and QFT is right. A new theory for gravity is needed that shares a long list of properties with GR.

3. GR is correct and QFT is wrong or incomplete. A new theory for atomic processes is needed that shares a considerably longer list of constraints put in place by QED and QCD.

4. GR and QFT are wrong or incomplete in the scientific sense of needing to be replaced by proposals that are only ever so slightly better in specific technical situations than GR or QFT.

5. Other than the above, because right or wrong does not feel like the right way to characterize the issues between GR and QFT.

Please feel free to cite the numbers in the comments. Where would one place work on strings? Loop quantum gravity? Non-commutative geometry?

A two strike rule is in place for those who wish to claim they have figured out everything already. I would prefer that if you are of that opinion of your work, you should sign up and start blogging at Science20.com. That's what I did and now I don't have a theory on the table, which is better than selling t-shirts that are wrong.

Doug

Next Monday/Tuesday: 4-Parameter Analytic Animations, Solid Man

## Comments

It sounds strange to say this to someone who loves quaternions so much, but I've noticed it looks like you need to hear something many times before it somehow clicks: a quaternion is a NUMBER. By trying to equate a quaternion to something that is NOT a number (namely, a four-vector), you then destroy what makes a quaternion a quaternion.

Try translating this to math if it will help you:

Let's say I have N apples. If I rotate my coordinate system, does N change?

Now let's say I have some complex valued field, and at point x the value is C. If I rotate my coordinate system so that the coordinate label I give x changes, does that change the value C? Does it change the real or imaginary component values of C? Would you call C a number, or a 2 component vector (or both?)?

Now let's say I have a quaternion valued field... how are the components affected by my coordinate choice?

Now let's say I have a vector field... how are the components affected by my coordinate choice?

Do you now understand geometrically the difference between a number that can be written in multiple components, and a vector?

Since you seem obsessed with that first component in the quaternion multiplication, let me point out (as David will probably explain much better in his SR articles) that you are assuming a metric of a certain form to make this comparison. This means you can't even describe the space-time of SR, because you are trying to ignore the metric. So it will only work if you restrict yourself to specific coordinate systems where the metric happens to have the form the quaternion multiplication requires for this coincidence. You can't properly describe flat space-time by pretending a quaternion is a standin for a four-vector... because they are two different beasts.

A small story…I still remember where I saw the Lorentz invariant interval for the first time, in the book "Concept of Number: From Quaternions to Monads and Topological Fields (Mathematics and Its Applications)" on the second floor of the Cabot Science Library. I was so excited, I decided to not do anything with the observation for about six months. Then I wanted to test the hypothesis that quaternions could be useful to solve physics problems. I audited a course at MIT, 8.033, "Classical and Relativistic Mechanics". My test was to solve any and every assigned problem two ways: the way it was taught with the Lorentz group, and the way I was making it up using quaternions exclusively. If there was one problem I could not solve with quaternions, then I could shelve the project. Quaternions worked for 52 of the 52 assigned problems. That is my experience. And I was not solving the problems the "right" way with a global transformation. Many of the quaternion solutions were local (simple problems involving the origin and a boost in one direction were global with the quaternion method). Today, I could solve things with a real-valued quaternion way of representing global transformations.

Choose a crazy metric if you must. Quaternions are a division algebra, they can follow the crazy metric if requested. There is always a way from here to there.

A 4-vector can be added, subtracted, or multiplied by a scalar. If I choose to represent a quaternion over the manifold *R ^{4}*, then I can add, subtract, or multiply it by a scalar. I can also do more with a quaternion than I can with a 4-vector. Quaternions come equipped with both multiplication and division. I can also decide to work on the quaternion manifold and play all kinds of games with automorphisms. I do look to extensive work physicists have done with 4-vectors to figure out fun things to do with quaternions.

Real numbers are a subgroup of quaternions. It is the case where the 3-vector is zero. Complex numbers are a subgroup of quaternions. It is the case where all the quaternions happen to point in the same direction. Whatever I can say about real and/or complex numbers applies to a subgroup of quaternions. Complex numbers can be represented on different manifolds. Complex numbers are introduced as having two numbers on the manifold *R ^{2}*. That certainly works. Yet complex numbers can also be represented on the complex-valued manifold

*C*as one complex number at a time. Really, one number, not something with two numbers hiding on the inside (some people resist the idea). To cover all the possibilities seen on the

^{1}*R*manifold requires z and z*, the conjugate. Many of the insights in complex analysis require one work on the manifold

^{2}*C*.

^{1}A similar story will happen for quaternions. They are introduced on the manifold *R ^{4}*. There is the 4x4 real matrix representation… Few mention the quaternion manifold, let's call it

*H*. That requires more than one kind of conjugate, but that is easy enough to do.

^{1}A 4-vector is a tensor. A quality of tensor expressions is that they are true no matter what the choice of the coordinates happens to be. A similar statement can be made about quaternion expressions. So I can see how to do more with quaternions than 4-vectors, but that should not represent a problem.

My test was to solve any and every assigned problem two ways: the way it was taught with the Lorentz group, and the way I was making it up using quaternions exclusively. If there was one problem I could not solve with quaternions, then I could shelve the project. Quaternions worked for 52 of the 52 assigned problems. That is my experience.Do you have any such calculations in your previous blog posts? I'd like to see a simple calculation done both ways. A tutorial on how to do that would be very interesting.

Applying a Lorentz boost to quaternions is an odd operation. It is as silly as trying to arrange a Lorentz boost in Euclidean space.

It is my understanding that, 4-Vectors can be made into spinors and a representation of the Lorentz Group can constructed in spinor space such that any element of the Lorentz Group, in spinor space acts, on the spinor as a boost. Transforming the spinor back to a 4-Vector would give the same result as if it were merely boosted. Is calling the spinor space boost a "lorentz boost" an improper use of terminology?

Or am I incorrect?

If I'm correct, why can't this be done with quaternions?

Time becomes significance when an observer is modeled that is located in a reference frame that moves with respect to the observed item. Thus the timespace model is attached to the HBM model. It always covers a set of HBM pages.

A Lorentz boost is an observed rotation free displacement that is taken relative to the observer.

Thus it covers a set of HBM pages and the observed item obtains a different position.

In fact a Loretz boost is a displacement in one dimension. The quaternion approach is better suited for 3D changes, such as isotropic or multdimensional flows. Lorentz boosts are better treated with complex state functions.

Hello Barry:

Thanks for the blog suggestion. I audited the course more than a decade ago, so have trouble recalling the details. Fortunately I put up all the answers

One thing was clear: quaternions did not in any material way make solving problems easier or harder than the standard methods.

I often don't understand what Hans writes. There should be zero qualms about doing all the standard things one expects to do with the Lorentz group using real-valued quaternions. It is a cute variation on the 3D spatial rotation that quaternions are known for. One does the same thing with hyperbolic sines and cosines, realizes some additional things must be added while others subtracted. A short search finds the additional triple products. What is neat about the "correction" is it sounds consistent with 3D rotation being a compact Lie group while the Lorentz group is not.

Once you make a clear distinction between space and progression it becomes easy to handle them separately.

Space and progression can easily be combined into 1+3D Euclidean space. So, it fits in a quaternion. What is more convenient, you can construct a quaternionic distribution that represents curved space-progression and represents the dynamics of the whole universe. This is impossible with spacetime.

The principle trouble I have with "string theory" is that I have known of it too long, long before the attempts to apply it to a "Theory of Everything". The problem is that it has gone through one or two previous iterations: An idea in search of becoming a theory of "something or another".

The latest iteration (its attempt to become a "theory of everything") is (to me, at least) a "last ditch effort", so to speak, after all previous failures. Worse is that this attempt is so desperate (after all, where else can it go if it fails now?) that the "theory" built up around the "string" idea has become an "everything, including the kitchen sink" sort of beast.

Sorry, I don't hold out much hope for this beast.

David

Susskind did not reject string theory but he stated that it is quite probable that it would never be shown in the labs. (I took this from one of the youtube lectures)

What you say is indeed true. However, much of the elegance, in my opinion, at least, comes not so much from the "string idea", as from the Kaluza-Kein (KK) theory, and super-symmetry. (The "strings" really just give them additional degrees of freedom: Freedom to fit just about anything! And if the additional freedom of a one dimensional "string" isn't enough, hey we can go to 'branes, by golly!)

In fact, one of the things the "string idea" was supposed to "alleviate" was the singularities of point particles. However, as anyone that understands singularities can see immediately, it does no such thing, it merely increases the dimensionality of the singularities by one or two (or a few) dimensions (depending on whether one has strings, or 'branes).

Additionally, they still can't get away from the unobservable "base space", or "background geometry", even after years of trying to create a background independent formulation. This is absolutely the antithesis of "beautiful" or "elegant", in the opinion of many.

So, while it has enough degrees of freedom it just about can't help but be able to fit whatever we may observe, no, "sorry, I don't hold out much hope for this beast."

David

1. The derivation of the graviton.

2. The derivation of the entropy of a black hole.

3. The theory of interactions as strings knotting together (using complex analysis and conformal mapping)

4. The derivation of Einstein's Equations!

5.The prediction of the number of dimensions of space (ok the dimension is weird, but at least it contains the machinery to predict that.)

6. The Lagrangian is simple.

I wouldn't call the issues you point out "inelegant" (and that is merely a semantic choice on my part). Elegance isn't about being right or wrong. What I meant by elegance was the relative simplicity of the particular derivations I've listed above. What you point out are failings of the current theory. Concerning the problem of background independence, I've heard diametrically opposing views and I can't really decide between them because I haven't studied that issue at all. It could be beyond my current level of understanding.

The most interesting equations are the general elementary coupling equation and the equation for the coupling factor in section 4.1. Next, the restricted elementary coupling in 4.2. Further the quaternionic form of the Dirac equation (formula 5 in 4.2.1).

Also interesting is figure 5 in section 3.13.1 and the table in section 4.9.

Many items are rather unconventional and controversial, so I have to spend much attention to properly justify the steps that are taken. Much of the math is put in the appendix.

I am careful not to use the T---- word for work on strings. A scientific theory is our best intellectual structure to understand a broad range of issue and makes specific predictions which are backed up by experiment. At this point, there is no experiment proposed to vote on work on strings being correct or not. Much of the work assumes some sort of super symmetry is true. In another blog, Tommaso is again reporting that again that no SUSY particles have been seen.

Are the days of work on strings numbered? I don't think so, too much inertia to stop it. Only a "correct" theory would put a stop to it. Those folks are not in agreement over what the "M" stands for :-)

I find it interesting that you refer to (super-)string "theory" in the past tense. Freudian?

By the way, there is one very big difference in the criticism of General Relativity vs. that of (super-)string "theory": General Relativity made testable predictions (that proved correct) right from the start, and made additional testable predictions (that proved correct) very early on.

Where are the testable predictions of (super-)string "theory" that have even come close to being correct (besides those already predicted by simpler theories)?

The one thing it almost does is reduce to Quantum Field Theory in an appropriate limit (though there are "fudging" aspects in how it does so: Can you say wholesale incorporation of Many-Worlds Interpretation of Quantum Mechanics [QM]?), and General Relativity in another (though in a sense, it's simply the usual QM reduction to the "classical" limit).

Unfortunately, it also does this with a thoroughly distasteful*, unobservable prior geometry that still clings like so much fecal matter.

David

* Distasteful to anyone that understands Einstein's principle of general covariance/invariance (that, unfortunately, as far as I can tell, has yet to gain a thoroughly acceptable name—one that doesn't involve confusion with less powerful principles).

Yes, I agree that the Kerr-Newman vacuum solution, in General Relativity (GR), is interesting. In fact, here we have a "classical" model for fundamental particles that doesn't suffer from the issues of previous "classical" models, such as being "too large", or "blowing itself apart".

The Kerr-Newman solution, with the known parameters (mass, electric charge, and spin angular momentum [per mass]) of all known fundamental particles have the interesting features that none have event horizons (which is likely a good thing, since any such event horizons could easily lead to odd scattering behavior, compared to what we observe), and even the close up size (let alone the "far away" size, due to the curvature of spacetime) is on the order of the Planck scale.

Additionally, the "shape" of these particles (close up) is that of a ring, or disk. (There is disagreement, among Relativists, as to which it is. The coordinate systems "blow up" in this region, so trying to determine which it is is exceedingly difficult.)

Now, does this "mean" anything, considering the likely "breakdown" of GR (and likely even Quantum Mechanics as well) in this regime? That, perhaps, is the most interesting question.

David

There exists a lower limit of the parameters for which an event horizon can be formed. Below that limit the contents of the item appears naked. Since all three parameters play a different role their importance changes with the radius of the item. At very small radii the spin becomes the most important player, far more important than mass. As far as I know all elementary particles are naked.

If #1 were the case, I would expect we would have "been there" by now. Admittedly, many consider the "problem" to be that the "languages" seem to be quite different. However, it is easy to show that the language of Differential Geometry (the language of Riemannian Geometry and General Relativity) applies quite nicely to Quantum Mechanics (at least at the so called "first quantization" level). (One simply allows the "fiber space" to be a complex vector space. However, I believe there is some "platinum" yet to be mined in taking a look at complex manifolds that can be described as real manifolds.)

The principle arguments seem to be waged by those in camps #2 and #3.

However, judging from the history of "scientific revolutions", it seems to me that such "revolutions" invariably involve the "sacrifice" of one or more "precious, long cherished belief(s)" of the science of the time. So, the question occurs to me: What "precious, long cherished belief" of present science will need to be "sacrificed"?

Now, before speculation runs rampant (especially from the "cranks" and "crackpot" crowd): This has, historically, never been some "new fangled" "belief" that came as a result of the previous "scientific revolution". The "belief" has always been something that is so "ingrained" that it typically goes as an unspoken assumption that (practically) no-one has ever seriously questioned.

So, what is it? (I do have my idea, but I don't think this particular forum is the right place, at this time. [I expect to 'blog on such sometime in the not too distant future. So don't fret too much.])

David

I don't think we've fully learned our lessons from interacting quantum field theory yet. I'd like to hear impressions from people in the field, but it sounds like in QCD, we still can't predict parton distributions for the collider experiments, and the only interactive method we have is lattice methods which there are still mathematical questions of whether the limit even converges properly to the quantized continuum theory.

It's also possible we (or at least I) haven't even fully learned the lessons from basic quantum mechanics. I saw a foundational talk at an APS conference once, where they perported to show QM not only required dismissing local-realism theories, but also required dismissing hope of non-contextuality. That last part was news to me and breaks my brain too hard to think about, since we tend to think or even demand a notion of separability when solving problems.

"This has, historically, never been some "new fangled" "belief" that came as a result of the previous "scientific revolution"."

I don't think this applies anymore.

QM originally was studied in depth using non-relativistic equations. We knew this needed to be changed, and led to Dirac's ansatz which taught us about anti-matter and other useful things, but had many problems itself. Which then led to trying to quantized fields as well, which naturally required treating everything as fields, which was an interesting insight. But this too had problems, which led to new mathematical techniques to solve. We are still at this point now, as suitable mathematical techniques for truly dealing with interacting quantum mechanical system are still being worked on. Even so, the possibility of a 3/2 spin particle leads to inconsistencies, which investigation of (I don't think this is historically how it was found, but I heard a prof lament they didn't follow up on this themselves) practically demands super-symmetry. Studying super-symmetry finds gravity naturally falls out. Many such ideas lead to inconsistencies, unless one considers multiple dimensions. What findings are useful interesting clues? Without experimental guidance, that question only leads to bickering.

The theoretical problems have become notoriously difficult with very 'ambiguous' leads from experiment for a long time now... theorists have essentially been left to their imaginations. The only leads we seem to have currently to guide theory are:

- strange behavior in the neutrino sector (large experiments can't even agree on mixing parameters, possibly due to invalid theoretical expectations ... but what the theorists come up with in response often looks unappealing)

- dark matter

- cosmological constant

- dark energy

- possible inflationary epoch evidence in cosmic background

- quantized theory of gravity (although some push the notion that gravity is just statmech on steriods, and so isn't a field theory that should be quantized)

Basically, the "beyond our current theories" hints have been way too vague for way too long. Look how much discomfort QED caused people at first with renormalization procedures, etc. Without useful experiment to guide and give confidence, we'd probably still be arguing about that. We unfortunately have been in an experimental desert for a long time, which is unlikely to change soon (except for possibly the neutrino sector, there's also a small chance the higgs field has some surprises up its sleeve). So theorists have been wandering blindly with more and more radical ideas for quite awhile. I have seen no evidence of theorists considering any principle sacred. It's not like every idea has been exhausted, but it is likely any wondering about breaking this or that principle has been at least given an initial discussion. The papers wondering about the nature of time, often feel like useless meta-physics philosophy.

I wish we had more specific experimental data to go by. We have big problems, but only vague guidance.

That being said, it's not like science has done nothing. A lot IS known, and it is frustrating to see people using Science2.0 to promote pet theories that don't even understand the question they are trying to answer. We need more people posting articles that are "reports from the field" like Tomosso, or like you (hopefully soon) teaching math and insights regarding "complex manifolds that can be described as real manifolds".

I have often quipped that the single most beneficial thing that could happen to theoretical (and experimental) physics would be the discovery and study of a microscopic black hole, in our neighborhood! We could truly learn a lot!

I think there are other useful experiments, as well, that we can do, at our present technological level. Unfortunately, simply going to ever higher energies is not especially likely to be of great help. However, as we have seen, historically, such can often lead to surprises no-one expects. ;)

DavidIt will also be fun to watch some theorists then explain how it shouldn't have been a surprise at all because ... :)

I had to fill in for an intro quantum mechanics lecture/problem seesion once, and I was surprised how "obvious" the students felt everything was. They've heard all these pseudo-quantumy things so much growing up, that they didn't find it weird. It really bothered me. If they don't find quantum mechanics at least a little strange, I'm not convinced they are really understanding what the math is saying. I ended up using the "qauntum reflection from a potential drop" problem to convince them that it was NOT obvious to their intuition. It was fun to watch the moment of realization to those that clearly were blindly working through the math, until pointing out what it meant ... imagine trying to drive a car off a cliff and instead bounced back!

I think that may also be a problem with SR. To teach it properly, it almost feels like one has to take the extra step of doing the work to pull away all the popular science things they've misunderstood, and _then_ proceed forward.

But maybe, just maybe, the answers to all these questions will seem obvious to people in the future (probably only because their imaginations will be blind to all the possibilities that were eliminated well before their time).

Quasi particles are a very different thing, as the name is intended to convey. They aren't even as "fundamental" as composite particles (bound states of fundamental particles).

However, they can provide a testbed for many ideas, especially the *possibility* that all things we call "particles" may be mere excitations in some non-directly observable system (making all "particles" "quasi particles").

The "problem", however, is that they are only a testbed, and not looking at the fundamental levels or foundations of our present understanding. For instance, they suffer from inheriting the structure of our physical system.

One way or another, we need to "drill down" to the deeper fundamentals, not playing around with "models". (This is not to say that models don't help our understanding of actual and hypothetical systems. The problem is that they can never substitute for getting "down and dirty" with the "real thing".)

David

I'm quite aware of what the semiconductor does and does not rely "heavily" upon, having minored in Solid State Physics in my Ph.D. program. Only in a certain sense does the "semiconductor industry rel[y] heavily on quasi-particles." You see, none of the "quasi-particles" that the "semiconductor industry relies heavily on" need be considered anything other than "standard" electrons and "holes" (quite analogous to the positrons as "holes" in the Dirac "sea"). The modifications are not so different from the way photons are modified when traveling through a medium.

However, as I have cautioned before, don't confuse the model with the reality, *especially* using models as analogy to "reality".

David

If that is true, what then defines a hole?

"Choose a crazy metric if you must. Quaternions are a division algebra, they can follow the crazy metric if requested. There is always a way from here to there."

The definition of quaternion multiplication leaves no freedom. If the quaternions a=(a0,a1,a2,a3) and b=(b0,b1,b2,b3), then quaternion multiplication ab=(c0,c1,c2,c3) will always have c0=a0*b0-a1*b1-a2*b2-b3*b3. If you want to claim that a quaternion can represent a four-vector and the "c0" term from multiplication of two such quaternions gives you the spacetime interval, then you are restricting yourself to a particular metric.

Convert this to math to learn more:

Consider for instance the components of the metric for an inertial frame using cartesian coordinates in flat-spacetime. Your "c0" term matches your expectations well here.

Now consider the same spacetime, but instead of cartesian coordinates, use a spherical coordinate system for the spatial coordinates. If there is a four-vector A=(a0,a1,a2,a3) at point X=(x0,x1,x2,x3), then what is the magnitude of a? Does it match your expectations of taking a quaternion with components A=(a0,a1,a2,a3), and looking at the first component of AA?

Now, consider the same spacetime, but instead use Rindler coordinates for an accelerated frame. If there is a four-vector A=(a0,a1,a2,a3) at point X=(x0,x1,x2,x3), then what is the magnitude of a? Does it match your expectations of taking a quaternion with components A=(a0,a1,a2,a3), and looking at the first component of AA?

And regarding your comments on the "manifold" of quaternions, note that quaternions already come with a built-in sense of distance. How far is A from B? It is the magnitude of (A-B). This is a Euclidean space, as the other quaternion person in this thread has pointed out to you multiple times.

The sticking point between your perspective and mine may have something to do with an old theorem by Frobenius where he showed that quaternions are the only 4D division algebra _up to an isomorphism_. http://en.wikipedia.org/wiki/Frobenius_theorem_(real_division_algebras). The key part I wrote in bold, the "same shape". I could multiply two quaternions in R4 in flat spacetime, and would get the flat Minkowski metric interval, along with a 3-vector that has no name (I refer to it as a 3-rope for want of a better name). I multiply the same two quaternions in R4 in a curved spacetime, and get a result that has a slightly different interval and can in certain situations have exactly the same 3-rope. It feels like the curved spacetime situation would have the proverbial same shape.

Thanks for making the issue all about math. One starts with a 4-vector in Cartesian coordinates:

Now I wish to transform this to spherical coordinates:

These allow the coordinate transformation:

If I square A

_{Cartesian}I do expect to see the same thing as A

_{Spherical}in the first component. I have rewritten x, but it has the same value, as is the case for y and z. A basic trig identity will get one to the same spot using the radius and angles.

I had to lookup the Rindler coordinates. It has a metric like so:

In a daring and bold move, I decide to develop a new 4D division algebra, the Rindler numbers. It has this as the scalar. In the definition of numbers, I would use the same rules as are in play for quaternions, then hit them all with a minus sign.

This might start a minor skirmish (That is NOT the multiplication I was taught!), but once one took a more general view - it is an operation that takes two numbers and returns one which this does - then perhaps people would see the generalization. Some may protest loudly enough to get me to write out the Q8 table which has 1*1=-1 (because now -1 is the identity for Rindler numbers), but it can be done.

Then I get slapped around by the math wonks because the Rindler numbers are the same "up to an isomorphism". Thus quaternions can be used in the place of Rindler numbers so long as one understands the additional linear transformation that needs to be done.

Doug

Unfortunately you did the problems wrong. Maybe you misunderstood the questions. For the "spherical case", you just wrote the the same (t,x,y,z) in terms of spherical coordinates (t,Rcos(phi)sin(theta),Rsin(phi)sin(theta),Rcos(theta)) and then substituted what R,phi,theta are terms of x,y,z to just get back where you started. That shows nothing useful. The point was to use the spherical coordinate system ... so you have fourvector=(t,R,phi,theta), and now also the metric components are different. Get it? Try redoing the problem.

Second, you got the Rindler coordinates wrong (actually all you wrote was SR in inertial cartesian coordinates with a different sign choice that your usual one for the metric). Rindler coordinates are for an accelerating frame. The inertial coordinates T,X are related to the Rindler coordinates t,x by T=x sinh(g t), X=x cosh(g t) where g is the acceleration constant for the spatial origin.

Third, I think you may have opened a new can of worms here:

You seem to be claiming that you can relabel the quaternion multiplication however you want, since 'up to a label' is just 'up to an isomorphism'. While I contest that claim, realize that even if you want to stand by that claim, it would mean the first component in the quaternion multiplication is arbitrary ... so quaternion multiplication doesn't pick out the spacetime metric, your arbitrary choice of "representation" did. This destroys your initial excitement about quaternions, which was because it just gave you the spacetime interval "for free".

So please think this over some more.

and this metric:

then one can calculate the Lorentz invariant interval in spherical coordinates this way:

The question is, how can this calculation be done with quaternions? The expected answer is that obviously it cannot be done. To my eyes the question itself is a bit off. Look at the 3-vector, {R, theta, phi}. Can one do a spatial rotation of a distance R into an angle phi? Not allowed by anyone, not even me :-) This is pretty clear when thinking about the units: the first two terms have units of time, the angles are dimensionless. The metric has an opposite issue, the first two non-zero terms being dimensionless while the last two have units of time squared.

What if I start with a different vector V, one that has the units all the same:

Now I view that as a quaternion over the R

^{}

^{4}manifold. Square it:

The first term looks like the one.

Must go out the door to buy a door. Will try to find time to think about Rindler, but I cannot make a promise on that.

Up to an isomorphism is about the group structure, not about labels.

You demanded that the units match. But then you fudged the result to get what you want.

You didn't just change the units, you actually changed the coordinate system.

If you wanted everything to have the same units, you could have just made the components unitless:

V = (t / t_planck, R / L_planck, theta, phi)

Or if you insist on changing the coordinate system, then why didn't you use:

V = (t,R/c,t theta,t phi)

or

V = (t,R/c,R theta/c,R phi/c)

Don't fudge the answer you want. Actually think about the problem.

Currently your answer restricts you to only some coordinate systems (when you are trying to claim that quaternions can give you the spacetime interval for any coordinate system), so it a clear contradiction. Also, your current "trick" only works if the metric is diagonal. It wouldn't work for example if considering a rotating frame, as the metric has some non-zero off diagonal components.

Your entire motivation for thinking quaternions are useful for spacetime comes down to your misunderstanding of this issue. I am glad to see you are starting to think about it, but please don't brush it off. This may be a useful topic for you to devote one of your blogs to figuring out. Either way, please do follow up on it.

The more interesting point is the off-diagonal terms. They most certainly exist, but the question is where should they go? In the Riemann-world view, there is one and only one place to put them. And based on that world-view, I can see the logic of dismissing my efforts. That path is not long. And I considered mentioning it in my earlier reply, but didn't have the energy to dive into it.

I recall a give-and-take I had with David about quaternion triple products and the gamma matrices that show up in quantum field theory. I had read other people claiming that quaternion triple products could do exactly the same thing. Such sources were almost true. The difference was five factors of

*i*between the two. So quaternions cannot represent the gamma matrices. The question is whether this difference changes any calculations. I don't know, but it is my backlog of things to explore.

It has been clear for some time that I may have to bite the bullet and ditch metrics. The reason can be seen by thinking about those off diagonal terms, say one like g

_{tx}. What does that element of a metric tensor tell us? It is about how the changes in time in one spacetime vector are related to changes in space in another spacetime vector. I view the time part as a real number and the space part as an imaginary number. Therefore the product of a real and an imaginary number is another imaginary number. The metric tensor however takes the product of a real and imaginary number and puts it in the real number spot because the metric tensor has no other choice. That strikes me as wrong. The path here is not long either, but it is like saying Riemann was wrong oh so many years ago and Riemann cannot be wrong, he is a math guy and there are all kinds of proofs. I have no interest in such a battle. His work works in arbitrary dimensions so will have an endless list of passionate defenders. My stuff only works in 4D where there is a logical place to put variations in

*tx*. From my perspective, the metric tensor give one part of the answer of what it means to multiply two 4-vectors together.

In the quaternionic format of the Dirac equation two quaternionic probability amplitude distributions (QPAD's) play a role. One is the quantum state function of the electron. The other represents the local superposition of the tails of the quantum state functions of distant particles. It represents the influence of inertia. The quantum state function of the electron acts as a drain. The coupled QPAD acts as a source. The drainage compresses the space in the environment of the electron.

QPAD's combine "charge" density distributions with "current" density distributions. In a static QPAD the carriers of the "charges" move uniformly. They can be interpreted as tiny patches of the shared parameter space of the QPAD's. The "charges" form a scalar field. The "currents" form a 3D vector field.

Both for complex quantum state functions as for quaternionic quantum state functions the squared modulus of the distribution can be interpreted as the distribution of the probability of the presence of the carrier of the properties of the considered particle. When restricted to a single spatial dimension the two versions offer the same functionality.

As a bystander, since quaternions already have a well defined magnitude, it is very clear how Hans' stance is supported by math. I find it curious that Doug admits he doesn't understand Hans, and despite all the help from others, Doug seems to not really see the flaw in his claims, or see enough to think the problem is something to worry or even care about.

Doug, since your blog appears to be an exploratory one rather than a teaching one, maybe you should resolve to learn from at least one of your larger mistakes each week? Just think, in a year, you'd have corrected over 50 mistakes in your understanding! Well before then all the big misunderstanding should be gone and you'll interatively be getting close to just working out the last little details in your ideas!

This value is positive definite, meaning that the only way it can equal zero is if all the parts equal zero, otherwise it is a positive number. Hans makes the claim that therefore one should not even bother seeing if problems can be solved in special relativity using quaternions. He doesn't bother, the logic is too obvious. That was easy. Or lazy.

One subtlety overlooked is that requiring the conjugate operator with a product makes the multiplication non-associative:

These point in different directions although their norms are the same.

To actually solve problems in special relativity using quaternions, I never used the conjugate operator on a product. This is what the square of a quaternion looks like:

Apple, meet orange.

Conjugation is the switch of the sign of the imaginary parts. Reflection is the switch of the sign of a single direction in imaginary space. Three independent reflections exist. Quaternions have eight different sign selections. In a continuous quaternionic distribution the sign selection does not change. Therefore a quaternionic distribution exists in eight different sign flavors. One of them is the base sign flavor. It has the same sign flavor as the sign flavor of the parameter space of the distribution. See the table for significance of these sign flavors (http://www.scitech.nl/English/e_physics/#_Table_of_Elementary particles/waves)

For quaternionic distributions this results in eight sign flavors. The first is taken to be the sign flavor of the parameter space. The last column in the picture lists the handedness. The fore-last column lists the "colors". The red and blue squares represent the directions and anti-directions (up-down) of the imaginary base vectors of the quaternion. The real base vector is ignored in this picture.

In elementary particles two quaternionic distributions that are sign flavors of the same base sign flavor are coupled. The sign flavors determine the type and the properties of these particles. The number and direction of switched legs determine the charge, but only a switch of handedness provides charge. Otherwise the particle is neutral. The above mentioned table lists 64 couplings of which 8 have coupling factor equal to zero (they are waves rather than particles). The waves couple quaternionic distributions that have the same sign flavor.

One of the best things to come out of my collection of retractions is an appreciation of how any proposal to replace GR has to be absurdly close as a metric theory - something I knew from working through Taylor and Wheeler's two books on Special Relativity and Exploring Black Holes - but also must be absurdly close to the linear theory known as gravitomagnetism. I can see spending a few months fishing for a different Lagrangian that ends up at the same field equations. The nice thing about such a quixotic quest is a Mathematica notebook can tether the work to the real world.

I know Feynman was not a happy camper about the process of regularization and renormalization. There are too many constraints to propose something like QED is wrong. Instead, it might be incomplete in the following way. There is an emphasis in physics on symmetries. Perhaps we need to strike a balance between symmetries and conserved things with things that don't have nice symmetries and are not conserved. A complete set of equations has those terms that are conserved and those that are not. The first equation I ever looked at constructed with quaternions had that property, the square of a quaternion:

The first term is the Lorentz invariant interval of special relativity fame. The other three terms are, what, quaternion detritus? [My dad was a fan of fancy word choices, in this case detritus "is rubble or debris and detrition is erosion by friction"]. I have derived the Schrödinger equation, the Klein-Gordon equation, and the Dirac equation, each time generating extra quaternion debris. There might be platinum it those equations-with-no-names, but I am well aware of the limits of my skills to extract such works.