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

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.

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?

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.