New Quaternion Math Leads to New Reasons Why Physics Work

[tjakal]

Stumbled across this wanting to know more about quaternions from a background of artistic computer animation. *Mind = blown* The specs on my brain do not allow for this. Must be real interesting solving natures deep mysteries, truly admirable work.

[QuaternionEM]

Doug,

I always believed that the quaternion was appropriate for modelling

spacetime , but I felt the imaginary 3-vector should represent space, and the real part time. What you may have here is watershed event that makes the quaternion palatable for all calculations.

However the space-like interpretation relates to the very small

instead of the very large as one might expect..?

This means events in the quantum realm can only be linked to the classical realm by traveling faster than light???

[sweetser]

@QuaternionEM Quaternions can do much, but I got stuck trying to reach out to gravity. That required hypercomplex multiplication, which is like quaternions except there are no minus signs anywhere.

Travelling faster than light is NEVER allowed. One can do math with spacelike events without travelling faster than light. Straight old calculus doesn't work, but you can take the norms of derivatives - the average amount of change - that can be known.

[StillOnMars]

Why bother with all this? All you need in life is Jesus!

[sweetser]

@StillOnMars Praise be Martian, speak the truth! I should spend more time reading the good book and sniffing glue, but neither is part of the scientific method.

[avrilinblood]

My question to you is:

Can quaternion math explain quantum mechanic predictions? I have a very serious issue going for the present day approach of QM - "it works because it works" - QM predictions have been verified experimentally but with hardly any logical insight of WHY and HOW. I think the incompatibility of QM with classical physics is due to the wrong direction physics has been taking the last century, and that there should indeed be a simple, logical unity between classical and quantum

[sweetser]

@avrilinblood My hope is doing calculus correctly in spacetime may explain why classical physics - ordered in time - is different from quantum physics - cannot be order in time. My dream is to have animations the correspond to every equation in physics, whether it be classical or quantum mechanics. Our minds would be happier if the visual processor could be fed QM animations.

[avrilinblood]

Thank you so much for this video. I am still to dumb to get it all, but unlike most theoretical science this kind of complexity doesn't provoke subconscious unease, I have a very strong intuition even when it comes to areas I fail to understand. I have stumbled onto quaternions trying to figure out where did science went wrong, seems like Maxwell's Aetheric universe and quaternions equations were the last true path before his equations got vectorized, simplified and so on...

[allseevid]

Campinas is the Brazilian capital of queers

[MegaAstrodude]

I need more information.

[Kcmasterpiece347]

@MegaAstrodude YUP

[MegaAstrodude]

Mendel Sachs obtained a similar 16-component tensor that unifies electromagnetism and gravity. I'm trying to check for one-to-one correspondence with yours. I think you've got it!

[MastersOfInfinity]

Oh so, more proof that God is real?

Nice.

[bvssvni]

Thanks! That was informative. I got a formula x = e^q/(e^q+1). Can this be inverted using quaternions? The product or sum of x's is used to create new vector-spaces by manipulating the gradient which actually creates the pseudo-effect of a gravity center from particle distances. If quaternions can be used then perhaps there is a shortcut around GR complexity.

[bvssvni]

@bvssvni Sorry, I typed wrong: e^g/(e^g - 1) thanks.

[himbrom]

Why the cosine of a straight line in quaternion space looks like a straight line again: Writing cos(-i*x)=[exp(x)-exp(-x)]/2, the timelike part of x commutes with the rest and can be pulled out of the exponential, explaining the "spiralling out" kind of behaviour. But the rest, a spatial quaternion, will give a negative real upon squaring, just like the complex i. Thus Euler's formula exp(i*x)=cos(x)+i*sin(x) is applicable resulting in a constant direction "i" times an oscillating prefactor.

[sweetser]

@himbrom The "i" in a quaternion animation context is a 3-vector in space. It is the same 3-vector for all. That is why all the points line on the same line in 3D space. Sines and cosines in an animation are about time playing with space. That is why the complex planes have spiral patterns. It is just darn difficult for our brains to think about time playing with space!

[david0aloha]

It's cool to see you keep working on this. I'm transferring to undergrad physics and mathematics from another program and I'm intensely interested in complex and quaternion geometry, as well as fractals (especially visualizations of them). I'm a very visual thinker so I find myself trying to map out these scenarios and images in my mind. My goal is to be able to work with the quaternion version of topics like maxwell's equation(s) - I currently understand the concepts better than the math.

[sweetser]

@david0aloha Best of luck in your transfer. The only way I figured out this stuff was practice, practice, practice. Feel free to email me outside of YouTube (sweetser@alum.mit.edu) with any questions.

I am doing the post production work on comparing my proposal for gravity with the linear perturbation of general relativity.  Turns out that exactly half the terms flip signs, nice.

[Paulginz]

When you are dealing with functions of a single quaternion you are actually just taking the usual complex functions of a complex number on a certain quaternion subalgebra or plane. (as you say, you keep feeding the same quaternion that points in the same direction). Hence it shouldn't be surprising that the graphs lie on a plane with one time dimension and one space dimension. I think attempting to visualise these in 4D only serves to increase confusion.

[sweetser]

@Paulginz Complex functions are always graphed statically in the complex plane. There is no difference between the real axis and the imaginary axis other than a 90 degree rotation. Animating the real line changes this issue. A reflection in time does not look like a reflection in space.

A bit of confusion may be a good sign, indicating new visualizations. Integral dimensions are certainly easier than non-integral dimensions, but now there is a new area of study, fractals.

[Paulginz]

@sweetser A physicist and a mathematician walk out of a conference on 11-dimensional string theory.

MAT:"Great talk."

PHYS:"It went right over my head. I can't visualist geometry in R^4, let alone R^11."

MAT:"It's really easy. I just visualise R^n and set n=11."

What I'm trying to say is that, yes, relying on an algebraic symmetry to reduce quaternions to complex numbers doesn't really help visualise the quaternions on an intuitive level. Choosing a better camera angle in the still image would.

[sweetser]

@Paulginz Good joke :-) I don't think of quaternions as R^4 since that suggests all dimensions are identical. Time and 3D space are different based on our own experience. You, like Euclid, want points to last forever. You are in great company. The analytic animations are an effort against that long running tradition (the plea to the intuitive level). I need to see 3 complex numbers that share the same real number (time) interacting, and stills alone cannot do the job.

[infinummjb]

@sweetser Does your work present any new math/physics when combined with fractals? I have a feeling it might be interesting to merge those two together.

What do you think?

[sweetser]

@infinummjb The Mandelbrot set is generated by z_(n+1) = z^2 + c with the complex number z not going off to infinity. It is natural extension to swap in quaternions for the z, and see what happens. I have not invested enough time to do this. I did write some code, but not enough code, my bad. I need to wander around the quaternion plane some sensible way.

[robertsdionne]

sweetser, I'm 7:33 minutes into your video and you've blown my mind. Classical physics is the stuff in the light cone. Quantum mechanics is the stuff outside of the light cone. Without the norm, the event derivative has two outcomes when taken outside the light cone! I'm going to watch the rest now. :)

[2point718281828]

Sweetser!

I remember reading your stuff on physicsforums, keep up the good work.

[sweetser]

Will do! This stuff is profoundly fun. Working on another video. (twitter acct: visualphysics). I even resubmitted to the Independent Forum, see if I can reopen that line of communication which was helpful to me.

[jannonymous]

read your Quaternion history page on your site... I think you omitted something worthy of mention. A great physicist used Quaternions, Nikola Tesla.

[mike4ty4]

Question: Why is it so important to be able to add the dipole moment gravity when as you say experimental evidence suggests it has to be at least a quadrupole moment?

[mike4ty4]

And as for not giving up GR, the problem is not that it is too "threatening" (what does that mean?) but that there currently isn't any theory that agrees as well with the experimental evidence collected over the past many decades that works like GR does. Can you formulate or do you know of, an alternative that would pass the muster of the data?

[mike4ty4]

Also is there any experiment that could be done to test your proposed "unifying" of gravity+EM? Something that doesn't require a particle accelerator of unfeasibly huge size?

[Subspace4d]

please make a video to show what quaternion in fact do thank you.

[helalsinbeabi]

Quaternion math is something like a sugar, crystallized heptagonal surface leads us to think about the coordinate system of degrees of freedom, author shows a great effort to explain us something but make sure that he shows nothing. Sugar dissolves in hot water, molecules are dispersed in solution, however, crystallized shape forms again with the evaporation, this is called flexible sudden change, like hopf bifurcation, please think about it, try to think 3D, most of you knows nothing but I am.

[marsCubed]

@helalsinbeabi I am pretty sure that I did not understand your point here.

However, you made me think of an article in new scientist some years ago.

Water was shown to form crtstal like structure at extremely short times scales. while it is liquid.

This seems to relate to your point and i wonder whether it is complementary to your suggestion. it seems to be a factor which might make more sense of this.

[marsCubed]

typo, crystal*

[ortega24024]

As my mentor once said: "Hamilton tried to invent vectors - but being Irish, came up with quaternions instead". I went through a quaternion-obsession phase once (hooked by connections with Riemann sphere and Mobius transforms) but non-commutativity frustrated some of my efforts. How does your work relate (if at all) to twistor/spinor theory? Reformulations are 10 a penny...need to get the LHC up and running to settle SO much debate!

[sweetser]

A quaternion is the union of a scalar and a 3-vector. Physicists call this a 4-vector because it transforms like a 4-vector, a coverup of the difference between scalar and 3-vector.

This work has no relation to twistor theory which works on the manifold C^4, while this is H^1 and Hypercomplex^1.

I have not animations of U(1), SU(2), and SU(3) in any context.

I though you would be interested in the part on a 4D commuting division algebra, a counter-example to a famous theorem of Frobenius.

[jannonymous]

non-commutativity is natural for 3+ dimensions. take an object and rotate it in space and see if those operations commute. They dont!

[Matrix29bear]

Great video. Nice to know other folks are aware of "Directional Spacetime".

The term "Dimensions" is soooo wrong a term to use on infinite spatial zones. Dimensions are tiny line segments indicating one spatial direction. Silly that people think of "Length" "Depth" "Width" as "Dimensions" when they are mere 90 degree angular vectors within an infinite sphere in 3-directional-space.

You are mentally coherent in the realm of Nikola Tesla. I think you need to mention "Spin" directions as well

[sweetser]

Dimensions do not have a size, although folks who work on strings promote that idea. They do represent degrees of freedom, of which there are 3 for space, one for time. I do have a bias for x, y, and z, but all this math applies to spherical coordinates, cylindrical coordinates, and bizarro coordinates, so long as the space part has 3 degrees of freedom.

[mangopowerade]

I found this deeply fascinating, and I hope your talk was a success. :D

[tyrbolo]

Interesting. Looks like you might have something

don't worry about what other people think LOL

[n2liquid]

I think his problem is a little more complex than we can think. It's not something that he can simply "not worry". He needs to show that he is right to the world, so he won't be another one who dies and have his discoveries re-discovered a century later, you know. Also, his hat is really nice.