The coincidence you cite I am not sure of. Where things went I think a bit awry is with the interpretation of time as a scalar being different than that of relativity.
I am going to try to carve out some time to work on this. I will present a bit here.
The book by Hestenes and Sobczyk is good. Hestenes writes in a straight forwards way that avoids overly pedantic stuff.
Cheers LC
Branko,
Many thanks for reading an commenting upon my essay. Perhaps I have given you an idea? If so, please use it wisely and freely.
Regarding the proton diameter, the value that I estimate is in the lower part of the NIST data range. It is very close to the value determined using muonic hydrogen. Having said that, I should also say that my value is outside the range published by the Paul Scherrer Institute as part of their study. I also claim that the Earth's reference frame is in absolute motion. I have not yet determined the correction for this motion.
I have briefly read your essay but not carefully enough to make any sensible comments. I'll study your work more closely and comment.
Best Regards and Good Luck,
Gary Simpson
Satyavarapu,
Thank you for taking the time to read and study my essay. Do we really need imaginary things? I will simply say this ... if the universe is truly 5 dimensional, then the extra two dimensions are not imaginary. The evidence that I offer in support of this is the proton diameter calculation. Admittedly, this is only circumstantial evidence, but if correct then it seems to be pretty good evidence.
In some ways it is unfortunate that we use the term "imaginary" when speaking of a certain set of numbers. It is better to think of the complex i as an operator that causes direction to be reversed when applied twice.
Think of it like this ... in the Michelson-Morley experiment, no aether drift was observed. The presumption is that motion with respect to the aether is the same as motion between two physical objects. But what if we really don't know how to describe or measure this motion. What if motion with respect to the aether is in the direction of the complex i? What if we are moving in such a way that we do not even notice the two extra dimensions? I realize this seems very speculative. I will simply ask how does the proton diameter work if there are not 5 dimensions?
Best Regards and Good Luck,
Gary Simpson
Hi Gary,
Your work with quaternions is fascinating. You begin by noting that time is viewed as either a scalar or a vector ("dimension"), and choose scalar, with complex 'i' viewed as a dimension.
Since we have discussed Geometric Algebra, I thought you might be interested to know that a recent book Understanding Geometric Algebra for Electromagnetic Theory, by John W Arthur, treats time both ways. He first develops Maxwell's equations viewing time as a scalar. After this he treats time as a vector and shows how this leads far more naturally to the fully relativistic formulation compatible with Hestenes 'Space-Time Algebra'. In neither of these is the complex 'i' a vector.
What's so special about geometric algebra is that every term has both a geometric and an algebraic definition, unlike every other branch of mathematics. In this scheme it's probably more correct to say that the complex 'i' does not appear in GA, since, although it has the algebraic value of the square root of -1, it is a pseudo-scalar, which is a different beast. Even so,in 2D it reproduces the complex plane very nicely.
I think of the 'i' in GA as an operator, and specifically an analog of the Hodge duality operator of differential geometry. It has the effect, when operating on a term, of transforming the product into the dual of the term being operated on. This is quite different from the scalar 'i'.
I don't know if this is any help at all, but I find viewing 'i' as the operator in GA to be quite helpful to my own understanding. I don't know that it applies to your immediate work, but since GA includes quaternions, and since you seem to be moving in the direction of GA, I thought I'd put my two cents in.
And thanks for your comments on my paper.
Best regards,
Edwin Eugene Klingman
Edwin,
Many thanks for reading my essay. I am planning to post one more paper to viXra.org to show the inversion of the matrix that I present in this essay. Then I plan to spend as much time as needed to learn GA and Dr. Hestenes' work. The extra reference will prove useful I am sure. I also think of the complex i as an operator.
I think there is a connection between the consciousness field that you propose in your essay and the scalar field that I propose in mine. It might be a way to add the observer to the picture.
Do you think that the 6*pi^5 observations are coincidence, or might they contain some truth? Or is the proton size calculation simply wrong?
Best Regards and Good Luck,
Gary Simpson
Gary,
I worked this up a couple of years ago. It is not difficult to understand and is related to your work. It also has content in your equations 3.
Your paper is not at all related to my paper here, but if you are interested you might find it interesting. It has little bearing on quaternions in a direct was. If you are interested I can send a paper I published which does illustrate a Clifford algebraic format for the equivalency of spacetime geometry and the Tsirelson bound of quantum mechanics.
Lawrence,
Many thanks for the post. Yes, please send a copy of the file that you mention to the email address on the cover sheet of my essay.
I've studied the note already and part of it is helpful to me. However, part of it is not consistent with Hamilton's definitions. You state that ij = jk = ki = ijk = -1. That is not what Hamilton stated. He stated that i^2 = j^2 = k^2 = -1 = ijk. As a result, ij = k, jk = i, and ik = -j. These identities impose handedness onto the system. We agree regarding anti-commutation of the unit vectors.
BTW, does anti-commutation of the complex i with the unit vectors seem reasonable to you?
Best Regards and Good Luck,
Gary Simpson
Here is the paper I published a few months ago. I intend to publish an extended version of this in something like Annals of Physics.
You are right in that ij = k, jk = i and ki = j, it is cyclic. I wrote wrongly there, even though I know otherwise. As for commutation, you do have ij = k and ji = -k. So this looks like commutators, not anti-commutators. One could potentially of course look at Jordan products or a graded quaternion system with
E^i = e^i + iσ^_{ab}(\bar θ^aψ_b + θ^a\bar ψ_b) + F_{ab}\bar θ^bθ^a,
which would be a sort of supersymmetric version of this.Attachment #1: LawrenceCrowell_V7N13.pdf
Dear Garry Simpson
Thank you for nice reply , which made me to go into deep thinking.... Best wishes....
I am reproducing the reply for the question you asked on my essay...
see there for the attachments....
Thank you very much for studying my paper so thoroughly and giving esteemed questions. I am just giving two reported cases of Galaxies / Clusters of Galaxies which are being generated after Bigbang
[35] Rakos, Schombert, and Odell in their paper 'The Age of Cluster Galaxies from Continuum Colors' Astrophys.J., 677 , 1019, DOI: 10.1086/533513, e-Print: arXiv:0801.3665 [astro-ph] | PDF arXiv:0801.3665v1 [astro-ph] 23 Jan 2008
[36] C. PAPOVICH et el, CANDELS OBSERVATIONS OF THE STRUCTURAL PROPERTIES OF CLUSTER GALAXIES AT Z=1.62, https://arxiv.org/pdf/1110.3794v2.pdf
See the CANDLES web pages also for simple language explanations.
There are many other papers and websites also if want them I will give them,
By the way, see the attachments to this post, to see these files for your quick reference...
Best Regards
SNP. Gupta
Hi Gary. It occurs to me that the Higgs field is supposed to be a product of complex and quaternion fields, so perhaps the multiplication in your first equation is relevant there.
I have become comfortable with the idea of 3 temporal dimensions from working with Fourier analysis of plane waves, but your essay reveals another possibility - complex time. Quite an interesting idea. Had to chuckle at your comment about nature being devious, having asked myself that a few times.
But I have to question the hypothesis of an absolute speed relating to the Mp/Me ratio in the way you suggest. If the cosmic microwave background defines the rest frame, the speed of the solar system is about 0.0012c, which is about 1/5 the absolute speed, 0.006136c, required to produce the Mp/Me ratio. A practical researcher does not ignore hunches based on numerical coincidence, but sometimes that's all it is. In any case, failure of this final speculation would not affect the rest of the work.
By the way, I think I first read about three time dimensions in Milo Wolff's book about space resonance and matter waves which seems to reflect some aspects of the Higgs mechanism - the book with Milo (presumably) and his motorcycle on the cover. His tripod website is still there. Sad to hear of his passing.
Anyway, nice work! - cw
Colin,
Thanks for reading and commenting upon my essay. If my memory is correct, you are familiar with quaternions and such. Hopefully this was an easy read for you.
Also, thanks for the heads up regarding the Higgs. That is exactly the sort of clue that I was hoping to get by posting this essay. I'm not sure where it will lead me but I will study the idea.
I think nature is more devious than we can even imagine ... I'm glad you liked the humor:-)
I was not aware that the cosmic background radiation implied a velocity of 0.0012 c. This is another good clue. There are certain types of average whereby a value is divided by the number of degrees of freedom that a system possesses. Since I argue for 5 dimensions, then it is possible that both arguments are true. I will need to refresh my memory on this. I seem to remember it from statistics. In any event, as you note, the mathematical structure does not depend upon the hypothesis presented as Equation 2.
You can definitely get three time dimensions simply by multiplying the complex i by the three spatial dimensions. But that implies to me that the complex i has the dimension and that time is simply a scalar that operates on the complex i.
Thanks again.
Best Regards and Good Luck,
Gary Simpson
Dear Gary Simson,
But previously also proton diameter was calculated, did that method also used the 5 axes grid ...? I dont know how they calculated earlier. Hope you can show some light on that....
Best Regards...
=snp.gupta
Gupta,
The calculation that I present is based upon a simplified 5-D model.
To the best of my knowledge, QED can calculate the size of the proton by solving a system of simultaneous differential equations. I have read that it is an extremely difficult task. I do not have the ability to determine the proton size using QED.
Best Regards,
Gary Simpson
Dear Sir,
You have correctly said that this topic is about mindless mathematics and that time is not a dimension. But do extra-dimensions exist? We are hearing about it for over a century. But it has never been found. In fact, the term dimension is still to be unambiguously and scientifically defined. Quaternions are a complex number of the form w xi yj zk, where w, x, y, z are real numbers and i, j, k are imaginary units that satisfy certain conditions. Is it a physical description? If yes, why cannot complex numbers be used in computer programming? If i stands for square root of -1, what does j, k stand for? This type of mindless mathematics are being addressed in this contest. Kindly clarify.
Regards,
basudeba
Basudeba,
Thank you for reading and commenting upon my essay. I will answer your questions to the best of my ability.
You agree that time is not a dimension and you question the existence of extra dimensions. You ask for proof of extra dimensions. I have presented an accurate calculation of the size of the proton based upon a model that uses 5 dimensions. I argue that the accuracy of that calculation is circumstantial evidence in support of extra dimensions.
There are two i's in my essay. One is the complex i. The other is the unit vector i. They are not the same although they both satisfy i^2 = -1.
The thing that is puzzling you also puzzled me for a long time. Regarding quaternions, you state that i, j, and k are imaginary units. You believe this because they satisfy i^2 = j^2 = k^2 = -1. This thinking is incomplete. The unit vectors i, j, and k represent the x, y, and z axes respectively. The unit vectors make it possible for the axes themselves to be a part of computations. The unit vectors and the complex i can also be operators. For example, ij = k and the application of the complex i twice causes direction to reverse.
As an example, consider two simple addition problems. Let x = 1 and let y = 1. It follows that x y = 2. This is a simple scalar result. Now let x = i and let y = j. It follows that x y = i j. This is a vector from the origin of length sqrt(2) at an angle midway between the x and y axes.
Is this a physical description? Yes. Quaternions and complex numbers are used in computer programming. Quaternions produce rotations in video games.
Essentially, you must train your mind to think of the unit vectors as the axes used in geometry.
What my essay does, is combine physical space, in the form of an arbitrary unit vector, with the complex plane used by QM and GR. For the case of an inertial reference frame, it is very easy to visualize that combination as a 3-D universe.
I hope that clarifies your questions.
Best Regards and Good Luck,
Gary Simpson
Fun stuff Gary!
Your work appears to have tie-ins with recent work by Stephen Adler. Here he talks about complex-valued spacetime foam..
Here he suggests we need to test for quaternionic values in QM experiments.
Also notable is recent work by Hyun Seok Yang asserting that non-commutative spacetime is inherently emergent. See these papers for starters.
I already know I want to give you a high score, but I also know that elevating you now will make you a target. So I'll wait until your rating dips down a bit, before I rate your essay, and then boost it back up.
All the Best,
Jonathan
Jonathan,
Many thanks for reading my essay and for the numerous references for further study. That will probably take me awhile.
Dr. Klingman also chooses to vote late. You are both wise men.
Best Regards and Good Luck,
Gary Simpson
Gary,
Your work starts with the wave function and ends with a very nice acknowledgement to the late Dr. Milo Wolff, founder of the Wave Structure of Matter. In between, your understanding of math to explain the proton is what this FQXi contest is all about - well done and good luck to you.
A quick note about Dr. Wolff for those who are not familiar with his work. Dr. Wolff started a revolution for those that are working on a theory of matter that can be explained by wave energy. There's a lot of work remaining to prove this theory, but it has been an inspiration to some, and hopefully a call to action to others to explain the mysteries of the universe.
Jeff
Jeff,
Many thanks for reading my essay. Yes, Dr. Wolff was a major influence in my thinking. He is sorely missed.
Best Regards and Good Luck,
Gary Simpson
Dear Sir,
You say: "The unit vectors i, j, and k represent the x, y, and z axes respectively. The unit vectors make it possible for the axes themselves to be a part of computations". In that case why complicate things by adding terms i, j, and k? The x, y, and z axes could have been sufficient by treating them as unit vectors. After all, vectors are different only because they have movement (energy) and direction. The axes provide direction. The axes have no meaning without something to represent. We also use mobile coordinates. Thus, what is the justification of adding i, j, and k? Further, x, y, and z are real, whereas i, j, and k satisfy i^2 = -1, which means complex. Why should we use complex numbers at all? They do not have physical presence. Anything that has no physicality cannot be a part of physics. Your statement that you have tried to "combine physical space, in the form of an arbitrary unit vector, with the complex plane..." presupposes that both do exist physically. Is there any proof in its support? Can you give examples?
Your statement: "Let x = 1 and let y = 1. It follows that x y = 2. This is a simple scalar result. Now let x = i and let y = j. It follows that x y = i j. This is a vector from the origin of length sqrt(2) at an angle midway between the x and y axes" only conforms our views. Addition is linear accumulation, which is possible between similars. Here x and y have the same value and belong to one class. But x = i and let y = j shows that they belong to two different classes. You cannot add 5 oranges and 3 apples. You can add them only as fruits. We have submitted an essay to physically explain 10 dimensions. Unlike your 5 -D inferred space, we have shown direct correlation, where we have used the same logic as you have shown here.
Regards,
basudeba
