I see you have an essay here. I remember an essay last contest with similar content, it dealt with quaternions. I presume you are the same fellow. I will have to read it tomorrow.
As for spelling, I often write from and form and similar errors, drop s from plural words and so forth. It is just a glitch.
Cheers LC
I just gave your paper an 8. The reason I did not give a 10 is that I think things go awry at the end. Up until equation 4.2 and page 8 things look very interesting. It is at the end when you write that the time is a scalar and not GR or SR and so this departs from GR. Actually this is a form of Clifford algebra CL_{3,2}(C) ~ SO(3,2) based on the interval
s^2 = i^2 + ω^2 - (\bf i}^2 - (\bf j}^2 - (\bf k}^2,
where for one of the "time" components constant (say i) this defines the anti-de Sitter spacetime AdS_5 ~ SO(3,2)/SO(3,1). This is general relativity! Spacetime can be described according to the spinor variables A and b with the interval
s^2 = a·b + axb, x = times,
so the scalar a·b plays the role of time and the "vector part" axb is spatial part.
The equations 3.3 and 3.3.1 are a "biquaternion" version of the electromagnetic or Yang-Mills field tensor. The biquaterion form comes from the 2x2 matrices A_0 and A_i etc in eqn 3.3. I worked out a couple of years ago how the derivative
d(f{\bf i} + g{\bf j} + h{\bf k}))
----------------------------------
d(x{\bf i} + y{\bf j} + z{\bf k}))
leads to the field tensor. This approach leads to a bi-quaterionic form that I think is connected to the Petrov-Penrose tensor.
So there is a lot of good stuff here. Given that you are not trained to be a math physicists it is pretty remarkable.
LC
Hi Gary,
I don't recall seeing the complex 'i' in GR for the time dimension, it is just c*t from my experience and thus a scalar time. Have you introduced the i in your work? I know it is in the Schrodinger wave equation and thus features in standing waves (i.e. for particles), but I don't see why the same notion of time cannot be used in an understanding of GR and QM (from a WSM standing wave perspective of the Universe).
Regards,
Declan Traill
Lawrence,
Many thanks for taking the time to read and comment on my essay. In the previous essay contest, I had an entry that dealt with using quaternions as a basis for Calculus. There were several other essays that dealt with quaternions also. Perhaps mine is the essay that you remember.
Can you recommend a suitable textbook(s) for Clifford Algebra and de Sitter Space? Perhaps the time has come for me to learn these things. Keep in mind that I am an engineer by education and engineers do not study things like Group Theory ... perhaps there are some prerequisites necessary prior to Clifford and de Sitter ...
Do you think that 6*pi^5 and the resulting proton diameter are coincidence, or could they have a physical significance? Or is it simply wrong?
The concept of scalar time that I am presenting is a little different from what you have interpreted, but I did not really make it very clear. I am proposing that there is a "master" scalar time that is multiplied by Euler's Equation. This produces two time-like values. The product with the cosine is hopefully the time used by SR & GR. The product with isine is hopefully the time used by QM. I then interpret all of this geometrically by having the "master" scalar time operate on the complex plane. That is where the 2x2 matrices become essential and the bi-quaternion form of Hamilton's work presents itself.
Best Regards and Good Luck,
Gary Simpson
Declan,
Your statement is basically like saying that you do not see the unit vectors i, j, and k in Euclidian Geometry. It might be true, but it also misses the point. Minkowski space-time is based upon 4-vectors and that means i*c*t. The complex i is there whether or not you see it presented in writing. This is a prime example of how Wikipedia can unintentionally mislead people.
Best Regards and Good Luck,
Gary Simpson
Dear Mr. Gary D. Simpson
As you said:
Even if you don't think you are a top thinker (hey, I'm not for sure) it is worth participating. You just never know how your thoughts might affect someone else who might then affect someone else ...
If someone has only ten percent of good ideas, it's better than prosaic essay that supports the prevailing opinion. Your article is full of ideas and is useful and supported with math. Your math is certainly not numerology. But section, Proton Diameter leads to wrong conclusions. Your Calculated value is close (but not in the middle of the interval) from the CODATA value which is known to very few of significant digits. This means that the calculated value is likely wrong, but true.
In my article "Cycle towards Methodology of Everything", http://gsjournal.net/Science-Journals/Research%20Papers/View/6731 I got value for the proton radius 8.764-16m.
I have it determined, as well as all other relations without using dimensions.
Therefore, I argue that the debate about the number of dimensions is waste of time.
In this year's essay I confirmed the importance of Mach principles and values of Proton shift, which I calculated. I would appreciate if someone finds errors in formulas.
Regards,
Branko
Not necessarily. There is nothing magical about Tensors, they are simply 4 dimensional matrices - a way to neatly encode equations involving 4 dimensions. The time dimension need not involve the complex i. If you say that i, j and k are unit vectors in the 3 physical dimensions, you could say that complex i is a unit vector in the time dimension, but it need not be a vector at all. The equations work just fine treating time as a scalar even though it occupies a 'dimension' (in the mathematical sense, rather than a physical sense) in the 4D matrix.
I rather think that the complex i vector comes into play when 3D standing waves are involved (i.e. in the case of particles comprised of 3D standing waves). In standing waves the medium moves in circles. In the case of Electromagnetism these circles are traced out by complex number vectors (this is where the complex i comes in) and so time itself is scalar, but as it determines the rate of these complex vector rotations in particles the complex i and t are closely coupled.
Anyhow, I have given your essay a community score of 10 because your essay is well written and I like your work on complex numbers and quaternions and I think there should be more of it.
Best of luck in the competition...
Regards,
Declan Traill
Dear Simpson
Your Harmony essay is very good. You envisaged, 5 axis coordinate system. , which are x,y,z,t and i the imaginary. You used it to derive the Diameter of proton and other results nicely.
Is that really necessary to have that i axis? Why should we need imaginary things to derive real things.....? Both the classical Physics and QM are for real matter at different scales.
Hope you will clarify me....
Dear Simpson
I modified the text and FQXi server did not take that change(modifications) even after 10 times trying and restarting my computer.... So I am putting another post...
"Is that really necessary to have that i axis? Why should we need imaginary things to derive real things.....? Both the classical Physics and QM are for real matter at different scales."
Hope you will clarify me....
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
