Euclidean Space and Time Transformations, Quantum Gravity

[Yablon]

I mentioned that while making improvements to my paper about fractionalized Dirac monopoles I have come across a form of Euclidean transformation between space and time which preserves the Minkowski invariant dt^2-dr^2 and has certain quantized constraints which may provide the basis for a quantum theory of gravitation.

I am still completing this redrafting, but in accordance with my belief that anything brand new should be posted as soon as possible after it has been double checked, I now have written up these new and very fundamental results about space and time and quantum gravitation in a draft I have posted to my blog, at:

https://jayryablon.files.wordpress.com/ ... or-spf.pdf

This new material about spacetime and gravity and will be found near the end of section 7.

My next step will be to go back into SU(2) and show how to generalize the half-integer charges discussed in section 5 into fractional charges of all denominators, using the basic methodology of section 5. When that is completed, this paper will be done, except that I may then develop a few specific, explicit examples for roots through the sixth root of unity.

Have fun!

Jay

[Matthew]

Jay, I have been reading this paper and I don't see anything wrong with the fundamental premise.

In the last section you get into a lot of explicit matrix calculations. I wonder if it could clarify matters to rewrite these formulas "coordinate free" as it were, using geometric algebra. You are going back and forth between real 3D and complex 2D matrices and vectors, writing out all the elements with respect to some chosen basis.

Of course this is the way to find exciting things in mathematics - by inspiration and imagination, following a hunch, sweating hard and doing the hard work and seeing patterns emerge! But once you are onto something, it's time to look for a way to express the essence. Moreover, if many of your own calculations turn out to be "just" instances of a well known and well understood mathematical facts at a deeper mathematical level of description, you make it easier on the reader (some readers, anyway) to switch to that already available language.

[Yablon]

Jay, I have been reading this paper and I don't see anything wrong with the fundamental premise.

In the last section you get into a lot of explicit matrix calculations. I wonder if it could clarify matters to rewrite these formulas "coordinate free" as it were, using geometric algebra. You are going back and forth between real 3D and complex 2D matrices and vectors, writing out all the elements with respect to some chosen basis.

Of course this is the way to find exciting things in mathematics - by inspiration and imagination, following a hunch, sweating hard and doing the hard work and seeing patterns emerge! But once you are onto something, it's time to look for a way to express the essence. Moreover, if many of your own calculations turn out to be "just" instances of a well known and well understood mathematical facts at a deeper mathematical level of description, you make it easier on the reader (some readers, anyway) to switch to that already available language.

Hi Matthew:

I appreciate the feedback. I only just now saw your post, because my computer crashed a couple of weeks ago and I have been somewhat cyber disabled since then. I expect my new computer next week.

Since the original post about a month ago I spent quite a lot of time trying to see if the premise that these imaginary space coordinates are time coordinates a la Minkowski could work, physically, based on what is observed in the natural world. While I can get the mathematics to all hang together nicely, and I love the three time dimensions, there is a problem with the physical results which have caused me to believe that that these Euclidean space and time transformations do not exist in the real world. But in its place, I now have an interpretation (of but four days vintage after several weeks of struggle) that I will momentarily detail wherein what I found in (7.17) of https://jayryablon.files.wordpress.com/ ... or-spf.pdf in fact represents time evolution in the manner of the Heisenberg equations of motion, and I am exceedingly confident that this will stand up. I hope to have a new draft with this revised view posted in the next few days. First, let me explain why I feel it necessary to abandon the Euclidean space and time transformation idea.

The most important problem with Euclidean space and time transformations arises from the fact that for worldlines of non-relativistic or even mildly-relativistic material bodies such as electrons with v/c~, the spatial length r traversed over a given time t is exceedingly less than the time elapsed as represented by the ratio . And it arises from the well known fact that all atomic radii are within an order of magnitude of the Bohr radius. Specifically, these Euclidean transformations – if they were physically real – would result in atomic radii that are orders of magnitude larger than the Bohr radius owing to the fact that along the worldliness of material bodies such as electrons, whereby even a small component of t being converted into r via a Euclidean rotation would greatly increase the atomic radii in a manner that has no observed support. If my premise was physically true, we would have some rather big atoms. And we don't. Consequently, I have discarded as a viable interpretation of these imaginary space coordinates.

However, the same (7.17) of https://jayryablon.files.wordpress.com/ ... or-spf.pdf contains space coordinates multiplied by an exponential with . If you think about it, by promoting the to Heisenberg position matrices , these exponentials act on the space coordinates in exactly the same way that time evolution is brought about in the Heisenberg equations of motion. The divisor m, which is associated with the roots of unity and which also governs the fractional Dirac monopole charge, also compresses or dilates the time intervals in the equation of motion. The times in the evolution equation become , which inherits the factor of 3 in . Thus, larger m roots accelerate the time, and for anything periodic, bring about a higher frequency (cycles per time) in proportion to .

While the Euclidean space and time transformation proved difficult for me to sustain, I believe this new understanding of (7.17) in https://jayryablon.files.wordpress.com/ ... or-spf.pdf will stand the test of time. Just look at (7.17), promote the to matrix operators, apply all that is known about matrix mechanics, and you will see that this complex factor multiplying the space coordinates represents quantum mechanical unitary time evolution.

Jay