Is Quantum Theory As Fundamental As It Seems? by Michael James Goodband

Michael,

Sorry but I cannot agree with your position (I am sure shared by others). You wrote down the standard way of thinking about the metric tensor when you formed the double contraction of the dx_i and the metric matrix as though all elements were scalars and the coefficients of dx_i are simply a scalar 1x8 matrix. This is where you went off the rails, for in O this is not the case. The differential form dx_i should be considered an O algebraic element, i.e. when using O algebra over the field of real numbers, real coefficients attached to, and inseparable from the O basis elements, not ignoring the basis elements as you appear to have done. Any multiplications are completely under the definition of the particular algebra in play. Thus generally speaking, the form of the desired scalar ds^2 is completely determined by the algebra and in fact whether or not it even admits a scalar product form.

The native O space is intrinsically (natively) rectilinear *because* the O basis elements are orthonormal, not due to any characteristic of the coordinates themselves. So there will always be a rectilinear representation of anything O. I derived the ensemble derivative from the position of a general curvilinear transformation off of the native rectilinear system and that is how it was introduced. I then presented the rectilinear ensemble form as the trivial transformation of the rectilinear to itself, a unity transformation. So I am unclear on your statement "... your derivative is written out in rectilinear co-ordinates".

This is not tensor calculus as you mention, for tensor calculus is all about matrix algebra, and O cannot be represented in a matrix form. Also, tensor calculus a priori assigns forms to be invariant, and then forces differentiation to work out by adding in the Christoffel stuff. My approach defined differentiation straight up as a diffeomorphism by determining what the curvilinear form for the limit of the O product of an O functional algebraic element with the O surface normal algebraic element divided by the enclosed volume as the volume approaches 0. This is a direct extension of the integral definition of 3D divergence, gradient and curl, after understanding the three different product types on the surface normal separately within them are all combined inside the more fundamental H (extendable to O) algebra.

There was no slight of hand, no changing of anything behind the scenes. I have not "derived the space-time split within (my) algebra". This split is the child of the less fundamental 4D space-time/tensor calculus approach and some choices within. It does not exist in O. It does not have to because O allows things to work out without it. This is purely the outcome of understanding the differential operator is an algebraic element, and its application is effectively a multiplication governed by the algebra. Take the time to go through it, you will find the results are as presented.

You stated "... but a space-time split as S1 is time and S10=S3*S7 is all the space." Are you requiring the split because of your belief this is how time and space *are* out of orthodoxy, or do you actually require it as an independent requirement of your specific work? This is unclear to me.

Rick

Michael,

Wouldn't volume be implicit in an "entropy change with energy?"

Hi Michael,

I think you are mistaken about the apparent over-counting of R in S15, both from the physical point of view and mathematical point of view.

You wrote: "If we look at all the normed division algebras, R, C, H, O the total dimension is 15, but R is occurring here 4 times: once on its own and once in each of C, H and O.

I am afraid this is a very strange way of looking at division algebras. There aren't any redundant occurrences of R in S15, because the algebras R, C, H, and O do not coincide on R. This is seen most clearly by noticing that S7, for example, is a non-trivial, non-product bundle of S3, and similarly S3 is a non-trivial, non-product bundle of S1. Identifying R's of O and H or those of H and C within such bundles is thus like identifying different animals because they appear to have the same number of heads. But the heads of R, C, H, and O aren't interchangeable like the heads of the mechanoid Kryten in Red Dwarf. So I think you are not only unjustifiably over-stretching the principle of "make no preference" in reducing the number of dimensions from 15 to 12, but also making a mathematical mistake in doing so.

But suppose we overlook this metaphysical and mathematical mistake and look at the issue purely physically. Well, then the evidence of quantum correlations suggests---and I have demonstrated this with a number of non-trivial examples of quantum states as well as a general theorem---that our physical space respects the symmetries of a parallelized 7-sphere in a manner different form that in particle physics. If we ignore this evidence and consider only S10 based theory, then the resulting Unified Physics would necessarily be Nonlocal (NonUP), because it would not be possible within it to reproduce, say, the GHZ-type four-particle correlation in terms of explicitly written down local functions. The only way to arrive at a manifestly Local Unified Physics (LocUP) is by recognizing the central role played by the manifold S15 in the edifice of division algebras.

Now Sedenions, as you point out, do not form a normed division algebra. But why should that stop us from considering them when they do in fact give us the normed division algebras at all physically relevant scales? Moreover, as a topological manifold S15 is no different from S10 or any other analytically manageable manifold. So there can be no mathematical reason for rejecting S15.

Finally, as you point out, there remains a question of time within an S15-based theory. But I think this problem is easily solvable in the manner I have suggested above. Therefore I remain optimistic about the S15 program.

Best,

Joy

That was me above.

Michael,

" Note an implicit dependence on space has been slipped in under relativity - (causation, time, space, event)."

Isn't volume a measure of space?

The gold standard would be the idea the energy of the universe, from galaxies to background radiation, is being dispersed as the universe expands, ie. increases in volume.

As I recall, entropy is the amount of unusable energy in a system. It seems a useful idea for mechanical systems, but why is it so important for physics? In a truly closed system, the energy isn't lost, but simply dissolves any order within that system ie. the useful part. In order to actually lose energy, ie, drop the temperature of the entire system, it would have to radiate it out of the system, thus into an increased volume.

I don't think one can discuss energy without time, temperature, or the various dimensional descriptions of space; distance, area, or volume.

Personal note; The daughter was accepted into Johns Hopkins today. Big news around the family.

Rick,

"My approach defined differentiation [... ] with the O surface normal algebraic element divided by the enclosed volume as the volume approaches 0. This is a direct extension of the integral definition of 3D divergence, gradient and curl, [...] understanding the differential operator is an algebraic element, and its application is effectively a multiplication governed by the algebra."

Having studied the Arfken reference you earlier pointed me to I understand what you're saying, and once again thank you for the insight.

Edwin Eugene Klingman

Hi Michael,

Let us follow your logic of determining number of dimensions by identifying the "redundant" dimensions in the algebras R, C, H, and O. As is well known, C, H, and O can be constructed from R by the Cayley-Dickson doubling process:

C = R R i,

H = C C j,

O = H H h,

where i and j are the unit imaginaries appearing in quaternions

q = w i x j y k z

with k = ij, and h is a new unit imaginary satisfying h^2 = -1 and anti-commuting with i, j, and k. The basis { 1, i} of C is thus complemented to a basis { 1, i, j, k } of H = C C j with k = ij; and similarly the basis { 1, i, j, k } of H is complemented to a basis { 1, i, j, k, h, ih, jh, kh } of O = H H h. Thus O is spanned by one scalar belonging to R and 7 imaginaries i, j, k, h, ih, jh, and kh.

Let us now apply your logic of counting dimensions. There appears to be only one real basis element belonging to R shared by all division algebras---so that is one real dimension. Then a total of 7 imaginary basis elements, with i belonging to C, H, and O; j and k belonging to H and O; and h, ih, jh, and kh belonging to O---so that is 7 distinct imaginary dimensions. This gives a total of 8 dimensions.

My question to you then is: Why are you not considering a unified theory based on S8 rather than S12 ??? According to your logic you have four redundant dimensions in your theory.

Best,

Joy

Hi Michael,

I am sure you recognized the illogic of your dimensions counting from my previous post, but just in case you missed it, let me reiterate that just as R = { 1 } occurs 4 times in the 15 dimensions of the algebras R, C, H, and O, the complex C = { 1, i } occurs 3 times, once in C, once in H, and once in O; and the hyper-complex H = { 1, i, j, k } occurs twice, once in H and once in O.

So just as over-counting of R reduces the total dimensions by 3 giving 15 - 3 = 12, the over-counting of C reduces the total dimensions by further 2 giving 12 - 2 = 10, and the over-counting of H reduces them by yet another 2 giving 10 - 2 = 8. Thus, no matter how you count the dimensions your way, your logic dictates that your theory should have been based on S8 instead of S12.

The broader conclusion here is inescapable. Either you embrace the 8D octonion approach of Rick's work (and my work on Bell) if you want to maintain your logic, or you acknowledge the absurdity of your dimensions counting and embrace the full force of division algebras encapsulated in S15.

It is either 8 or 15. There is no justification for 12 or 11.

Best,

Joy

Hi Michael,

"Time, space and mass - the things of the fundamental unit system - can be dumped in the metaphysics bin, which cannot be directly derived in physics without effectively assuming them in some form to start with."

Exactly so. I think most people, who muse on creation and their place in it with benefit of mathematics, have developed a kind of intuition for what is created from time and space alone. These can only be metaphysically real, i.e., independently objective though fundamentally "distant."

"I've experienced the same thing in Tai-Chi and rock-climbing, where the extra information you take in without the filtering is a definite aid in attempting to evade the downside to gravity."

I was also involved in an Eastern form of martial arts some years ago, and understand the sense of immediacy. I think perhaps, as you mention it, the training may have contributed to my recognition today of how information is networked continuously to a single source of task management that manifests to the agent as a (locally real) continuous range of variables.

Side note to John: congratulations to your daughter.

All best,

Tom

Hi Michael,

Thanks for the summary. In your post addressed to me you have correctly identified the point where we disagree. But in the post just above that one you wrote:

"...there is no experimental evidence for macroscopic extra dimensions."

Well, the evidence is in the eye of the beholder. To my eye there is overwhelming experimental evidence---namely the routinely observed quantum or strong correlations---which can be explained local-realistically ONLY by recognizing that there are 4 "hidden" and 3 manifest macroscopic dimensions of the physical space, constituting the total of 7 dimensions. If one sticks to 3 macroscopic dimensions only, then there is absolutely no way to explain the strong quantum correlations local-realistically. Thus any physical theory based on only 3 macroscopic dimensions must acknowledge the magic of non-locality in the world, just as Bell claimed. So the choice is between 4 hidden, macroscopic extra dimensions, or manifestly non-local physics in which my choice of raising my arm over here could have an instantaneous effect somewhere in the remote parts of the universe. 4 macroscopic extra dimensions or magic, there is no other way out of it.

The above of course does not exclude the possibility of any number of compactified extra dimensions, such as the 7 in your theory. I have no doubt that with 7 compactified extra dimensions your theory predicts ALL of the QT predictions --- including ALL the spin correlation results, without exception. The only trouble is that your theory is then necessarily non-local in the sense claimed by Bell, in the same manner that orthodox quantum theory is non-local. Thus if your theory indeed predicts absolutely everything QT predicts, then it also predicts non-locality. This is way I prefer to call your theory NonUP --- or Non-locally Unified Physics.

To my mind there is a beautiful way out of the non-local magic of your theory, which is also at least one way forward for my non-relativistic framework to become relativistic. That beautiful way lies in S15.

On the other hand, I do appreciate your point about the possibility of the extra dimensions being "ghost" dimensions, and I agree with you that the presence of quantum correlations in my rotation experiment would indeed be the first experimental evidence that the extra dimensions in question are real. That is why I have invested so much in that experiment, against almost universal scepticism.

Best,

Joy

Michael,

On space-time split, meaning a split signature, not a distinction between time and spatial, I see your response as answering my question as yes, it is orthodoxy driving your stated acceptance of space-time split signature.

Your 1): orthodoxy on previously applied math used to describe the experimental data without any attempt to demonstrate if and only if, non sequitur on the error

Your 2): aligned with the orthodoxy of GR, where intrinsic curvature has never been shown to be necessary for the description of gravitation.

Your 3) non responsive to the question at hand

The remainder could be written perhaps more accurately without a space-time split, and if included, begs the question on S^n radii being represented as positive definite being in conflict with the assumption. Perhaps you should directly address this to clear things up.

Not space-time split - not physics, sorry but not my opinion. Frankly, other than following orthodoxy, I see no reason for it in your work. But you know your work better than I do, which is why I ask. Whether or not is is a requirement is a very important question, also why I ask when someone like you demands it.

Rick

Further to my post above: To my mind the strongest evidence in favour of sedenions in the present context is that they permit creation ex nihilo:

zero = [ a non-zero ] x [ a non-zero ].

Something out of absolutely nothing!

Hi All,

It is easy to see in Joy's macroscopic model that the radius of the sphere is equivalent to time. When I said divide by c above, that would only be for photons in an EPR scenario. In an EPR-Bohm scenario, time from creation to detection would be divide the radius by the magnitude of the particle velocity.

I suspect the S15 topology is compaticified relating to Michael's particle "space". And the extra dimensions of it don't exist macroscopically. I will explain more later.

Best,

Fred

Hi Joy,

Here is something from Tony Smith titled "Why not Sedenions?". It is quite lengthy but make sure to scroll to the bottom to where he answers the question.

Ok, have to go do some work for awhile.

Best,

Fred