Pretty unbelieveable...

[alexjrgreen]

An example of a tensor that can be thought of as composed of dyadics is the inertia tensor of a 3D rigid body.

Something of this kind should apply to the behaviour of the Polywell.

[blaisepascal]

If you can accept that Gibbs-Heaviside vectors (those commonly taught in high school and undergraduate college courses) are (more or less) derived from the more general quaternions (which are actually a pairing of a scalar with an "axial" or "pseudo" vector, pseudovectors being able to stand in for an ordinary or "polar" vector with no loss of information, but definitely not being the same thing), and ignore Gibbs' and Heaviside's denials that this borderline thievery actually took place, then I hope you can see that if a "flat space" restriction applied to quaternions it would also apply to ordinary vectors, hence to tensors. This does not seem to be the case, since tensors (presumedly based on dyadics) are often applied to curved spacetimes in GR.

It's really a confused mess that should have been hammered out over a century ago.

The thing I pick up from reading stuff by quaternion boosters is that they tend to feel that (a) Heaviside & Gibbs dropped important stuff from quaternions in their work, stuff of real physical significance, and (b) that more modern work in GR and QM are based on Heaviside & Gibbs vectors, and are thus missing the dropped stuff from Hamilton.

The trouble is that in my view, both propositions are dubious at best, outright wrong at worse.

The flaw is that "vector", as a mathematical and physical concept, has been manipulated, generalized, and overused over the years to the point of confusion between the different meanings.

To Hamilton, a "vector" was a quaternion of the form xi+yj+zk, and could be used to represent the position of a point in 3-dimensional space. You could scale a vector by r to form the new vector rxi+ryj+rzk, and you could rotate the vector about some axis by multiplying it by a general quaternion a+bi+cj+dk, where (a^2+b^2+c^3+d^2) = 1. Rotation is conceptually simple, but a tad tricky due to the issues of computing the proper quaternion for the desired rotation -- but combining rotations is easy, given the properties of a quaternion. The length of a vector was sqrt(x^2+y^2+z^2).

To Heaviside & Gibbs, a vector is an "arrow" in 3-dimensional space which has a direction and a length. You can scale a vector by changing it's length, add them using the parallelogram rule, compute a scaler from a pair of them (the "dot product") based on their lengths and the cosine of the angle between them, or compute another vector from a pair of them (the "cross product") based on their lengths and the side of the angle between them. It is possible to represent a vector by a triple of numbers (x,y,z) and the four operations have simple formulas based on the triples. Rotation of a vector can be done can be done using formulae involving the cross product.

Both Hamiltonian quaternions and Gibbs vectors are designed for and work with 3 dimensions, which was fine when the general belief was that physics was 3 dimensional (and Euclidean). But Einstein and Minkowski skewered that with relativity and space-time. SR and GR can't use Gibbs vectors because they don't work in (3,1)-space. In particular, the cross product giving a vector is a distinctly 3-dimensional construct, as a 4-dimensional analog of a cross-product would have 6 components, not 4, so can't be represented by a 4-vector.

Quaternions can't be extended to 4 dimensions either. Quaternions are generalized from complex numbers, but that method of generalization doesn't continue very far. The next, and last, generalization along those lines are the Octonians, but they don't have the same sort of geometric interpretation that quaternions do. They certainly can't represent 4-space transformations.

The modern use of vectors is a generalization of the idea that a Gibbs vector can be represented as a triple of numbers. Instead of cross vectors, a rich panlopy of linear operators and tensors exist which accomplish the same goals, and more, and can be generalized to any number of dimensions.

When I hear people talking about how quaternions are so much better for EM, GR, and QM than Heaviside/Gibbs vectors, and how this should have been resolved a century ago, I feel like they are missing a century of mathematical physics.

[alexjrgreen]

When I hear people talking about how quaternions are so much better for EM, GR, and QM than Heaviside/Gibbs vectors, and how this should have been resolved a century ago, I feel like they are missing a century of mathematical physics.

This is a straight contest between mathematical models. Some models are easy to test, some can only be tested during an unusual event like an eclipse, and some (like string theory) cannot yet be tested.

Doug Sweetser has put a lot of teaching material on the web about quaternions. Are quaternions directly equivalent to vectors or tensors: no. Does that difference result in testable hypotheses: possibly.

The most useful property of quaternion dynamics is that it keeps account of bulk properties. For plasma physics, with its often counter-intuitive behaviour, this may be important.

[DeltaV]

The thing I pick up from reading stuff by quaternion boosters is that they tend to feel that (a) Heaviside & Gibbs dropped important stuff from quaternions in their work, stuff of real physical significance

Do you mean "real physical significance" as in traditional 3x3 rotation transformation matrices (the columns of which are the transformed G&H unit vectors) completely ignoring the physically demonstrable fact* that rotating by one full turn (or an odd number of full turns) about a fixed axis is topologically NOT equivalent to rotating by two full turns (or an even number of full turns)? Quaternions collapse to integer +1 for even numbers of full turns (strings not entangled) and to -1 for odd numbers of full turns (strings fully entangled), which corresponds to observable physical reality. G&H-based rotation matrices do not even address the intrinsic topology of rotations in 3D.

*Dirac strings (no intended connection to string theory):
"Some mechanisms related to Dirac's strings", Edgar Rieflin, Am. J. Phys. 47 379 (1979)
http://www.evl.uic.edu/hypercomplex/html/dirac.html
http://www.science-project.com/science- ... -body.html [Edit - fixed dead link, twice... Big Brother?]
http://www.google.com/patents/about?id= ... dq=3586413

[Edit - added picture]

and (b) that more modern work in GR and QM are based on Heaviside & Gibbs vectors, and are thus missing the dropped stuff from Hamilton.

See Sachs' books/papers, where he says quaternions are the building blocks ("irreducible representations") for the "most general" form of GR. He's not throwing out Einstein's work, but rather extending it to its full potential. Sachs' GR metric has extra, rotational aspects ("spin degrees of freedom"), and he asserts that QM falls out of this new formulation in the linear, low-energy limit.

The flaw is that "vector", as a mathematical and physical concept, has been manipulated, generalized, and overused over the years to the point of confusion between the different meanings.

I agree completely.

Rotation is conceptually simple, but a tad tricky due to the issues of computing the proper quaternion for the desired rotation -- but combining rotations is easy, given the properties of a quaternion.

I think most of the trickiness about the "proper" quaternion results from the above-mentioned topological property (odd vs. even number of full turns) not being taught in high school, as it ought to be. Trying to express a rotation (or a rotation-dilation-entanglement - "Gravitation", Misner, Thorne and Wheeler, Ch. 41, I think) with a method that doesn't account for the topology means giving up some information, causing ambiguity. Taking an active rather than a passive view of rotation, when feasible, sometimes makes things easier too.

Both Hamiltonian quaternions and Gibbs vectors are designed for and work with 3 dimensions, which was fine when the general belief was that physics was 3 dimensional (and Euclidean).

I believe that mathematics (at least the deep parts) is discovered by humans, not designed/invented, but if you want to think of it as the latter, go ahead. G&H vectors are intended for 3D. Agreed. I however disagree that quaternions are intended only for 3D. That was Hamilton's original intent, which failed, and he then realized he needed 4 dimensions (3 of one sort and 1 of another sort) for things to work out. On the family tree of mathematics, there are four allowed "division algebras": real, complex, quaternion, octonion (dimensions of 1, 2, 4, 8). The last two are non-commutative and the last one is also non-associative. It has been proven (1898) that there is no further progression of this sequence.

SR and GR can't use Gibbs vectors because they don't work in (3,1)-space.

Agreed.

Quaternions can't be extended to 4 dimensions either.

Wrong. Quaternions are very similar to spinors and Pauli spin matrices, but don't need any extra "i"s. Many researchers have used quaternions for spacetime since 1914. A few of the many more recent writings are M.J. Walker or J. Kronsbein (Am. J. Physics, 1950s-60s), Misner-Thorne-Wheeler (1973), S. DeLeo ( http://arxiv.org/PS_cache/hep-th/pdf/9508/9508011v1.pdf ). My own preference is for approaches that don't use complexified quaternions (aka biquaternions). Complexifying them makes them no longer a division algebra.

The modern use of vectors is a generalization of the idea that a Gibbs vector can be represented as a triple of numbers. Instead of cross vectors, a rich panlopy of linear operators and tensors exist which accomplish the same goals, and more, and can be generalized to any number of dimensions.

The goal should be to find the irreducible representations that give the most general expressions for spacetime physics, like Sachs does.

When I hear people talking about how quaternions are so much better for EM, GR, and QM than Heaviside/Gibbs vectors, and how this should have been resolved a century ago, I feel like they are missing a century of mathematical physics.

Funny, that's how I feel when I hear them say the opposite.

[alexjrgreen]

I also notice that a number of people are trying to make Polywell arguments using classical collision models. The dominant mechanisms for transferring energy between the ions and the electrons are collective mechanisms, not classical binary collisions. Our experience is that you have to do full-up kinetic simulations if you want to understand these mechanisms and their effects. We've been doing that for the past 1.5 years, and we plan to be doing a lot more simulations over the next 2 years.

If quaternions can help with this, that would be a step forward.

[MSimon]

I also notice that a number of people are trying to make Polywell arguments using classical collision models. The dominant mechanisms for transferring energy between the ions and the electrons are collective mechanisms, not classical binary collisions. Our experience is that you have to do full-up kinetic simulations if you want to understand these mechanisms and their effects. We've been doing that for the past 1.5 years, and we plan to be doing a lot more simulations over the next 2 years.

If quaternions can help with this, that would be a step forward.

A link to the Nebel quote:

viewtopic.php?p=20032#20032

[UncleMatt]

I have yet to see proof that capacitor stacks actually gain mass when energized. I have been following this technology for many years, but can't seem to find clarity on that question. Please let me know if I have missed something, and if possible please point me to a duplicated research study where they found proof such a mass fluctuation is actually produced by capacitor stacks. That would increase my confidence in this concept actually going somewhere.

[alexjrgreen]

I also notice that a number of people are trying to make Polywell arguments using classical collision models. The dominant mechanisms for transferring energy between the ions and the electrons are collective mechanisms, not classical binary collisions. Our experience is that you have to do full-up kinetic simulations if you want to understand these mechanisms and their effects. We've been doing that for the past 1.5 years, and we plan to be doing a lot more simulations over the next 2 years.

If quaternions can help with this, that would be a step forward.

A link to the Nebel quote:

viewtopic.php?p=20032#20032

Something like this: Quaternion-based rigid body rotation integration algorithms for use in particle methods

[DeltaV]

I have yet to see proof that capacitor stacks actually gain mass when energized. I have been following this technology for many years, but can't seem to find clarity on that question. Please let me know if I have missed something, and if possible please point me to a duplicated research study where they found proof such a mass fluctuation is actually produced by capacitor stacks. That would increase my confidence in this concept actually going somewhere.

It's not just when they are energized. They have to see a changing acceleration, properly phased, while the energy is changing.

http://physics.fullerton.edu/Woodward.html

[DeltaV]

If quaternions can help with this, that would be a step forward.

A link to the Nebel quote:

viewtopic.php?p=20032#20032

Something like this: Quaternion-based rigid body rotation integration algorithms for use in particle methods

Molecular dynamics researchers also use quaternions quite often, and they've developed some interesting approaches to numerical integration for simulation.

My post about Sachs was initially in response to Paul March's comment, but since Sachs claims his formulation also describes nuclear forces, there might be some utility there for p-B11 fusion interactions. That's waaay outside of my league though.

[DeltaV]

These are also sorta interesting:

http://www.rwgrayprojects.com/Universe/ ... -Pratt.pdf

http://arxiv.org/PS_cache/arxiv/pdf/081 ... 1185v1.pdf

[UncleMatt]

I have yet to see proof that capacitor stacks actually gain mass when energized. I have been following this technology for many years, but can't seem to find clarity on that question. Please let me know if I have missed something, and if possible please point me to a duplicated research study where they found proof such a mass fluctuation is actually produced by capacitor stacks. That would increase my confidence in this concept actually going somewhere.

It's not just when they are energized. They have to see a changing acceleration, properly phased, while the energy is changing.

http://physics.fullerton.edu/Woodward.html

Forgive my crude phraseology. My point remains, however, in that I have not seen enough proof in the form of duplicated experiments with direct confirmation that the transient mass fluctuations claimed actually exist that are the basis for this concept. I have read the papers about the experiments performed by March and Woodward, but still looking for experiments from their peers with supporting data and conclusions. I would love for it to be true, and make this concept practical, but still haven't seen enough evidence.

[alexjrgreen]

Forgive my crude phraseology. My point remains, however, in that I have not seen enough proof in the form of duplicated experiments with direct confirmation that the transient mass fluctuations claimed actually exist that are the basis for this concept. I have read the papers about the experiments performed by March and Woodward, but still looking for experiments from their peers with supporting data and conclusions. I would love for it to be true, and make this concept practical, but still haven't seen enough evidence.

It's a direct consequence of e=mc^2 and Newton's second law...

m= e/c^2

F = dp/dt = d(mv)/dt = m(dv/dt) + v(dm/dt)

[UncleMatt]

Forgive my crude phraseology. My point remains, however, in that I have not seen enough proof in the form of duplicated experiments with direct confirmation that the transient mass fluctuations claimed actually exist that are the basis for this concept. I have read the papers about the experiments performed by March and Woodward, but still looking for experiments from their peers with supporting data and conclusions. I would love for it to be true, and make this concept practical, but still haven't seen enough evidence.

It's a direct consequence of e=mc^2 and Newton's second law...

m= e/c^2

F = dp/dt = d(mv)/dt = m(dv/dt) + v(dm/dt)

I understand the theory behind it, I am waiting for experimental evidence to show that one can measure the mass variances with regard to this concept, and then in a practical way produce propulsion effects. I'm not against this reasearch at all, just looking for more verifiable evidence than is currently available, thats all.

[Betruger]

Apparently it's relatively cheap to test out. Maybe you could see it with your own eyes?