Of course, one can use Clifford Algebra Cl(3,1) in electromagnetism. But then, one is not working with Gibbs vectors anymore (NOTE 1.1, 1.2) , but with multivectors (NOTE 2.1, 2.2, 2.3). Better yet, we can try to check with tensors. There is a physics fundamental reason (NOTE 1.31P) why tensors are better than Cl(3,1) when one transforms coordinates.
Also, every element of a geometric algebra, such as Cl(3,1) and spacetime algebra by Hestenes, can be identified with a tensor, but not every tensor can be identified with an element of a geometric algebra. In that sense, tensors are more general than pure grade multivectors. The rank of a tensor is not restricted by the dimension of the base vector space like the grade of a multivector.
This means that there is no way to represent any 2nd-rank tensor by a bivector, vector, and scalar. Or, in particular, no way that any 3×3 matrix could be represented by a 2-vector in 3D!
So, it is not a 3D here and 1D there, but a union of the two, in 4D, that is desirable in electromagnetism spacetime. One can transform length into time, and vice-versa.
Tensors allow that, Cl(3,1) does NOT do that. And electromagnetism may be at least 4D, maybe 6D or more, which tensors can represent, but Cl(3,1) would not.
Similar question: what's the relationship of tensor and multivector . The short answer is that all multivectors are tensors, but not all tensors are multivectors, so geometric algebra is not and cannot be isomorphic to tensor algebra.
Therefore, geometric algebras (Clifford Cl(3,1), Hestenes) seem insufficient for electromagnetism, in all possible generality. What is your view?
NOTES:
1.1. The 3D vector cross product is not a vector, but a tensor. This is well-known and the reason to invalidate its use in Maxwell's equations, and physics equations. This is both a physical and a mathematical reason, commented below. But, would it not serve a restricted purpose adequately? No, it can create mistakes upon reference frame change, for example, simple mirroring, gives us wrong units in the SI MKS, worldwide, and this is all old news that have to be somehow continuously repeated, with a "life of its own," as a misconception in even current college books at competitive US universities.
1. 2. MATHEMATICAL REASON: The 3D vector cross product is not a closed operation in the 3D vector space, it produces a member that does not belong to the same set, the 3D vector space, although it may look like it in some cases.
1.3. PHYSICAL REASON: Both sides of an equation representing a physical relationship, such as A = B, must change equally when the frame of reference changes and the so-called inertial condition is obeyed, as already stated by Galileo, Newton, and Einstein, that the laws of physics are the same for all uniformly moving observers. But if one writes an equation using a 3D vector cross product, such as A= B x C, the left and right side may transform differently if the coordinate frame of reference changes, while still inertial. In the past, this was accommodated, not solved, by considering spurious things such as polar and axial vectors, and pseudoscalars. This is solved using tensors, which maintain the form A = B under inertial reference change.
2.1. Multivectors, in geometric algebras, or Clifford algebra, or Cl(n,1) algebras, or STA Hestenes algebra, solve the mathematical reason (1.2), creating a closed space in all operations. No geometric algebra operation is mathematically unsound.
2.2. Multivectors do not solve the physical reason (1.3), but tensors do. This is another reason, besides lack of isomorphism with tensors and no use of time as a coordinate in spacetime, that invalidates the use of geometric algebras in equations of physics, including electromagnetism.
2.3. With multivectors, If one eliminates the physically wrong results, by requiring an additional step of "filtering" through, let us say, a Hamiltonian, this will not produce those results that are physically valid but were ignored in the first place, using just multivectors. A sequence of filters cannot filter less than the first filter, well-known in physics, math, and engineering. This appears, more easily to see, in non-euclidean spaces, such as anything larger than, let us estimate in general, a few Planck lengths.
3. This thread arrived at a first conclusion, which is stated in the NOTES 1-2 above. There is no room to refuse to notice an obvious thing, or to re-explain here, the reasoning and references are available above, to anyone.
4. We are now moving on, to non-mathematical aspects of using Clifford algebras, even beyond the physical reason given in the NOTES above, where using Clifford algebras would be detrimental to special relativity. We are talking about time.
Ed,
Thanks you for your attention about scoring...and your attempt to police RG. But me, as well as Hans and others have been on RG for a long time, and we know to ignore rude people "who know all", or repeat like parrots what they believe to have understood in the books of their masters. I spend a good deal of my time on RG, less to praise my articles or books, than to help people to understand some difficult issues. Of course read my papers can help, and it seems that you have some reading to catch, notably on Clifford algebras, and mathematics.
In my post I underline 3 points :
- Clifford algebras are algebraic structures built on any vector space endowed with a scalar product. If you want to establish a relation with tensors in Relativist Geometry you need fiber bundles...even in Special Relativity.
- Rather than focus on the many denominations of types of vectors in a Clifford bundle, it is good to understand, from a general point of view, how they work. Notably transposition, adjoint map, exponential,...And to notice that the vector space from which the Clifford algebra originates has a special role.
- it is usually (in QTF) acknowledged that the EM field is represented by U(1), which is a commutative group. And not difficult to see that the Clifford algebra has a center, which includes for even dimensions, the scalars. In a unified theory of fields the natural representation of the EM field is through the scalars in a Clifford algebra.
Hello Christian Baumgarten Thank you, yes. The Clifford product for spacetime algebra gives basic benefits of its resulting algebraic structure.These mathematical benefits produce a considerable amount of physical insight about the electromagnetic theory in a straightforward way.
Evidently, as an apparently unappreciated concern, as I wrote in the question, lurks in the treatment that there is no way to represent any 2nd-rank tensor by a bivector, vector, and scalar. Or, in particular, no way that any 3×3 matrix could be represented by a 2-vector in 3D! Geometric algebra is not and cannot be isomorphic to tensor algebra.
There IS a missing structure in geometric algebra. The question is, does it matter in electromagnetism? Not enough to answer by citing results using geometric algebra itself, that would be a circular argument.
Hello Robin A. Damion and Christian Baumgarten , Thank you, Robin. The "problem" arises when trying to avoid Minkowsky's spacetime of four dimensions. As an alternative to Minkowsky spacetime, one replaces the unit imaginary i with the Clifford bivector I for the plane, that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis, trying for 3 + 1 D.
All seems to go very well, but there is no isomorphism between tensors and multivectors, medium with a varying dielectric constant cannot be used, collective effects are not represented, the electron is just a point particle, and one misses the important fusion of space and time, to 4D. This is strong evidence that we do live in at least 4D, as it must be used, and that geometric algebra is NOT sufficient for electromagnetism?
Cheers, Ed Gerck
Hello Christian Baumgarten , Thanks. The book by Jancewicz is in introductory level, but reveals that 4D spacetime is not going to go away by using the Clifford algebra to map in lower dimension, an interaction that it cannot represent fully. As the book says, in page 28 , "All considerations in this book are performed in euclidean and pseudoeuclidean spaces, so .. the use of multivectors is sufficient."
This is a result in topology, and even though the path in 4D is continuous, the resulting path in lower-dimensional geometric algebra must be discontinuous. So, the intimate link between magnetism and the electric currents as its sources is shrouded by an "illuminating" bivector, when none is needed, a simple Lorentz trasformation will do.
By avoiding tensors, and the real questions that this avoids, the book offers a much more complicated and unphisycal view, e.g., of a bivector representing a magnetic field, and promotes the use of the vectorial product of Gibbs, which result is not in the same space.
This book is similar to consider platonic solids to model the orbits of planets in the solar system, or use octonions to model subatomic particles. It works to some extent, then the underlying strong mathematic structure diverges, as it has no reason not to. It can be useful as a simpler model, but it does not represent nature. No one can use it to affirm what it denies on p. 28, a fuller model.
Your answer was timely, though, because this is exactly the question here -- Are geometric algebras insufficient for electromagnetism? The book by Jancewicz says YES, in that page 28.
Cheers, Ed Gerck
Hello Christian Baumgarten , To clarify, it is stated above. To make 4D spacetime "go away", as an alternative to Minkowsky spacetime, one replaces the unit imaginary i with the Clifford bivector I for the plane, that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis, trying for 3 + 1 D.
Neither Cl(3,1) nor tensors allow logically representing the mutual electric vs magnetic induction that underlies Maxwell's electromagnetism (not to be confused with electrodynamics).
CL(3+3+3,1) possibly could. To be explored.
Hello all: Any Cl(n,1) for n>0, and geometric algebra, will be insufficient for electromagnetism, due to disallowing time as a dimension, a par with n>0 spatial dimensions. Tensors, because they are structure-neutral, should be the natural choice for a more comprehensive description of electromagnetism. This is according to the lack of isomorphism between multivectors and tensorial spaces, which position is supported also by the findings in Jancewicz (op. cit.).
Another conclusion, including Jancewicz (op. cit.), is that simpler descriptions, such as Cl(3,1) or STA of Hestenes, can be used in restricted domains, such as euclidean spaces. However, any assignment of physical quantities, such as a bivector to magnetism, or scalar or vector potentials to measurable phenomena of electromagnetism, may entirely be a fiction of the geometric algebra or vector formalism employed.
This will likely also apply to quantum mechanical effects in electromagnetism, such as the Aharonov–Bohm effect.
Cheers, Ed Gerck
Hello Christian Baumgarten , You seem to diverge, it is a sign! And you are carrying at least two others in recommending your posting.. Your posting, however, is false, consider deleting. All my quotes, that you cited and tried to baseless refute in unscientific way, typical of RG false controversies, but good for traffic in a not so recommended way, are referenced, sometimes verbatim, in the established literature in the field. Just Google them! You do not have to react this way.
For example, to make 4D spacetime "go away", which of course I wrote that it does not work in practice and that is why Cl(n,1) is not ultimately valid, although it can be used in a limited way in euclidean spaces, as you did in your papers. It has the "argument" exactly as I cited. You will find it published in the open literature, December 31, 2012, https://doi.org/10.1371/journal.pone.0051756, Revisiting Special Relativity: A Natural Algebraic Alternative to Minkowski Spacetime, by James M. Chappell t. al., as:
"As an alternative to a planar Minkowski space-time of two space dimensions and one time dimension, we replace the unit imaginary i, with the Clifford bivector I for the plane that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis e1 and e2. We find that with this model of planar spacetime, using a two-dimensional Clifford multivector, the spacetime metric and the Lorentz transformations follow immediately as properties of the algebra."
This is a well-cited result, and was, at least indirectly, used by you in your papers. It has other unfortunate errors, and circular arguments, that I did not cite yet, but that deny further use of Cl(3,1) in electromagnetism, not just in non-euclidean spaces. Be warned. The same error happens in any Cl(n,1), n>0.
Hello Christian Baumgarten , Your contributions are welcome, and any comments help, even incoherent postings. Of course, you can edit any posting you wrote, as you have done in the recent past in this thread, and as the RG platform allows for anyone. Sometimes, there is a double entendre to avoid, sometimes a "not" is omitted. It is certainly fine if you do not delete, and often I ask so people tend to not do it, people are contrarians. It is a technique that allows for a continuation, to hear more, and I thank you for your added time.
What I understood from your papers, and I remember I recommended one in RG, and used it in a class, is useful. Your insights were noted.
But, there is a part, pretty much summed by yourself, when you wrote somethings that seem to need repair, to avoid drinking too much of the propaganda by Clifford algebra, geometric algebra, and by STA of Hestenes, which is useful but is attempting to be a general language in physics and mathematics, and yet falls short.
I see them as an overuse of good ideas by Grassmann, Clifford and Hestenes, trying to have mathematics dictate physics. It looks good on paper, and there are lists of "advantages" of using Clifford algebra in spacetime, while denying spacetime, a fusion of space and time, that one can transform into another.
You may wish to comment on that, for example, your words:
"CAs, as long as they can be obtained from a conservation law via Hamiltonian formalism, can not produce non-physical results. They simply describe the full space of physical possibilities that is compatible with the respective conservation law."
So, this is very temerarious to say:
1. "can not produce non-physical results" is logically false, an illusion by mathematics as if physics.
2. "They [CA] simply describe the full space of physical possibilities that is compatible with the respective conservation law" is to agree with the paper I cited, where spacetime is reduced from 4D to a concocted reality without time as a dimension. You are denying what the words say, in spacetime, a union of the two. It is not relevant, you can verify, whether it is Cl(2,1), Cl(3,1), or Cl(n,1), time is not included as a dimension, even though one still speaks in lip service of "spacetime".
Structure should come from physics, mathematics can represent but not define. Even topology can be denied. Anything made by humans, compared to billions of years of nature, is horribly fallible, and already wrong, although useful to humans. Geometric algebra fancy to represent nature, but it is their nature that they are representing, not the one out there.
To imagine that Cl(3,1) can represent spacetime while denying time as a coordinate in fusion with space, is asking to be hacked, by a spacetime approach, or even higher. It can be useful, in restricted euclidean spaces, but that is not the field of electromagnetism.
Hello Christian Baumgarten , Thanks. I prefer to abandon an approach that will not work, as early as possible. There is less "cost" this way, and the more expensive cost is logically the earliest -- the other costs follow. Some prefer to entertain an experimentally false result, and churn out papers, it is useful.
I see no doubt in the literature that geometric algebra, including Clifford and STA, is insufficient for electromagnetism, and that electromagnetism cannot be limited to euclidean spaces, as the discussion also explains. At least two observations are missing: isomorphism with tensors, and using spacetime at least as 4D. There are more reasons. But tensors are structure-free, they do not impose any limitation on the physics, they allow nature to be represented as she capriciously may want :-) And electrodynamics is not euclidean, not even locally.
The Maxwell equations (the part that is not faulty) can be a logical consequence of the Lorentz force and 4D spacetime. If one restricts to Cl(3,1), with no actual 4D spacetime, that choice will miss non-euclidean spaces, and more. You should be able to confirm it, even if you try to disprove it.
The idea of using the Hamiltonian treatment as a "filter" of multivector solutions that do not work physically, although working mathematically, which you called HCA, cannot be used to decide on the solutions that do not fit in multivectors, in CA. The "source" is incomplete. The missed part includes non-euclidean spaces, and electromagnetism.
Difficult to apply a final coat of paint on a car that has not yet been assembled.
Similarly, electromagnetism should be clearly understood before trying to define which mathematical approach will best describe it.
Difficult to mathematically describe what has not yet been clearly understood.
Hello Christian Baumgarten , Your posting is false. About your work, I am powerless to change the truth. If Clifford algebras Cl(n,1) can represent electromagnetism, you just denied special relativity. There is no spacetime in space and time. Since that is not what we can measure, there is something incorrect in thinking so. I suggest you look first into the lack of isomorphism between multivectors, which is all one can use in Cl(n,1), and tensors. Since that is true, you can use it to reconstruct the reality you assume.
Hello Christian Baumgarten , You wrote, "... you provide little support for your assertions - besides a single reference to a single paper,...".
You omitted that I cited also p.28 of your own reference by Jancewicz, stackexchange, well-known facts in the field, including lack of isomorphism between multivectors and tensors (a simple matter of counting!), all contrary to the use of Cl(n,1) in non-euclidean spaces, which spaces we do find even in elementary applications of electromagnetism. I also suggested you to search in Google, it is that easy to find supporting references to my assertion that Cl(n,1) should not be used in this case, it is insufficient.
Besides, you created two notions of time, while denying at least 4D spacetime. This sequence of errors are not that important, though. You can take time to use the few suggestions, and they were offered because I considered that they are valid reasons, with a sincere intent to lead you to reconstruct the reality you assume.
Hello Igael Azoulay
and all: We measure that time and space can exchange -- it is impossible to measure time as "space and time," as in Cl(n,1), but we can measure ONLY a mixture of the two, spacetime, at least 4D. As said over a century ago, the dividing line between past, present and future is an illusion.
How can you make clear in Clifford algebras that quaternionic number systems exist in many versions that are distinguished by how Cartesian and polar coordinate systems can sequence them?
Physical reality applies these versions to determine the electric charges, color charges and spin of elementary particles.
64 Shades of Space;: http://dx.doi.org/10.13140/RG.2.2.28012.46724
I would argue that electromagnetism is a phenomenon in 3D space and that the only algebra that is suitable is the 3D vector cross product. The reasons are explained in this article entitled "Pythagoras's Theorem and Special Relativity" with reference to the 7D vector cross product,
Article Pythagoras's Theorem and Special Relativity
All basic fields obey the same quaternionic field equations, which are partial differential equations. These fields can only differ in their start and boundary conditions. Our living space (the universe) is a basic field that exists always and everywhere. It differs fundamentally in its start and boundary conditions from the electromagnetic field that is described by Maxwell equations. This means that each Maxwell equation has a quaternionic equivalent. Some quaternionic field equations have no Maxwell equivalent. See BehaviorOfBasicFields; http://dx.doi.org/10.13140/RG.2.2.15517.20960
I agree that the 3D vector cross product is the key to full mechanical description of the mutual induction of the electric and magnetic aspects of all localized electromagnetic energy quanta. Suffices to better understand the true meaning in 4D space of the E and B vectors.
Clearly put in perspective in this recently published paper titled "The Hydrogen Atom Fundamental Resonance States":
http://file.scirp.org/pdf/JMP_2018042716061246.pdf
Hello all, The 3D vector cross product is not a vector, but a tensor. This is well-known and the reason to invalidate its use in Maxwell's equations, and physics equations. This is both a physical and a mathematical reason, commented below.
MATHEMATICAL REASON: The 3D vector cross product is not a closed operation in the 3D vector space, it produces a member that does not belong to the same set, the 3D vector space, although it may look like it in some cases.
PHYSICAL REASON: Both sides of an equation representing a physical relationship, such as A = B, must change equally when the frame of reference changes and the so-called inertial condition is obeyed, as already stated by Galileo, Newton, and Einstein, that the laws of physics are the same for all uniformly moving observers. But if one writes an equation using a 3D vector cross product, such as A= B x C, the left and right side may transform differently if the coordinate frame of reference changes, while still inertial. In the past, this was accommodated, not solved, by considering spurious things such as polar and axial vectors, and pseudoscalars. This is solved using tensors, which maintain the form A = B under inertial reference change.
NOTE: Multivectors, in geometric algebras, or Clifford algebra, or Cl(n,1) algebras, or STA Hestenes algebra, solve the mathematical reason, creating a closed space in all operations. No geometric algebra operation is mathematically unsound. But, they do not solve the physical reason, but tensors do. This is the other reason, I mentioned, that invalidates the use of geometric algebras in equations of physics, including electromagnetism. If one eliminates the physically wrong results, by requiring an additional step of "filtering" through, let us say, a Hamiltonian, this will not produce those results that are physically valid but were ignored in the first place, using just multivectors. A sequence of filters cannot filter less than the first filter, well-known in physics, math, and engineering. This appears, more easily to see, in non-euclidean spaces, such as anything larger than, let us estimate in general, a few Planck lengths.
Ed, Are you sure that you are not looking deeper into all of this than is necessary? The 3D vector cross product arises in electromagnetism in the case of the F = qvxB force. It's simply a way of expressing the fact that the induced force on a charged particle moving at right angles to a magnetic field is mutually perpendicular to both the field and the motion. Doesn't it serve the purpose adequately?
Hello Frederick David Tombe , and all: No, it can create mistakes upon reference frame change, for example, simple mirroring, gives us wrong units in the SI MKS, worldwide, and what I explained is old news. But some prefer to keep using what is mathematically and physically wrong, or "just" physically wrong, entertaing the results, and even publishing them as a if they are a "valid" approximation. They are not, and nature shows otherwise.
And this happens not only in physics, or maths. A well-known science magazine editor said that if all neuroscience published works would disappear, neuroscience would advance. More than half is wrong, see my talks on it, and by others, and US NIH declarations. This irreproducibility crisis continues to today, and was recently covered by The Economist. See Presentation The Big Idea in Physics and Science: The Absolute
"... Also discussed is the current irreproducibility crisis in the bio-sciences and how medical research results are provably wrong most of time, how that negatively affects health-care decisions, and how the Big Idea in physics can be used to provide a new foundation for research and verification in the bio-sciences."
Dear Ed,
You wrote: " The 3D vector cross product is not a vector, but a tensor."
This is not correct. I think you do not understand Maxwell's discovery.
The electromagnetic tensor is not a correct representation of Maxwell's discovery because it inappropriately merges into a single field the two mutually inducing components of electromagnetic energy, which is something that the electromagnetic E and B cross product does not do.
The electromagnetic tensor is ok for providing general overviews, but is meaningless in actually describing the actual nature of electromagnetic energy of which all electromagnetic quanta are made.
You need to beef up on the exact meaning of the Lorentz equation which is exactly conform to Maxwell's electromagnetism and to observed physical reality.
Best Regards
André
Hello all, (edited, with added context, thanks for questions also by PM)
It is incorrect to assume that E and B in electromagnetism might be independent or have different dimensions or units in the SI (that is why in electromagnetism we cannot really use the SI MKS, and prefer Planck units or the SI CGS, not that we prefer millimeters over meters, or c=1 makes any real difference with numeric calculations in hand-held computers).
Unfortunately, for historical reasons, MKS made the wrong choice in electromagnetism. A new system of international units is certain to come, as CGS already has different versions! Meanwhile, we are not waiting and are happy to use Planck units, not the human-based SI, in electromagnetism and many other areas in physics, such as lasers and particle physics. The telescope Hubble would be in service earler and better, check history, if Planck units were used in engineering as well, instead of using the wrong distance in inch conversion, and launch Hubble "as blind as a bat". What is the importance of a few dozens of millions of dollars, and space walks, added in correction? Let us not make the same mistake, and use the Planck units that give us c=1, at least numerically it is easier to get the right numbers :-)
Electric and magnetic fieds have never been measured, we just measure forces, called Lorentz force. But they are a good model mathematically, and they are just different aspects of the same field. Our equations tell us that, sice Maxwell, and before. Potentials, also, have never been measured, and they depend on further assumptions in mathematics, some of which are in dispute even today, and this becomes crucial in terms of modeling the reality of this universe, it is not a matter of personal choice. The universe is not in the sphere of personal beliefs, and we can travel time backwards, with a simple telescope, and see today things billions of years before us and the Earth even existed. That is one of the reasons this dicussion can help.
There is no magnetic source in electromagnetism, magnetism is caused by the movement of electric charges, even for a permanent magnet at rest, in the laboratory frame. Electromagnetism complies with special relativity even before Einstein was born, check the dates.
But not all Maxwell's equations are correct, especially when using the 3D vector cross product, which is a tensor of another space, masquerading as a vector, but revealing its true nature upon simple mirroring, for example. That is why we can no longer use them, not that they were not useful. We also do not use flogiston or other incipient ideas, we evolved in our understanding, and our language changes with us, we do not use Latin in writing scientific papers anymore, even though it can be shorter and more accurate, so we use Latin in expressions sometimes.
Dear Ed,
You wrote: " It is incorrect to assume that E and B in electromagnetism might be independent or have different dimensions. They are different aspects of the same field."
"Field" is just a geometric representation invented by Gauss to help visualize the Coulomb force.
The Gauss electric force field concept is just a conceptual omnidirectional representation of the decrease from maximum "potential" intensity of the Coulomb force that would apply at the location of a single mathematical point assumed to have unit charge "if a second test unit charge was involved at this zero distance", "potentially" decreasing as a function of the inverse square of the distance at which "this second test charge" would be present at any distance.
The test is easy to make. Just plug in a second unit charge in the E field equation, and you instantly end up with the initial Coulomb equation that will then provide the exact adiabatic energy induced in each charge "as a function of the distance separating them".
The E and B are not different aspect of the same "field". They are two different mathematical representations of two different mutually inducing configurations of the real oscillating energy being represented, one of which displaying measurable electric characteristics, the other displaying measurable magnetic characteristics.
All Maxwell equations are correct, as well as the Lorentz and Biot-Savart equations, and they all are in complete harmony with the E and B vector cross product. The proof is that they all have been extensively and successfully used in all working applications that they allowed to develop.
You write: "magnetism is caused by the movement of electric charges,"
This is only part of the story, just like electrodynamics and the electromagnetic tensor are only parts of the story. This is confusing electrodynamics with true electromagnetism.
Special Relativity never complied with Gauss's equation for the electric field, so it never was Maxwell electromagnetism compliant.
Best Regards
André
Hello André Michaud , If you continue interfering with every posting I write, this will be innocuous to physics and against the rules of RG. Your postings are false and off-topic, but we can continue to skip, you see that RG made it easier.
Your electromagnetism and your special relativity seem to make sense to you. However, this universe has a different nature. This question has to do with this nature. Please stop posting on your nature, it does not apply to this discussion.
You said once that you would like to help, please do and observe this thread if you want, but please do not interfere anymore. We are using these postings and it is simple to filter. There is no effect from your interference to us, but others in RG may disagree. Continue with your ideas, they will lead you to comparisons and further reflection.
Ed, You said that there can be mistakes with F = qvxB on reference frame change. This equation defines a helical path, and the helix will have a radius. What kind of reference frame change do you think would alter the helical path such that two observers would be observing a different path of motion?
Hello Frederick David Tombe , A simple reflection. The right-hand rule becomig the (false, e.g.) left-hand rule. This is not a matter of personal belief, anyone can test, it is well-well-known, and the cause of "corrections," not solution, by using spurious things like polar vectors, axial vectors, and pseudoscalars. The mathematical solution would be Cl(n,1), STA or geometric algebras, but that does NOT solve the physical problem, see NOTE 1.3 in the discussion text. Tensors solve it, as explained in NOTES above.
And, as you asked before, would it not serve a restricted purpose adequately? No, it can create mistakes upon reference frame change, for example, simple mirroring, gives us wrong units in the SI MKS, worldwide, and this is all old news that have to be somehow continuously repeated, with a "life of its own," as a misconception in even current college books at competitive US universities.
Dear Ed,
Yes I want to help by sensitizing the community to the need for basic electromagnetism to be clearly understood for progress to resume.
It is a wonder to me that so many in the community think understanding electrodynamics is the same as understanding electromagnetism.
Anyhow, I don't use to say more than once in a thread what I have to say on any issue.
I don't think that my 2 posts to you really were "interfering". Simply rectifying.
I have nothing more to add on electromagnetism.
Best Regards
André
Dear Ed,
Strange reaction to the partaking of information meant to re-induce formal study of electromagnetism, that never should have been neglected in textbooks.
Fundamental physics has nothing to do with politics.
If you think that the info I contributed was meant for you, you are mistaken. It would not even cross my mind to try convincing you or anyone else. My info is meant for whoever finds that it makes sense, which is obviously not your case. So you can rest in peace.
Contrary to Paul Marmet, I have no status in the community that I could be deprived of to shut me up and to chastise me for stigmatizing deficiencies in physics teaching in the community, so I can fully exercise my right to free thought and speech.
This will be my last contribution to this thread.
Best Regards
André
Hello André Michaud : Your posting is off-topic and false. We are using the idiom ‘as blind as a bat’ to describe this, someone who refuses to notice an obvious thing.
Though some species have poor eyesight, particularly during daylight hours, bats are not truly blind. Thus, we believe, like the Hubble Telescope, that one can be fitted with "lenses" to help see, and use other channels to be more useful. Do not be like Johnny Appleseed in the story, this "land" is not empty nor yours, you have no such freedom you fancy, and planting in people's land is an offense, actually he would face legal action for polluting, and your words apply to your electromagnetism only.
Ed, When would there ever be an occasion to do a reflection transformation of F = vxB? Once we've identified what the velocity v is measured relative to, then what more could we possibly need to know? There will be no frame of reference from which the helical path will appear any different. The 3D vector cross product serves adequately for the purpose of expressing the force.
Hello Frederick David Tombe and all: You seemed to have seen the issue with the 3D vector cross product, that it is not a vector, a member of the same space. There is no algebra, therefore it is not logical to continue to ask otherwise, but it is natural, let us call it a fact of human nature.
So, you naturally ask, "Once we've identified what the velocity v is measured relative to, then what more could we possibly need to know? There will be no frame of reference from which the helical path will appear any different. The 3D vector cross product serves adequately for the purpose of expressing the force."
Why is it NOT logical? Yes, to that observer, NOT to another observer who sees the "hand" (or, mutatis mudandis, the helix) through a mirror filter, and uses the "right-hand" rule while actually using the left-hand rule, and gets un-natural results. Children do it all the time, it seems, even in middle-school. It is worse for left-handed people.
The two observers are in disagreement. Who is right? The answer is none, what they see is an artifact of the formulation, it does not obey the physical principle that BOTH sides of an equation representing a physical relationship, such as A = B, must change equally when the frame of reference changes and the so-called inertial condition is obeyed, as already stated by Galileo, Newton, and Einstein, that the laws of physics are the same for all uniformly moving observers. And, of course, one does not control where the electron goes, so it is not just a matter of observing it or not!
But if one writes an equation using a 3D vector cross product, such as A= B x C, the left and right side may transform differently if the coordinate frame of reference changes, while still inertial. In the past, this was accommodated, not solved, by considering spurious things such as polar and axial vectors, and pseudoscalars. They are in the literature, search Google.
This is NOT solved (as Christian Baumgarten thought, see above) by using multivectors, Cl(n,1), Hestenes STA, nor using geometric algebras, (that solves only the mathematical issue of dealing with a closed space, where a multivector always stays a multivector) but by using tensors, which maintain the form A = B under ANY inertial reference change.
That is why we MUST use tensors in electromagnetism, or any other equivalent description such as maps and matrices, we cannot control where the electron goes, or who observes it.
Of course, in static, macroscopic, euclidean, mechanics (not electromagnetism, for example), where one can control BOTH the path and the observers, as well as the speed and curvature are zero, multivectors can be used in the laboratory rest frame. But the 3D vector cross product should be abandoned, even in those very restricted conditions. And multivectors are much better, and intuitive, where they apply. Tensors also work.
With movement, though, even if stepwise, sometimes Euler angles cannot represent all paths in 3D, producing what is called gimbal lock. In that case, even at speed zero in 3D, quaternions need to be used. These videos may help, as a possible reference, with context, is at https://youtu.be/mHVwd8gYLnI (long) and https://youtu.be/N5PDboNJwks(short). Tensors present no such problem.
A reference for tensors is at http://mathworld.wolfram.com/Tensor.html
Ed, It seems to me that you are giving physical significance to frames of reference in their own right rather than asking what physical medium a frame of reference is fixed in. When we have identified the physical medium which acts as the carrier of EM waves, then all the laws of electromagnetism can be expressed in their familiar form in the context of a frame of reference that is fixed in that medium. Geometrical reflections of what is observed with EM phenomena is not part of the real world anymore than what we see when we look in the mirror. We would never subject F = qvxB to a geometrical reflection.
Hello Frederick David Tombe : There is no room to refuse to notice an obvious thing. This is about logic. What you wrote is incorrect, note that the electron is NOT fixed, NOR the waves themselves, NEITHER the medium in the general case (thermal vibration, vibrations, currents, laser-dressing [1], etc.), and they are observers, plus anyone or anything -- you cannot restrict observers, or do you think that special relavity only works for fixed or inertial observers? Please read the above posting again, I added some comments that may help.
[1] On the laser-dressing case, we calculated first and later it was proved experimentally a case where the medium is stationary but subject to a laser. See Article Quantum well lasers tunable by long wavelength radiation
"It is suggested that the lasing frequency of GaAs‐GaAlAs quantum well lasers can be tuned when the quantum well structure is irradiated with a pulsed low‐energy long‐wavelength laser. This new tuning mechanism is based on the decrease of the electron binding energy, in the well, with increasing intensity of the external laser field. Typically, 0.60 mJ of 10.6‐μm radiation in a 1‐ns pulse may blue‐shift the quantum well laser frequency by approximately 20 nm."
From a private comment, that may be useful here. A teaching assistant says, "The world is so complicated -- the more I learn, the less clear anything gets. There are too many ideas to pick and choose from. How can I trust myself to know the truth about something? And if everything I know is so shaky, what on earth am I doing teaching?"
His professor, without turning the head away from the computer, responds, "You just have to do your best, and if that is not good enough, if more is needed, you can learn and improve. The perfect universal truth is different for the caterpillar after it leaves the cocoon as a butterfly."
Cheers, Ed Gerck
To all,
For a comprehensive study on Clifford algebras you can have a look to my paper on this site.
There is no magic mathematical tool which can answer all questions in Physics, all have their use. To understand fully the relation between Clifford algebras and tensors 2 ingredients are necessary :
- first fiber bundles and specially vector bundles. Using the category theory one can see that there are direct relations (functors) between the usual tangent bundle on a manifold endowed with a scalar product, and a Clifford bundle on the same manifold. So that one goes easily from tensors to vectors of a Clifford bundle, and even to forms valued in a Clifford bundle (which is the right representation of a field)
- a good understanding of the scalar product, which is an essential part in the definition of a Clifford algebra. Actually a Clifford algebra is nothing more than the demultiplication of the vector space endowed with this scalar product, and has a very specific role. For instance any morphism in a Clifford algebra is actually the extension of a linear map on this vector space. Moreover there is a Clifford morphism which goes from real Clifford algebras with signature (p,q) to complex Clifford algebras with dimension p+q. This enables to define physical real structure (consistent with the relativist time vectors) in complex Clifford algebras
Eventually, do not forget that the EM field is very special : it is represented by a group (U(1)) which is commutative. And in a Clifford algebra by the scalar component. Which makes the EM field ubiquituous but some times disturbing.
Tensor calculus and Clifford algebras are ways to obscure a theory. They do not help to clarify things.
Hello Hans van Leunen : Your posting is off-topic and false. Taken alone, one or a couple of your postings seem like a fringe researcher, perfectly acceptable. But, looking at all threads, it becomes obvious that you do not know what any of this is about. Please respect this thread and delete all your postings.
Hello jean claude Dutailly : Your posting is false. Clifford algebra is not defined by any scalar component, as you wrote, but by how the product of two vectors is defined, creating an algebra. Clifford algebra is about multivectors, not vector bundles.
Your profile is a phenomenon of RG, a RG-whale, with 66.54% of your score given by Answers like these. This tactic is easily avoidable, and RG helps now. You may help by confusing those who want to confuse, another phenomenon of the times, why would one want to do that? It helps detect them too, when they reply to you or recommend.
Ed,
Thanks you for your attention about scoring...and your attempt to police RG. But me, as well as Hans and others have been on RG for a long time, and we know to ignore rude people "who know all", or repeat like parrots what they believe to have understood in the books of their masters. I spend a good deal of my time on RG, less to praise my articles or books, than to help people to understand some difficult issues. Of course read my papers can help, and it seems that you have some reading to catch, notably on Clifford algebras, and mathematics.
In my post I underline 3 points :
- Clifford algebras are algebraic structures built on any vector space endowed with a scalar product. If you want to establish a relation with tensors in Relativist Geometry you need fiber bundles...even in Special Relativity.
- Rather than focus on the many denominations of types of vectors in a Clifford bundle, it is good to understand, from a general point of view, how they work. Notably transposition, adjoint map, exponential,...And to notice that the vector space from which the Clifford algebra originates has a special role.
- it is usually (in QTF) acknowledged that the EM field is represented by U(1), which is a commutative group. And not difficult to see that the Clifford algebra has a center, which includes for even dimensions, the scalars. In a unified theory of fields the natural representation of the EM field is through the scalars in a Clifford algebra.
Hello jean claude Dutailly : Your posting is good for your stats in RG, but false. It may seem to work in your RG score for a long time, but the end does not justify the means. It does not work for others and, eventually, not for you either. Life is a school.
Besides, the group already arrived at a verified conclusion. There is no need to "relitigate" it, or doubt it. There is no room to refuse to notice an obvious thing.
Hello all: This thread arrived, in the past, at a conclusion, which is stated in the NOTES in the body of the initial text. There is no room to refuse to notice an obvious thing, or to re-explain here, the reasoning and references are available above, to anyone.
We are now ready to move to non-mathematical aspects of using Clifford algebras, even beyond the physical reason given in the NOTES above, where using Clifford algebras would be detrimental to special relativity. We are talking about time.
A limitation on the idea of treating time as a fourth Euclidean-type dimension is that it does not appear possible, at least in the macroscopic world, to freely travel in the time dimension as is possible with space dimensions, according to Chappel et.al. [1]. Hence, in their view, the Minkowski formulation seems somewhat unhelpful in clarifying this aspect of time.
Instead Chappel et. al [1] propose to view time as a geometric property of space, using Clifford algebras to represent physical systems in special relativity. How this is proposed, and the un-physical consequences of the result, will be covered as anextension this discussion, where we intend to show the opposite of what Chappel et.al. intend in [1].
This also will show the futility of quoting in science, where previous views are treated as immobile scientific truths, whereas they should be considered in history only, not providing a scientific guidance of what would be truth necessarily.
Neither, we do not have to prove more than once that magnetism and electricity are not independent, for example, just because someone wishes to question it. We may just assume this known result is well-known by the participants, as in this discussion.
William Kingdon Clifford commented in the abstract of his paper “On the Classification of Geometric Algebras” communicated to the London Mathematical Society on 10 March 1876, “… the system is the natural language of metrical geometry and of physics”. The system which he was describing is his geometric algebra, which we now know as Clifford algebra. However, his words are, today false, as we intend to show here. Clifford was refering to the physics of 1876, before special relativity was kown in physics, or general relativity, or spacetime.
Therefore, it may NOT be correct to extend the use of his algebras, not just in four dimensions, as motivated in [1] today, but in models of physics, depending on the field.
This discussion has shown (see NOTES, above) that it does not apply for a physical reason, even though the mathematics is correct (unlike the 3D Vector cross product used originally in electromagnetism). Now, we will be considering another physical reason that Cillford or geometrics are insufficient, time.
[1] Article Time As a Geometric Property of Space
On the fusion of time and space in special relativity, and its difficulties conceptually, as we see by attempts to use STA and Clifford(3,1), and in the original formulation of special relativity, that do not use this concept.
How can special relativity “fuse” different things, time and space, experimentally, measured in a different way? The discussion presents a experimental and mathematical position, and intends to clarify this common question:
https://www.researchgate.net/post/On_the_fusion_of_time_and_space_in_special_relativity
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