Grothendieck, the great and strange mathematician, complained about the nature of mathematics:
"I think there's an inherent flaw in the mathematical way of thinking and I have the impression that it's also reflected in this Manichaean vision of human nature. On the one hand, there's the good, and on the other, the bad, and in the best case scenario, we see both living side by side."
Grothendieck, disappointed, has left mathematics and went into
seclude in Pyrenees. I do not understand it. Personally I do not see
anything wrong with mathematics, and with yes-no Aristotelian logic:
There is good and there is evil and there is the specific situation that determines which is which. What's wrong with this?
From The Guardian article about Alexander Grothedieck of Aug 31, 2024: "He was in mystic delirium’: was this hermit mathematician a forgotten genius whose ideas could transform AI – or a lonely madman?"
"I open a first page at random. The writing is spidery; there are
occasional multicoloured topological diagrams, namechecks of past
thinkers, often physicists – Maxwell, Planck, Einstein – and recurrent
references to Satan and “this cursed world”
whose ideas could transform AI
So I stay with the devil of the algebra and the angel of geometry, and with their happy marriage - geometric algebra. Which geometric algebra? This is a good question. Certainly time is not the same as space. Einstein and Minkowski merged space with time, but their reasoning was highly questionable. Maxwell equations were taken as a starting point. 10-parameters Poincare group came out rather easily as an invariance group, and it was set on a pedestal. It was given the absolute power over all other possible interactions, for no good reason. Then it was discovered that Maxwell equations are invariant under the 15-parameters conformal group, containing the Poincare group as a subgroup leaving the "conformal infinity" (Dupin cyclide) as the unmoved "absolute", but it didn't help us with understanding the nature of time. Is time "real" or "imaginary"? Or both? There is a lot that needs to be done in physics before jumping ahead into the future of fancy mathematics with no "yes-no" dichotomy. Some physicists promote the view that the universe is a quantum computer - like it would explain anything. It does not explain anything at all, it just sweeps problems under the carpet, since we do not understand what quantum theory is about. Quantum theory does not explain consciousness. We need to understand what "consciousness", "information", and "measurements" are first. Then, perhaps, we will be able to get some idea about quantum theory, the theory of "measurements".
Let's go back to algebra, geometric algebra. The mystery of spin is
hidden there. We do not need to mix space and time from the start (see Peter Woit blog "The Mystery of Spin"). If
time comes out of space and spin - that would be a nice surprise.
Geometric algebra A = Cl(V) of space happens to be
isomorphic to complex quaternions that can be faithfully represented by
2×2 complex matrices Mat(2,C). Unexpected meeting of
geometry with the formalism of quantum mechanics. Coincidence? Or there
is some deep meaning hidden in this fact? Can we read/decode the
message?
Hamilton invented quaternions when trying to make space into an
algebra. He was forced to add the fourth dimension. What is the meaning
of this fourth dimension? Is it just pure mathematics that has nothing
to do with reality? Certainly the fourth dimension of Hamilton is not
the "real time" of special relativity theory. So what is it? For a while
we have a toy that we are playing with, discovering new
functionalities. I talked to Laura about all these mysteries, and she
told me that as a little girl she owned a doll that could even pee. Of
course you had first to fill the doll with water. What can our Cl(V)
produce if properly filled with proper stuff? Shall we get spinors?
Feed 1.
First we have fed A with an orthogonal projector p = (1+n)/2, p = p* = p2. It produced the right eigensubspace
{u: up=u} = Ap,
which is a non-trivial left ideal of A. It carried an irreducible representation of A, where "spinors" live. We also got its orthogonal complement A(1-p). Spinors can reside also there.
Feed 2.
Then we have chosen a different approach, through the Gelfand-Neumark-Segal (GNS) construction. This time we fed A with a "state" - a positive linear functional f on A. It produced a left ideal If:
If = {u: f(u*u)=0}
which we didn't like (as it consists of norm zero vectors), so we have taken the quotient Hilbert space Hf = A/If, build as the set of equivalence classes [a], and we got a representation ρf of A on Hf with cyclic vector Ωf = [1] such that
f(a) = (Ωf,ρf(a)Ωf).
If f is a pure state, then the representation ρf is irreducible (we did not show it yet). So elements of Hf are also candidates for spinors. The relation to Feed 1 is still waiting for an explanation.
Feed 3.
Here again we start with a state f, but this time we add some salt - we use the fact that our A is not only a *-algebra, but also a Hilbert space on its own. This allows us to represent f through an algebra element, denoted by the same letter, so that
f(a) = <f*, a>. (1)
Taking the positive square root f½ of f we get
f(a) = <f½, L(a)f½ >,
where L is the left regular representation of A on A.
Connecting Feed 2 to Feed 3.
We can now happily use Proposition 4.5.3 from the GNS Promenade
to connect the two constructions. For this to take place, Feed 3 should
result in a cyclic representation. As it stands, the representation is
not necessarily cyclic in A. Let us understand this
problem better. Let us look at Eq. (1), but from the right to the left.
If f, on the right, is a positive element of the algebra A, then (1) defines a positive functional. What are the simplest possible positive elements of A? The answer is: they are orthogonal projections p=p*=p2. So, what happens if we take f = p = (1+n)/2? Since p = p* = p2, it follows that f½ = p. In that case L(A)f½ = Ap is our left ideal of Feed 1. It is certainly not the whole A - for this choice of f. Thus, in this case
f½ is not cyclic in A. But it is cyclic in the left ideal Af½ ! Define H'f = L(A)f½,
Ω'f = f½, ρ'f = L restricted to H'f. Then, according to Proposition 4.5.3, there exists an isometry U from Hf to H'f, that maps Ωf into
f½, and U intertwines the two representations:
Uρf(a) = L(a)U on Hf.
The
GNS construction (Feed 2) is not really needed in our case. The point
is that GNS construction is quite general, it works with any C*-algebra.
Here we have a particular case, when our algebra is already a Hilbert
space (perhaps too big), so we can apply Feed 3.
We have already made a connection to Feed 1 by taking f = p, but this needs to be made clear. We will do it in the next post.
As for spinors - they are lurking, always nearby, but still not as main characters.
P.S. 04-02-25 From my correspondence: "... Allais' effect only worked when his pendulum was triggered by a man. Allais' effect was officially recognized by the French academy of sciences."
Interesting. Quoting from Wikipedia
https://en.wikipedia.org/wiki/Allais_effect
Maurice Allais states that the eclipse effect is related to a gravitational anomaly that is inexplicable in the framework of the currently admitted theory of gravitation, without giving any explanation of his own.[30] Allais's explanation for another anomaly (the lunisolar periodicity in variations of the azimuth of a pendulum) is that space evinces certain anisotropic characteristics, which he ascribes to motion through an aether which is partially entrained by planetary bodies.
His hypothesis leads to a speed of light dependent on the moving direction with respect to a terrestrial observer, since the Earth moves within the aether but the rotation of the Moon induces a "wind" of about 8 km/s. Thus Allais rejects Einstein's interpretation of the Michelson–Morley experiment and the subsequent verification experiments of Dayton Miller.[31][32]
"If time comes out of space and spin - that would be a nice surprise"
ReplyDeleteArk, you are reading my vague thoughts and formulate them clearly. Yes, spinors should connect 4d space-time and 3d space. Something like that: creatures living on a 2d surface do not notice that it has TWO sides when observed from a 3d world. To do that, they should step over infinity. Similarly, living in 3 spatial dimensions, we can only suspect the 'unexpected two-valuedness', two sides of our 3d space, but can neither perceive nor grasp it completely. All hope is on algebraic geometry.
Ohanian, H. C. (1986). What is spin? American Journal of Physics, 54(6), 500–505. doi:10.1119/1.14580
ReplyDeleteLink:
https://sci-hub.se/10.1119/1.14580
Click save near down arrow
Abstract,"...in all cases the spin angular momentum is due to a circulating energy flow in the fields."
Pg. 4, "The second expression is dependent* on the polarization[+/-], and we must therefore identify it with the spin."
Pg. 5, "What is important here is not so much the numerical value of this result--there are a variety of ways of establishing that the Dirac spinors correspond to spin ħ/2--but rather the underlying physical picture of the spin as due to circulating energy flow in the Dirac field."
Pg. 6, "The physical picture of spin presented in the preceding sections has great intuitive appeal because it confirms our deep prejudice that angular momentum ought to be due to some kind of rotational motion. But the rotational motion consists of a circulation of energy in the wave fields, rather than a rotation of some kind of rigid body. The spin is intrinsic, or inherent, i.e., it is a fixed feature of the wave field that does not depend on environmental circumstances. but it is not internal, i.e., it is not within the internal structure of the electron or photon (of course, the structure of the wave field is crucial to the spin, but this is not what is usually meant by internal structure)."
Saša mentioned Ohanian's take on this (and also loosening the speed of light constraint) in comments on part 35: Rotating Vectors.
----
From: https://www.math.columbia.edu/~woit/wordpress/?p=11811
Delete"Alex,
It’s true that there’s a good way to think of the classical limit of “spin”, as adding a new factor to phase space of a sphere in 𝑅3. This idea is actually very general: if the values of your field transform under a group G as an irreducible representation of G, take as new factor in phase space the corresponding co-adjoint orbit in the dual of the Lie algebra of G. If you want, you can then use a phase-space path integral, and think of your classical spinning particle as moving along a trajectory in the usual phase space times the sphere (or co-adjoint orbit in general).
The problem with this is that a curved, finite volume phase space like the sphere behaves rather differently than the conventional phase space. Additionally, it’s only in the limit of infinite spin that you recover a classical limit. Things like spin 1/2 and spin 1 are at the opposite, truly quantum limit."
-Peter Woit
"I think what is going on with some people is that they don’t like intrinsic spin of an elementary particle because it doesn’t correspond to something they can visualize in a model of reality which consists of scalar elementary particles moving in space. They want to explain the spin 1/2 of an electron in terms of this model of reality. But when you do this, you run into an intractable problem: if you have spinless particles and only orbital angular momentum, all states will have integral total angular momentum, no possible way to get 1/2 integral values.
If you ignore the half-integrality problem and try and build a model of electrons as bound spinless particles anyway, you run into the problem that electrons exhibit pointlike behave on scales much smaller than their Compton radius. Then when you try to explain spin 1/2 in terms of orbital angular momentum you get into problems with your hoped-for model, since it would imply faster than the speed of light motion of the component particles."
-Peter Woit
[Faster than light may be implied by full quaternion treatment of Maxwell's equations, as previously mentioned in this series]
"The only interesting covering group of SO(n) is the spin double cover, other covers are trivial. Why should nature be happy with vectors and SO(3) and not exploit the more fundamental spinor representation and the double cover?
The classical situation is a little complicated to state, but here are some comments:
You can make a classical theory of a particle with spin in two ways:
1. Extend phase space by a sphere. You’ll then get an extra “spin” contribution to angular momentum, which can take any non-negative value (determined by the radius of the sphere). No integrality or half-integrality of spin until you quantize.
2. Extend phase space by an anti-commuting coordinate. This is “pseudo-classical” mechanics and gives you a spin-1/2 term in the theory.
If you work with classical fields, you can write down classical field theories where the field transforms under any representation (including spinor) of the rotation group that you want. These can only be half-integral.
Spinors are not inherently quantum. If you go from 3 to 4 dimensions, the story of 4d spinors is very much a basic part of the theory of twistors. Twistor theory is not a quantum theory, almost all the work in twistor theory has been based on applying the twistor geometrical setup to classical field theories."
-Peter Woit
If twistor geometry has it's spacetime points as spinor space it seems they're bundled together, though it's unclear if time is popping out of space and spin in that system. I don't know if the development of that geometry is analagous to or as rigorously developed as what's been explicated in this series.
DeleteBetween Ohanian and Woit's discussion on the issue I find it fascinating that this can have bearing on both classical and quantum physics. Spinors are not yet main characters, but I imagine the rest of the audience is eager for that development. Before that, though, I ask you astounding mathematicians just what are the devil and angel saying about this as a math object?
Upon a 360° rotation in a Euclidian space we're thought to be constrained to see the object returned to a physically indistinguishable position? But, because we keep track in phase space (a complex space?)we know this rotation is accompanied by a sign change? I can imagine a rather classical object which, taking a constant angular momentum (as the de facto generator of infinitesimal rotations), makes a perambulatory course, tracing around a sphere. If it mathematically mapped its course after "360°" to a point across a mirrored inversion plane, but with it's direction of travel pointing in the opposite direction(hence the flipped signage), that'd seem to me to fit the bill for the description of spin 1/2.
If it cannot be so, I await the story developments where this main character has the supposed mysterious "inherent" spin needed for the description of fermions, hidden in a complex Hilbert space, or something of that sort, as some of the folktales of the physicists say...
"...the spin of the electron is entirely analagous to the angular momentum carried by a classical circularly polarized wave." -Ohanian
"...you run into the problem that electrons exhibit pointlike behave[behavior] on scales much smaller than their Compton radius[I don't think that's a problem, I think it's the truth]. Then when you try to explain spin 1/2 in terms of orbital angular momentum you get into problems with your hoped-for model, since it would imply faster than the speed of light motion of the component particles."
A classical recipe, calling for FTL: just add aether?
Thank you for these extensive comments. In my own comment to "AI at work" I wrote:
Delete"P.S. 03-07-24 10:08 Reading the 1967 paper "Twistor Algebra" by Roger Penrose, on p. 348, there is the following footnote
5. To give a rigorous definition of a spinor which takes into account its sign, it is usual to appeal to the theory of fibre bundles. This is not essential, however, and an elementary (nonlocal) geometrical description will be given in an appendix to a forthcoming book by R. Penrose and W. Rindler on the applications of spinors in relativity.
It is interesting that this "forthcoming book" has been published only in 1984. Thus it took 17 years to prepare this book for publishing!!!"
There is a lot that I need to understand. I am planning to get to twistors in the future posts.
Anonymous, could you please tell who are you?
DeleteThank you for the link to Woit--Ohanian discussion, quite relevant to this Blog. There is much to think about. Woit is one of my favorite mathematical physicists who says intelligible things but still a bit too intricate for me.
Probably all erroneous, but i have a strong feeling that fermionic organization of matter is an evolutionary step forward compared to bosonic fields, since massive substance is much more diversified than the force fields. It is now agreed that evolution of matter is accompanied by the reduction of symmetry. And, indeed, when defining spinors we use factor groups, i.e., reduce the initial symmetries. Does this idea go in the right direction?
DeleteOne more general consideration: by some reason, spinors better reside in hyperbolic spaces than in (pseudo)euclidean ones. Of course, the universe must not have one of the specific geometries we know of, let it be one here and another there, but we should sew these two different geometries together somehow... Is it possible?
it it ->
ReplyDeleteit
Thanks.
Delete"... a doll that could even pee. Of course you had first to fill the doll with water. What can our Cl(V) produce if properly filled with proper stuff?"
ReplyDeleteCould the doll turn water into urine?
That would involve action-at-a-distance. The technology is not yet that advanced.
Delete"Taking the positive square root f½ of f ..."
ReplyDeleteWhy not negative square root?
Good question. Positive roots are sweet, negative roots are bitter. But if you prefer bitter, it will work as well.
DeleteIndeed, Peter Woit gives a great insight https://www.math.columbia.edu/~woit/wordpress/?p=13152:
ReplyDelete"Angular momentum is infinitesimal generator of rotations and ... rotations act also on the vector values not just on space ... not only on operators but also on states".
This works both for classic and quantum mechanics. And both for interger and half-valued spins. In the latte case, one should "involve not just vectors and tensors but also spinors". With their rotation group SU(2) double covering the usual SO(3) rotations in 3d.
It is interesting to translate this knowledge in terms of our discussion.
It is only today that I have discovered this 2021 paper by Peter Woit:
DeleteEuclidean Twistor Unification
https://arxiv.org/abs/2104.05099
Well, better late than never....
Some time ago i tried to read this paper but, though it is clearly written, cannot understand much of it.
DeleteFortunately, Woit has papers for various categories of readers. I liked his ‘Is Space-time Really Doomed?’ so much that translated it to Russian and it was published in our "Metafizika" last year.
@Anna Thank you. Yet another paper for me to digest! I am all excited!
Delete"Angular momentum is infinitesimal generator of rotations acting not only on operators but also on states".
DeleteHow to grasp this? Is it like differentiation of a product of two functions expressed by the Leibniz formula or like differentiation of a composite function... Or Woit means that rotation changes the very form of the wave function?
Perhaps it is just that, for instance in non-relativistic quantum mechanics, coordinates transform as
DeleteUx^iU*
and states as
U phi
A general rule for any space symmetry. But I am not sure if I am guessing correctly.
Another discovery: astrictly related paper by Peter Woit
Delete"Spacetime is Right-handed"
https://arxiv.org/html/2311.00608v2
"Yet another paper for me to digest!"
Delete@Ark, to my mind, ‘Is Space-time Really Doomed?’ is a bit aside from our present path of thought. But who knows? :)
Now i am excited about the 'general story of angular momentum'. Want to decode this message of Woit. Angular momentum is definitely the main actor in the play, at least because it is of the same units of measure as the action functional S (and the Planck constant) - not without a reason!
"Spacetime is Right-handed"
Delete@Ark, this may probably deal with your doubts about the role of right ideals and dual spinors...
"Spacetime is Right-handed"
DeleteThere is something of Kassandrov's right and left differentiation in this SU(2)L x SU(2)R 'chirally symmetric holomorphic representation of the complex Lorentz group' of Woit. Or it is only my wanting it to be so...)
Yes, Kassandrov is using left and right action to define his "analyticity". But there is something in his definition that he is evidently not too happy about: non-uniqueness of splitting an endomorphism into the product of left and right actions. I am also not happy about it. Something is missing here.
DeleteDear Anna and Ark, I like your readings ;=)))
DeletePeter WOIT, as Anna said, "is one of my favorite mathematical physicists who says intelligible things but still a bit too intricate for me."
I agree with that after reading some Woit's papers...
In the living world, many molecules are chiral (enantiomers). Living forms use only one (left or right). For example, the amino acids constituting proteins are levorotatory and all sugars are dextrorotatory. Nobody know why !?
I think this asymmetry comes from the very structure of our space-time (8D).
The arrow of time too.
The birefringence of space too (predicted by quantum electrodynamics).
We will find soon where come from this disymetry in Cl(3, 0) !
I'm very confident ;=))
Kassandrow, Peter Jack, C. J. Robson and Peter Woit... They all move us towards the solution.
DeleteIn the Woit's good book book, at the 28.2 "Real Clifford algebra" we can read that Cliff(r, r, R) = is isomorphic to Mat(2^r, R). It means Cliff(3, 3, R) = Mat(8, R)
DeleteIf you take the (3 vectors + 3 bivectors) subalgebra, so you have the signature (3, 3) = (+ + + - - -) and the complex symetry plane is made of (one real + trivector) with the trivector = {(i, 0) (0, i)} 2x2 matrix. trivector is identical to i (from complex)...
The two TIME(S) (one real and the other imaginary) looks like a plane of symetry separate two 3D subspaces (one fermionic-like and bosonic-like)...
It's funny ;=))
@Alain, i didn't understand why 'the plane of symmetry separates fermionic- and bosonic-like subspaces'. Should look in the good book of Woit, if only i have access to it...
DeleteVideo on YouTube is still unavailable here :(
@Anna This is the link to the Woit's book : https://www.math.columbia.edu/~woit/QMbook/qmbook.pdf
DeleteThe idea of a plane (complex with the scalar and the pseudo-scalar of Cl(3, 0)) is just an "idea", a sort of "intuitive view"...
Like a mirror plane ;=) between two 3D worlds...
As you said, all quantum measurement are Real numbers (or Complex number ?)
This plane could be a sort of projective plane...
In this page : http://www.madore.org/~david/.misc/20140711-split-octonions.pdf
We can see an interesting construct of split-octonions. I have the feeling that split-octonions are very close to Cl(3, 0), probably because of the 8d neutral metric (+ + + + - - - -)...
There is "divisor of zero" in this chiral construction.
It could be related to the creation/annihilation operators ?! I don't know enough to well understand...
@Anna You can download the video from here:
Deletehttps://drive.google.com/file/d/1iD2sFsfa3TPBLdp76EipeuM-kOjzqrjv/view?usp=sharing
@Alain
DeleteVery interesting paper:
https://www.linkedin.com/posts/jeanfrancoisgeneste_les-lois-de-la-physique-sont-elles-immuables-activity-7291069218316247040-g5Dl?utm_source=share&utm_medium=member_desktop&rcm=ACoAAAS1GHUBudtekvDgQVJytb51acG1UWKQ9Jw
Thank you, Alain, for the kaleidoscope of sparkling ideas. Thank you, Ark, for giving access to the video.
DeleteIt seems that spinors could not wait any longer in the shadowy corner and have already burst into the very center of our attention.
@Anna Forgotten to tell you: on the video you can select a subtitle track. I saved it with automatic subtitles. Sometimes they are funny, and probably unnecessary
DeleteThanks a lot, Ark for your concern. That was British English, which is understandable when not very fast. i have looked once without subtitles, it is ok, except i could not keep pace with all the text appearing on the screen. But this can be easily cured - by watching again. And again, if needed. The video is worth it.
Delete@Ark This text is abstruse or absurd. The guy who wrote this is way crazier than me ! Thanks ;=)
Delete@Alain "The guy who wrote this is way crazier than me ! "
DeleteCrazy? Ooh, no! Rather surreal. He has interesting ideas. For instance:
Jean-François Geneste - Gravity Modification & Zero-Point Energy-56-405
https://www.bitchute.com/video/mZYgX6JaB3tr/
Besides, "The mystery of spinors" video confirms my intuition that spinors deal with passing through infinity, which appeared to be simply involving the (n+1)th dimension.
DeleteIn the meantime, the addition or elimination of a dimension is no less than the birth or death of something, as is well seen with the example of quanta in the Fock space. Thus, the mystery of spinors is an ontological problem in some sense!
"So I stay with the the devil of" -> "the"
ReplyDeleteO.
Thenk you!
DeleteNext to last paragraph:
ReplyDelete"..., but this needs to be make clear." -> "..., ... made clear."
Thank you!
DeleteIn the Section "Connecting Feed 1 to Feed 3.", are the Feed references correctly attributed?
ReplyDeleteBeen a bit confused with them there, it seems Feed 1 got mixed with Feed 2 (GNS construction), and Feed 3 got called Feed 2 instead.
For example, at the beginning of the Section,
"We can now happily use Proposition 4.5.3 from the GNS Promenade to connect the two constructions. For this to take place, Feed 2 should result in a cyclic representation. As it stands, the representation is not necessarily cyclic in A.",
is that Feed 2 or Feed 3?
And in 2nd paragraph from the end,
"The GNS construction (Feed 1) is not really needed in our case. The point is that GNS construction is quite general, it works with any C*-algebra. Here we have a particular case, when our algebra is already a Hilbert space (perhaps too big), so we can apply Feed 2.",
is the first Feed reference to Feed 2 instead of Feed 1, and the last reference to Feed 3 instead of Feed 2?
Had a flashback of playing three-matchbox guessing street game with these Feed references.
DeleteHope that have guessed right. :))
@Saša Thanks. Changed the numbers here and there. Not sure if fixed enough. Indeed I messed up.
Delete@Ark
DeleteSecond sentence in that Section,
"For this to take place, Feed 2 should result in a cyclic representation.",
I think it should be "Feed 3" instead of "Feed 2", because Feed 2 as GNS construction already contains cyclic vector Ωf, and in the following sentences of that Section you invited us to discuss the Eq. (1) which was one of the main propositions of Feed 3.
FWIW.
You are right. Corrected. Thank you.
DeleteI recommend a film about spinors:
ReplyDeletehttps://www.youtube.com/watch?v=b7OIbMCIfs4
The author laments the lack of understanding of the subject.
And few people want to deal with the change in the phase difference between the magnetic and electric fields over distance for a simple dipole antenna.
"And few people want to deal with the change in the phase difference between the magnetic and electric fields over distance for a simple dipole antenna."
DeleteYou were not satisfied with the explanation given in the Comments several posts back?
"You were not satisfied with the explanation given in the Comments several posts back?"
DeleteTo a small extent. That "explanation" is merely qualitative. There is indeed something to be said for the harnessing of near-field and far-field considerations. However, that "explanation" does not provide for example an answer to the question of how fast magnetic (and electric) field nodes move along the distance from the dipole to reach the speed of light c at large distance.
@Bjab Concerning spinors video. Very nicely done. However, around min 53 he brings in the Lorentz group, which is unnecessary for spinors. Spinors are already in the nonrelativistic Pauli equation, we do not need Dirac equation. Of course he knows that, but it would be better if he would stress this point.
Delete@Bjab
DeleteHave just checked other textbooks and asked a couple of colleagues that are more in that field of research, and no quantitative answer to your question.
The reasons for that "behavior" might be that from academic point of view interest is mainly on the power irradiated by the antenna, which is calculated as a time average of Poynting's vector, and from the technical perspective focus is mostly on the signal, i.e. intensity as a square of amplitude.
In other words, you seem to be one of rare few interested in concrete relationship between the both fields as they propagate from the radiation source.
Another possible reason is that such a treatment involves exact calculations where usual approximations are not useful anymore, which leads to using numerical methods instead of usual analytical approach. So, maybe that might be a direction where to look next for your answer.
@Saša,
Deletethank you for your interest in the subject. One of the more interesting things is that since the electric field is phase shifted by 90 degrees with the magnetic field near the antenna, the pointing vector changes direction - sometimes it is directed outwards and sometimes towards the antenna. So the energy moves back and forth. The question arises what shape the antenna should have so that it does not emit the far field at all. Just as the electron vibration in the atom does not far-field radiate.
The motion of an electron in an atom is where classical electrodynamics falls short, as it gives that every pointlike charged body radiates if having non-zero dv/dt, i.e. acceleration, which implies that it loses energy and thus should not be possible to form a stable atom at all. That's also an explanation why so called planetary model of atoms with electrons as planets orbiting around the nucleus as a star in the center is essentially incorrect. FWIW.
Delete" That's also an explanation why so called planetary model of atoms with electrons as planets orbiting around the nucleus as a star in the center is essentially incorrect."
DeleteThe problem is: there is no other model. Quantum theory is not a model. It is a toll for doing some calculations. By some magic it often works if one knows where to use it and where not to us it.
Here is how Woit explains the origin of the second, unusual contribution to the angular momentum https://www.math.columbia.edu/~woit/wordpress/?p=11811:
ReplyDelete"If you define
'angular momentum = observable that generates rotations'
Then, angular momentum can have two different sources
1. Effect of rotating X in Psi(X). This is the angular momentum you are used to, cause by 'stuff moving around', it’s 'orbital angular momentum', formula
L = X x P
in its classical and quantum versions.
2. Psi may take values in a vector space on which rotations act non-trivially. Then when you rotate a wave function, you get a second contribution from this, the 'spin angular momentum'. In the case of spin-half, Psi takes values in C2, the spinors, and rotations act on the spinors (in a new way unlike the normal action of rotation on vectors and tensors, including the fact that it is a double-cover of the rotation group that acts, the usual rotations act only up to minus-sign problems)".
@Anna It is strange that Woit did not mention vector-valued wave functions where the rotation acts as R Psi(R^{-1}X) and not just through XxP as Psi(R^{-1}X).
ReplyDelete"P.S. 04-02-25 From my correspondence: "... Allais' effect only worked when his pendulum was triggered by a man. Allais' effect was officially recognized by the French academy of sciences."
ReplyDeleteInteresting. Quoting from Wikipedia
https://en.wikipedia.org/wiki/Allais_effect"
How much could rotating heliospheric magnetic field, i.e. the heliosphere that encompasses the whole solar system contribute to these effects?
Could the heliosphere be considered as a sort of an aether?
I mean, there are experimental indications that within the whole observed universe there are magnetic fields present, for example extragalactic ones of the order of microgauss. Could those fields play the role of aether as light would not be propagating through real vacuum because of the presence of these surrounding magnetic field?
DeleteI am confused. Isn't light an electromagnetic phenomenon? Or you, perhaps, see an essential qualitative difference between cosmic e-m fields and light waves (or photons)? Otherwise we woul have aether propagating through aether. This would have to be a non-linear effect. Anything particular you have here in mind?
DeleteIt is an EM phenomenon; light as an EM wave propagates through various rotating magnetic fields in the universe, not real vacuum, which means that as it traverses the fields it can also interact with them. Would that be non-linear type of interaction, don't know, maybe or even probably.
DeleteHave been wondering if those "background" magnetic fields that are present everywhere could play the role of supposed aether, without the need to really introduce the actual aether as a medium of propagation? In a very real sense, nowhere in the universe light propagates through an actual vacuum but "on top of" some background magnetic field, including our "sterile" laboratories.
Kind of like in solid state physics when photons acquire apparent mass due to interactions with atoms/lattice which affect their propagation or when electrons get their charge "shadowed" due to same reasons.
Haven't checked the Allais effect in details, but on the wikipedia page it said that Moon kind of produces "wind" of 8 km/s speed by its revolution around Earth. Could that be viewed and explained as a "simple" disturbance in the heliospheric rotating field like making sort of a whirlpool around Earth affecting the light propagation there and what we call and think the usual gravitational field is?
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