Oriented planes in R3 are images by stereographic projection of those oriented spheres in S3 that contain the point -e0. In this post we use the same method as we did it in Part 11 when looking for preimages of oriented spheres. At the end we will have to decide again about the value of ε to obtain the correct orientation. While for spheres it required some work with somewhat complicated algebraic expression, here it will be much easier - a merry and fast slide downhill.
With n a unit vector in R3, and h a real number, consider the oriented plane Πh(n) in R3:
Πh(n) = {y∈R3: y·n = h}. (1)
We expect it to be an image, by the stereographic projection, of a certain oriented sphere
St(m) = {x∈S3: x·m = cos(t)}, 0<t<π , (2)
that contains the point -e0=(-1,0,0,0) - the origin of the stereographic projection. Our aim is to find an explicit expression of (t,m) through (n,h). Let us stress that the formulas (1) and (2) should be understood having in mind the fact Πh(n) and St(m)are more than just "sets". They include the orientations, and orientations are contained in the pairs (n,h) and (m,t). Pairs (n,h) and (-n,-h) (resp. (m,t) and (-m,π-t)) define the same sets, but correspond to opposite "orientations".
We will proceed the same way as we have done it with spheres in Part 11. This time it will be even simpler, because the equation (1) defining Πh(n) is linear. Thus, with x=(x0,...,x3)∈R4, ||x||=1, we substitute the stereographic projection formula
yi = xi/(1+x0), (i=1,2,3) (3)
into (1) to obtain
x'·n = (1+x0)h = h+x0h, (4)
where x'=(x1,x2,x3).
Then we rewrite (4) as
- x0h + x'·n = h,
which suggests the candidate for m, namely (-h,n). But m should have the norm 1, while (-h,n) has the norm squared h2+n2=1+h2. We must also remember that (n,-h) and (-n,h) lead to the same sphere as a set, so our general solution is
m = (-εh, εn )/√(1+h2),
t = arccos(εh/√(1+h2)), 0<t<π
where ε=±1,
and we have to decide the sign taking into account the orientations. To
this end we follow the same path as we have taken in Part 11 with the
spheres. From Part 10, Proposition 1 therein, we know that to have the correct orientation we should have h=cot(t). Now, cos(arccos( εh/√(1+h2) ) )=εh/√(1+h2), while sin( arccos(εh/√(1+h2)) )=(1-(εh/√(1+h2))2 )1/2= 1/√(1+h2). Thus cot(t) = εh, therefore ε=+1.
This way we have arrived at the following Proposition:
Proposition 1. With n a unit vector in R3, and h a real number, the pre-image of the oriented plane Πh(n)with respect to the stereographic projection is the oriented sphere Sarccos(h/√(1+h2))( (-h, n )/√(1+h2) ).
Exercise 1. Verify that m0+cos(t)=0 as it should be. (Why?)
In the next post we will return to the 6D universe and we will place our 3D world of spheres and planes in there.
P.S. 30-04-25 7:24 One of the readers of Laura' subsrack articles left this comment:
Enon commented on your post Exploring the Hyperdimensional Hypothesis.
I first discovered your work many years ago, even before SOTT, IIRC when searching for one of your husband's papers. He might find this interesting: "The mysterious universal quantum holographic phase field - Beta in the Geometric Algebra Dirac Equation" https://enonh.substack.com/p/x-the-dirac-equations-lorentz-invariant
Indeed if falls into our domain if interest. I will have to discuss this Hestenes' idea in the future..
At the same time it shows that there is math and physics on substack. So, I may have reconsider Sasa'a suggestion.
The we ->
ReplyDeleteThen we
which suggest ->
which suggests
m = (-εh, n )/√(1+h2) ->
m = (-εh, εn )/√(1+h2)
cos(arccosh ->
cos(arccos
sin( arc cosεh/√(1+h2) ) ->
sin( arc cos(εh/√(1+h2)) )
Thank you!
DeleteOukh! Ark, your illustrations are one better than the other! Easy natural sliding down the slope in such a nice contrast to the previous hard work up. Although, at the second picture, the situation seems to be getting out of control :)
ReplyDeleteWhat is the Hestenes' idea? (I like his papers, they are partly understandable even for non-specialists)
What is the Sasha's suggestion?
These are not really MY illustrations. I simply tell AI what to draw and how. The first picture is by Grok, the second by Dall E3. I use another AI (deepseek) to create prompts.
ReplyDeleteSasha simply suggested that I should start posting on substack. Certainly it has more visibility than this blog.
Hestenes proposed an interpretation of a Dirac spinor. In my mind I think that Pauli spinors are more primitive, since in my mind spacetime is a secondary concept, that is constructed out of our experiences with space. Time is another specie. So far we have succeeded getting "time" out of Cl(3). Hestenes also mentions a particular left ideal of Cl(3,1), but he is not worrying about the meaning of such an ideal.
I guess you didn't draw these pictures with your own hand :) Let the AI do the routine work under your close guidance. It's not the execution that amazes me, but the essence of the pictures, their resonance with my own feelings.
ReplyDeleteNow it's time for me to delve into the math and not just look at the pictures.
Congratulations on the May 1st - the day of workaholics!
ReplyDeleteSure there will be a new post later today. It is being cooked.
DeleteExercise 1. Verify that m0+cos(t)=0 as it should be. (Why?)
ReplyDeleteThis is really easy to do with the ready formulas for m and t.
A bit anxious that i ceased to understand how we get
h=cot(t) from
m·y = cos(t) and n·y = h
in general case... But i will think myself, don't want to disturb you while cooking the next Part.
@Ark, i am sorry, want to understand what happens with the image plane when t=0 and t=pi.
ReplyDeleteIn this case, h = εcot(t) = +-infinity, the plane goes to infinity, and things seem to be getting out of hand. Fortunately, this only happens for special spheres that degenerate into points, and we decided to let this case aside since it doesn't do any harm in general, right?
"what happens with the image plane"
DeleteCan you be more specific? What image? What plane?
I mean the projection plane Πh(n) which is characterized by (y,n)=h. Should we worry about the case h=cot(t)=infinity for t=0 mod pi?
DeleteLet's talk about lines and circles instead of palnes and spheres. Easier to imagine but otherwise essentially the same. f take a 1D line on a 2D plane, it is an image of some circle through the South Pole. When we move the plane further and further, the circle becomes smaller and smaller. In the limit it would become the South Pole point. But stereographic projection is not defined there. Everything is well defined on the sphere, but using stereographic projection we loose information. So it is better to stay on the sphere. And still better to use Q or Q+.
DeleteAnother option, sometimes employed, is to use a pair of stereographic projections, one from the South Pole, and one from the North pole. So we have two coordinate systems on the sphere. Their coordinate patches cover the whole sphere. This is how we make the whole sphere a differentiable manifold. But this is another issue, not related directly to your question,
DeleteIndeed, in terms of lines and circles (instead of planes and spheres) i forgot that we consider only the circles that live on the sphere. And tried to imagine what happens to ANY arbitrary circle going through the South pole, in R3. That was a real headache. Thanks a lot that you have tamed the circles back to the sphere for me! Now i see better how they shrink to the South pole as the projection plane moves apart. Also, the role of the sphere's finite radius and, thus, the curvature (in case of 3d space) becomes clearer.
DeleteThere is one point that we need to remember. We are doing pure math, no physics yet. So, we are taking the sphere (S^2 or S^3) of radius 1. When moving to physics we would have to take its radius as some dimensional number R, probably very small for elementary particles, and very big for our universe.
DeleteOh, no! Why couldn't we use, even in applications to physics, some dimensionless values like total curvature defined by integrating the curvature around the surface? At least as far as the conformal symmetry is assumed to be valid. This way leads to topological invariants, which have found so intriguing physical applications known as topological dielectrics, etc.
DeleteWell, in physics we must take care of units. Mathematicians as a rule do not even know what these are. I know only one exception to this rule: my friend from Florence, Marco Modugno:
Deletehttps://www.researchgate.net/publication/225366680_An_Algebraic_Approach_to_Physical_Scales
Many thanks! The Modugno&others' paper hits the bull's eye: "For instance, ... they assign a dimension to the curvature tensor; clearly, this cannot be true from a mathematical viewpoint, because the curvature tensor is obtained from the connection by a differential operation which does not involve any unit of measurement".
DeleteJust as i wanted for curvature to be expressed dimensionlessly!
Dimensional analysis is also one of my favorite but still not mastered subjects. Will read the paper with pleasure and attentively.
"Will read the paper with pleasure and attentively".
DeleteAt least the introductory part :)
The general idea is very intelligible: to represent each particular physical scale as a point of a special field, which is considered as a base, and each particular-scale physical theory as a fiber over this point of base, as far as i understand.
DeleteThe fibration pattern is very powerful. Someone suggests to use it to explain even biological structures: the field of various environmental conditions is considered as a base, and various organisms (e.g., live cells) - as fibers over it; in such manner, both the common structure and changeability of living objects are taken into account. Fibration is trivial for inanimate and nontrivial for living (and for quantum) objects.
The same subject, the same team:
Deletehttps://www.researchgate.net/publication/1768796_Semi--vector_spaces_and_units_of_measurement
Marco was taking care of physical dimensions long ago. They are already taken care of in our 1994 paper:
Deletehttps://www.researchgate.net/publication/259041107_Galilei_General_Relativistic_Quantum_Mechanics
Unfortunately, the full text of the 1994 paper is not available: "To read the full-text of this research, you can request a copy directly from the authors".
DeleteCan I ask you for the text now please?
@Ark, I see that you are currently busy refining the opus on spheres. When you have time, I am still interested in your 1994 work with Marco Modugno but cannot get it from https://www.researchgate.net
DeleteYou can get it using the link here:
Deletehttps://www.mathnet.ru/rus/person/200923
It is interesting that Marco proposed using in this paper the Cyrillic character "Ch". In fact he has designed the font by himself using tex font editor.
DeleteOk, trying to download it...
DeleteThanks! I will have a look and try to understand as much as i can. I already know that somehow you managed to contribute to every really interesting subject.
ReplyDelete@Ark, I couldn't get that particular 1994 paper, but I read a similar one: "What is Time in Quantum Mechanics?" It was a pleasure to meet again your original way of thinking, the fundamental nature of the problems posed, and the non-trivial ways proposed to solve them. "Geometric quantization" sounds like a great program. Got the idea of "Event-Enhanced Quantum Theory": indeed, time is more about the moment we enter the river, rather than passively observing its continuous flow.
ReplyDeleteOne small technical question: you plot the total probability of detecting a wave packet versus sensitivity, and the curve is not monotonic, it has a peak, why?
Which figure you are talking about?
DeleteI mean Fig. 6 "Total probability versus sensitivity kappa". Usually, the higher sensitivity of the detector, the higher the probability of detection, right? Perhaps, this is too primitive viewpoint...
DeleteIndeed it is somewhat puzzling. But it is addressed in the second sentence of this comment in the paper:
Delete"Some particles (wave packets) will pass the screen without being detected, some will be reflected without triggering the detector. "
Oversensitive detectors reflect wave packets. At least this phenomenon is predicted by EEQT. Probably can be experimentally verified. I have never seen it mentioned elsewhere.
Of course what needs to be addressed first is the physical meaning of the sensitivity parameter κ. I didn't think about it at that time. Thank you for asking!
DeleteYes, i noticed the phrase "Some particles (wave packets) will pass the screen without being detected, some will be reflected without triggering the detector" and gazed at it for a while. But it does not explain the nature of this "reflection without triggering". Couldn't it be connected with some saturation (overflow) effect?
DeleteThe effect is similar to that of tunneling through the potential barrier. Some wave packet are reflected, some tunnel through. The detector acts much like a complex imaginary potential. The fact that there is an optimal detector sensitivity/shape, now that I think about it, has been noticed in the literature previously.
DeleteHere is my simulation. You can see some detectors reflect the wave packet, copied from my Youtube channel:
Deletehttps://disk.yandex.ru/i/rZcuDXCjO2pqsg
And here is the text for the video:
622 views Dec 9, 2013
Single quantum system under continuous observation?
Quantum theory is strange. It is often acclaimed as the most successful of all theories in physics, but at the same time it leads to never ending debates about its meaning and its ontology. Philosophers like to draw our attention to the fact that not all is "quantum", that there are also tables and chairs. Wave-particle duality can be described mathematically, but are we completely satisfied with this description? Evidently not all physicists are happy. Louis de Broglie, one of the founders of quantum theory, in his later years returned to the beginnings and tried to find a better solution in non-linear theories. CERN physicist John Bell was unhappy, not only just unhappy, he was literally angry at the status of quantum theory and the misleading terminology that is being used by his fellow physicists in order to hide the real problems.
How Nature does what it does? Is it a reasonable question at all? If it is, then how can we go about it?
Physics builds models of reality. Some of these models pretend to be fundamental, some other just phenomenological. Here is one such model. There is quantum wave, there are tables and chairs (red detectors, there are events -- when tables are overthrown, when red becomes white. And there are wave collapses. It all can be modelled mathematically. Probably the model is not realistic, yet it does its job. The wave is moving, it overthrows tables and chairs, and it leaves a real and visible track -- like those that can be seen in nuclear emulsions or cloud chambers. Each such even is accompanied by collapse of the wave and its rejuvenation. The process is governed by partly deterministic and partly random laws -- well known in this branch of the mathematical theory of random processes that deals with earthquakes and financial markets.
The animation presents one such history: a sequence of collapses and the track that we interpret as that of a passing "particle", for instance an electron. A different run of "the same" experiment would produce somewhat different track. During the animation the wave travels across the field of somewhat randomly distributed "particle detectors". Detectors fire using a (tamed) random algorithm . Those that have fired change their color from red to white and become non-active. All the data, including sensitivity of the detectors are on the atomic scale. We notice that there is a certain angle between the direction of the wave (directly towards the camera) and the track that is left. This is not unexpected, owing to the random nature of the whole phenomenon. The coupling between the wave and the detectors that is being modeled here does not include energy and/or momentum transfers between the wave and the detectors. Only "information-theoretical exchange", "I tell you where I am, and you will collapse me in exchange", is taken into account.
Tools used in this animation: parallel implementation of the time-dependent Schrodinger equation with complex potentials (CUDA), with Fast Fourier transform, OpenGL, piecewise-deterministic algorithm of EEQT.
Thank you! The animation is impressive though a bit complicated for me (compared to the circles sliding on a sphere). I believe that the optimal detector sensitivity/shape is due to some probabilistic element in the relations between detector and wave-packet.
DeleteAs far as I understand, the work on geometrization of QM is still ongoing, and some parts of it in the most adapted, toy version we see in the Blog.