wendy wrote:The notion of 'spin' is fairly recent (some time this century), and i don't necessarily completely understand it. But it is supposed to be attached to the notion of a rotating particle. For an electron, a 'spin-half' particle, means it assumes the same state after having rotated by 720 degrees. Half-spin particles obey the Fermi Dirac statistics and thus called 'fermions', while integer-spin follow the 'bose-einstein' statistics, are called 'bosons'. The essential difference is that no two fermions can have the same description: any orbital can only be occupied by exactly two electrons, one 'spin-up' the other 'spin-down'. This is the Pauli Exclusion principle.
A free particle in 4D would, by transfer of energy between modes, tend to assume a clifford-rotation. Clifford-rotations have a parity (ie left and right), which are mirror-images of each other. But even among the left-handed rotations, there is a separate rotation for each point on the sphere. The twelftychoron has rings of ten decagons on its surface. There are seventy-two such rings, and because each can be transversed either direction, 144 great arrows of order ten. A set of left-clifford rotations divides this 144 into 12 sets of 12. These 12 sets are represented on the sphere as the vertices of an icosahedron. There is another set of 12*12 representing the 12 right-hand rotations.
ICN5D wrote:That's an awesome render, quickfur! Yes, it blows the tesseract away by coolness factor.
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Wow, very fascinating! Every time I imagine something like that, I go to high-D thinking. Take the shape of electron orbitals, they're all over the place in symmetrical clumps of possibility. Those multiple locations for an electron of the same energy reminds me of multiple interceptions in our 3D plane. Surely some high-D stuff is going on there!
I imagine all atoms stretched along space-time would look like helices entangling with other helices, in cycles of cycles of cycles. As the universe cools, the helices tangle into larger systems, governed by new laws, while composed of the initial helical soup.
The d orbital with m = 0 is designated z2. The two orbitals created from the m = -1 and +1 orbitals are designated xz and yz. The two orbitals created from the m = -2 and +2 orbitals are designated xy and x2-y2. These designations arise from the mathematical formulas for the wave functions and indicate the orientation of the orbital.
quickfur wrote:Rotations in 4D have some unusual characteristics not present in 3D.
The most prominent is the Clifford isoclinic rotation, in which the two orthogonal rotations proceed at an equal rate. When the two rotation rates are not equal, some points in the object will trace out spirals whereas those that lie on one of the stationary planes of the other rotation will trace out circles. So there are two distinguished stationary planes, orthogonal to each other, where points on the object will trace out circles. Outside of these two planes, points on the object will follow a non-circular spiralling path. However, in the isoclinic rotation, all points of the object trace out circles, and each of these circles lie in a stationary plane! So then there is an infinite number of stationary planes, and the original two orthogonal planes of rotation become indistinguishible from all the other stationary planes. The rotation therefore loses its distinctness in orientation: you could have started out with any orthogonal pair of rotations that lie in any corresponding pair of these infinite number of planes, and you'll get an identical rotation. So a whole bunch of rotations that would have been distinct, if the two rotation rates were different, have collapsed into a single isoclinic rotation.
However, this doesn't mean the rotation is completely independent of orientation! In fact, there are still an infinite number of distinct isoclinic rotations. They are produced if you rotate the planes of rotation along a direction that doesn't lie in any of its stationary planes. Furthermore, the isoclinic rotation is chiral -- there are two distinct "senses" of rotation given the same (infinite) set of stationary planes, and they cannot be reoriented into each other using only rigid motion. They are mirror images of each other.
Now, Wendy has mentioned before that any object that begins with two unequal rates of rotation will eventually settle into an isoclinic rotation via energy transfer -- because there is a gradient of different momentums in different parts of the object, that will prefer to equalize over time in order to reduce the total potential energy. So given any object in its natural state, it will basically always be found in an isoclinic rotation, rather than having two unequal rates of rotation. This suggests that 4D particles would be found with isoclinic spin in their ground state.
The interesting thing about this, is that the chirality of isoclinic rotations means that there are two possible such ground states, corresponding with the two possible chiralities of the isoclinic spin! So this seems to suggest an inherent two-charge or two-spin system, that arises purely from geometry alone!
anderscolingustafson wrote:In our Universe one of the fundamental properties of particles is their spin. In 3d it's only possible for a spinning object to have one independent direction of spin but in 4d it's possible for a spinning object to have two independent directions of spin. I was wondering would the fact that things can have a double rotation in 4d effect the fundamental spin of the fundamental particles?
PatrickPowers wrote:
An electron is a point particle with zero radius, nevertheless it has quantum spin.
quickfur wrote:That diagram is certainly more accurate than tear-drop shaped orbital depictions (often employed in VSEPR theory). Nevertheless, the p orbitals are not ellipsoidal, as depicted here. They are more like dumbbell shapes. So this diagram probably still represents an artist's vision of the orbitals, rather than a mathematically accurate depiction.
For all intents and purposes, we can assume that the electron is a point particle. Experiments with single electrons in Penning traps have put an upper limit of 10−22 meters on the radius of the electron. That is 1/500000000000 of the Bohr radius of the hydrogen atom - it's very small. Being an upper limit, the electron could in fact be smaller than that, or it may actually be a point particle. Of course, a charged particle of vanishing radius is hard to describe theoretically, but nature is not bound by our theories.
For all intents and purposes, we can assume that the electron is a point particle.
Experiments with single electrons in Penning traps have put an upper limit of 10−22 meters on the radius of the electron. That is 1/500000000000 of the Bohr radius of the hydrogen atom - it's very small.
Teragon wrote:For all intents and purposes, we can assume that the electron is a point particle.
Not even that. Diffraction and interference is not explained by point-like particles. Matter is made up of standing waves with energy and momentum determined by the shape of those waves. A mere point-like particle wouldn't exhibit neither uncertainity principle nor Pauli principle and it wouldn't be stable orbiting an atom.Experiments with single electrons in Penning traps have put an upper limit of 10−22 meters on the radius of the electron. That is 1/500000000000 of the Bohr radius of the hydrogen atom - it's very small.
The electron radius is a really misleading term and it's worthless without a definition. Which radius is the guy talking about? What's the experimental setup and the theory used? There are many ways to assign a radius to an electron. Most of them have a practical meaning, but the picture of the radius of a solid particle should not be taken too seriously and might be more deceptive than it helpful.
Be that as it may, an electron is too small for quantum spin to be explained as ordinary spin in 3+1D.
Unlike spin, quantum spin is orientable. That is, quantum spin has the same sense no matter how the observer is oriented. Ordinary 3D spin does not have this property.
quickfur wrote: Now, Wendy has mentioned before that any object that begins with two unequal rates of rotation will eventually settle into an isoclinic rotation via energy transfer -- because there is a gradient of different momentums in different parts of the object, that will prefer to equalize over time in order to reduce the total potential energy. So given any object in its natural state, it will basically always be found in an isoclinic rotation, rather than having two unequal rates of rotation. This suggests that 4D particles would be found with isoclinic spin in their ground state.
wendy wrote: A free particle in 4D would, by transfer of energy between modes, tend to assume a clifford-rotation.
granpa wrote: so angular momentum is not conserved in four dimensions?
mr_e_man wrote:quickfur wrote: Now, Wendy has mentioned before that any object that begins with two unequal rates of rotation will eventually settle into an isoclinic rotation via energy transfer -- because there is a gradient of different momentums in different parts of the object, that will prefer to equalize over time in order to reduce the total potential energy. So given any object in its natural state, it will basically always be found in an isoclinic rotation, rather than having two unequal rates of rotation. This suggests that 4D particles would be found with isoclinic spin in their ground state.wendy wrote: A free particle in 4D would, by transfer of energy between modes, tend to assume a clifford-rotation.granpa wrote: so angular momentum is not conserved in four dimensions?
Because of conservation of angular momentum, and a glome's symmetry (making the inertia tensor a scalar), an isolated glome spinning would maintain its angular velocity. It would not decay toward an isoclinic rotation. This has nothing to do with energy.
Do you mean that a non-isolated glome would interact with other objects in a way that favours isoclinic rotation?
anderscolingustafson wrote:Be that as it may, an electron is too small for quantum spin to be explained as ordinary spin in 3+1D.
Although quantum spin is different from classical spin it is still effected by the number of dimensions which is why in 2+1 dimensions there are particles that have spin states that is neither integer nor none integer known as anyons but in any number of dimensions greater than 2 anyons cannot exist.
The reason ordinary spin cannot be orientated in 3+1 dimensions is because ordinary objects don't move at the speed of light so that their direction of motion depends is not the same for all reference frames meaning that its helicity is depends on the reference frame. For an object that is moving at the speed of light in 3+1 dimensions its direction of motion is the same in all reference frames and so its helicity is also the same in all reference frames and so the spin has a chirality to it. A massless particle can never move slower than the speed of light but it can interact with other particles and that interaction can change its direction and if it changes direction then it will take longer to get from place to place than if it moves in a straight line and so it will appear to move at less than the speed of light which is how the Higgs Field works. Particles with mass are interacting with the Higgs Field all the time and so even though without the Higgs Boson it would move at the speed of light and so have a helicity the interaction with the Higgs Field causes it to appear to move slower than light but the spin remains chiral in spite of the particle having mass from the Higgs Field. So spin that has chirality that is the same in all reference frames is different from ordinary spin it actually can be explained by there being 3+1 dimensions for something moving at the speed of light, which is what massive particles would do if there was no Higgs Field.
d023n wrote: Anyways, I was also considering that thinking in terms of angular momentum may be more helpful. It has the same units as energy after all, and I interpret it representing the amount of energy stored in the orbit. And then I remembered that in 4D there are 2 ways to simultaneously rotate, which made me think about orbitting 2 ways simultaneously. If an object is going around something using the XY plane, it still has the option to go around it using the ZW plane. Then it hit me.
It should have 2 orthogonal angular momenta! In fact, it makes sense to describe it as having 1 "double-angular" momentum that scales with the 4th power of the velocity, given that "single-angular" momentum scales with square of the velocity. This means that 4D still is not sufficient to achieve a stable orbit because gravity still scales with the inverse 3rd power of the distance, but 5D works!
However, even this should still heat the object, which would slowly radiate the heat, which would carry away tiny amounts of angular momentum until the object really did settle into true isoclinism. However, in odd dimensions, if the rotating object has different cross-sections along the leftover axis (e.g. a sphere but not a cylinder), then this angular momentum leakage by way of heating from internal stress created by the rotation could continue asymptotically toward fully halting the rotation because of that squishing effect.
mr_e_man wrote:Angular momentum does not have the same units as energy. Energy is K = (1/2)m(v.v) (using the vector dot product), and angular momentum is L = mx^v (using the wedge product, and position vector x). There's a missing factor of time; v = dx/dt. Maybe you were thinking of torque, which does have the same units as energy, being dL/dt.
mr_e_man wrote:This reminds me of the https://en.wikipedia.org/wiki/Yarkovsky_effect
The Yarkovsky effect is a consequence of the fact that change in the temperature of an object warmed by radiation (and therefore the intensity of thermal radiation from the object) lags behind changes in the incoming radiation. That is, the surface of the object takes time to become warm when first illuminated; and takes time to cool down when illumination stops.
mr_e_man wrote:Centre of mass has units of distance. Moment of mass has units of mass*distance.
PatrickPowers wrote:anderscolingustafson wrote:In our Universe one of the fundamental properties of particles is their spin. In 3d it's only possible for a spinning object to have one independent direction of spin but in 4d it's possible for a spinning object to have two independent directions of spin. I was wondering would the fact that things can have a double rotation in 4d effect the fundamental spin of the fundamental particles?
The first thing to know is that quantum spin is NOT spin. It has only a superficial similarity to spin as we know it. An electron is a point particle with zero radius, nevertheless it has quantum spin.
Quantum spin has some quite weird properties. See the Stern-Gerlach experiment, which is simple but inexplicable in Newtonian terms.
Unlike spin, quantum spin is orientable. That is, quantum spin has the same sense no matter how the observer is oriented. Ordinary 3D spin does not have this property.
It is entirely possible that higher dimensional spaces are involved.
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