GAME23 David Eelbode - Rotors and Spinors

[@rotgertesla]

That was a very great talk

[@theograice8080]

If attainable, I implore you to add links in the description of these videos to download: the Slides, the Video, and the Audio. I would love to share these (in whole, so they maintain attribution), with friends, peers, and colleagues.

[@theograice8080]

Are these slides available for download?

[@hamish_todd]

An example of this "able to distinguish sign flips" that is possible to bring about in a PGA context is single-sided application of translations to points. Let's say we're in 1D PGA Cl(1,0,1). Starting point is e1. Ordinary/two-sided application of a translation: T=1+3e01 is my translation: (1+3e01)e1~(1+3e01) =(1+3e01)e1(1-3e01) =(e1+3e0)(1-3e01) =e1+6e0 eg the point translated 6m Now try with T=-1-3e01: (-1-3e01)e1~(-1-3e01) =(-1-3e01)e1(-1+3e01) =(e1+3e0)(-1+3e01) =e1+6e0 So we got the same answer regardless of whether T was multiplied by -1 ("differently oriented") Try it one-sidedly though. T=1+6e01 (1+6e0)e1 =e1+6e0 And now we try T=-1-6e0: (-1-6e0)e1 =-e1-6e0 So T "imparts" its orientation to the point when applied single-sidedly. Double-sided application is great when you want to apply rotations, or translate things other than points, so I'm not saying not to use it generally. But it seems to have this "downside". Also note that the distance of the translation, 6, is in there, "readably", unlike with the double-sided translation which moved you by twice the distance. Note also that the geometric interpretation is weirdly intact in both cases. Double sided: move the page the opposite of your (6m) transformation which is (1-3e01); do e1; move the page back which is 1+3e01 Single sided: do e1; then move by your (6m) transformation which is 1+6e01 Both result in the impression of "having done e1 in a different place"

[@Achrononmaster]

@56:00 you can't just give a meaning to the coefficients in a superposition as 'probabilities', since they interfere (empirically). In classical mechanics the interference is just a wave phenomenon, in QM it's a single-particle phenomenon, so *_not_* a wave. So in QM if the spinor is your representation of a transformation (which respects all the symmetry and orientation as in the talk) then this cries out for a physical account for interference; afaik there is no account other than entanglement, and indeed in a GPT (generalized probability theory) entanglement and superpositions are inseparable (can't have one without the other). That's your explanation of the "mystery". Note that entanglement is non-trivial spacetime topology in the Clifford spacetime algebra, so cannot be modeled using simple position & rotation gauge (and conformal gauge) on a flat Minkowski spacetime (i.e., conformal GR as a gauge theory). You have to go beyond GR to consider non-trivial topology (which at low energy breaks conformal symmetry). That's why the SU groups show up as so-called "internal" symmetries, it's not that they are internal _per se_ , but that they are associated with local topological degrees of freedom of 4D spacetime. (Einstein really was the GOAT, he would have loved this... I reckon.)

[@ZeroInDaHouse]

I have been struggling for years to understand all of this including the significance of the euler identity, power series and so when it comes to spinors. I am starting to build a small intuition to what it does through KZbin videos like these. To me what is confusing is why are we talking about "4" dimensions. To me what I know so far is that spinors seem to come "bundled" as a two valued complex number where the real part can be seen as the "dimension" and the imaginary as the "phase" part aka a "bivector". In other words for 3 xyz dimensions you would need a 6 component "vector". Which after doing your calculation and particle interaction with you can "collapse" in its real part to figure out its origin and "orientation".

[@Zxymr]

I don't have a perfect grasp of the concepts too, but maybe my understanding might be able to help you. 3D Domain (Pauli Spinors) - Wavefunction domain is real-valued (spatial coordinates). - You'd apply the Schrodinger/Heisenberg equation to Pauli Spinors, which do not account for relativistic effects, making this unsuitable for high-velocity fermions. - When converting a 3D vector to a spinor, the solutions span a 1D space (i.e. 1 basis solution). - The Pauli spinor contains 2 complex-valued components: spin ↑/↓. Thus, the codomain has 4DOF (maybe fewer, if any part is dependent). - In summary: 3D coordinate (3DOF) → wavefunction → 2D complex spinor (4DOF). 4D Spacetime Domain (Weyl/Dirac Spinors) - Minkowski signature turns either space or time imaginary, but each dimension in the domain has 1DOF (spacetime coordinates). - You'd apply the Dirac equation to Weyl/Dirac Spinors, which accounts for relativity. - When converting a 4-vector to a spinor, the solutions span a 2D space (i.e. 2 basis solutions, both conjugates of each other). We refer to one as left-chiral and the other as right-chiral, since they are reflections/mirror-images of each other. - Since the energies in both chiralities have opposite signs, this informs us of the existence of the antiparticle. - The Weyl spinor contains 2 complex-valued components: spin ↑/↓. Thus, the codomain has 4DOF. The conjugate is completely dependent so it does not add DOFs. - The Dirac spinor simply has the conjugate appended to result in 4 complex-valued components. However, it is still only 4DOF. - In summary: 4D coordinate (4DOF) → wavefunction → 2D (Weyl) / 4D (Dirac) complex spinor (4DOF). To dispel misunderstandings: - You seem to have conflated the dimensionality of the domain and codomain. Nothing here has 6DOF since we don't have a 3D complex-valued vector in any domain or codomain. - Vectors, bivectors, rotors, and spinors all refer to different structures under geometric algebra. - The structure used to represent the state of a particle corresponds to its spin. Scalars for 0-spin (Higgs), spinors for ½-spin (fermions), vectors for 1-spin (gauge bosons), bivectors for 2-spin (gravitons). Although, do note that grade-½ multivectors are not a thing, despite how tempting it is to call spinors that. - The reason why spinors are used for fermions is because spinors transform via a single-sided rotor, whereas multivectors are used for bosons because they transform under a double-sided rotor. (I don't entirely understand why fermions are special like that.) - The "transform" mentioned above refers to Lorentz Transformations not including reflections (i.e. spatial rotations and Lorentz boosts). - This explanation implies matter/antimatter symmetry. The breaking of that symmetry requires more complicated concepts (e.g. weak isospin).

[@ZeroInDaHouse]

@Zxymr Let me try an summarize your explanation in a way me and others perhaps would understand more easily that do not speak the Physics language. All particles come in pairs, as "matter/anti-matter" pairs. The spin corresponds mostly to the "anti" pair. On our dimensional and elemental scale of things this would be the Hydrogen atom for example. Proton (matter) and Electron (anti-matter). However to exchange information (momentum) between both one needs an exchange field. This exchange field would be the quantum field wave field that arises between both at the interface which acts as a dampening force keeping both stable. Otherwise all that dark energy would continuously split them apart and never form stable elements let alone molecules. So its really all about the dampening which manages to balance dark energy vs the dampening force which is the field.

[@Zxymr]

@@ZeroInDaHouse 1. Spin and Chirality refer to 2 different properties of electrons, quarks, muons and neutrinos. The chirality of an electron determines whether it's an electron or anti-electron. 2. An anti-electron is NOT a proton, and an anti-proton is NOT an electron. Protons, electrons, anti-protons and anti-electrons are all different particles. This means that an electron will NOT annihilate with a proton. 3. An electron is a fundamental particle (i.e. it isn't composed of anything smaller), while a proton isn't. A proton is composed of 3 quarks. 4. An anti-electron is also referred to as a positron. An anti-proton is composed of 3 anti-quarks. 5. In a hydrogen atom, there is mainly 1 force between the electron and proton: the electromagnetic (EM) force, which pulls the electron towards the nucleus (where the proton is). The electron doesn't crash into the nucleus because it is orbiting at a constant speed at a certain distance from the nucleus (similar to how planets don't crash into the Sun even though they are attracted by gravity). 6. The EM force is not a "dampening force" because it doesn't slow the electron down. It instead diverts its path from a straight line into a circular motion (simplified description, the actual orbit is not necessarily circular). 7. The field corresponding to the EM force is the EM field. The other fundamental forces (gravity, strong force, weak force) each have their own fields too. 8. Without the EM force, atoms cannot exist simply because there will be nothing holding electrons in stable orbit around nuclei. They will just travel in a straight line instead of an orbit. 9. Dark energy does not generally apply at the atomic scale. It's negligible at these scales and we don't need it to explain why atoms cannot exist without the EM force.

[@ZeroInDaHouse]

@@Zxymr Any motion comes with friction, whether we like it or not but friction can always be reduced to the bare minimum too. The electron is not "anti-matter" but I believe it is the anti-proton present in our mirror universe. Essentially the "inverse" of the proton which manifests as spin. Or in the language of mathematics as the "imaginary" component.

[@Achrononmaster]

Possibly better to decompose a spinor (in QM context) into a scalar factor (the probability density) and the rotor part (double sided instruction to rotate, so respecting orientation, as per the talk. Or as Misha Gromov would say, "matrices are stupid.")

[@tablettorrensabellan]

This talk is pure pschycodelia for an engineer... Mathematicians are truly disconnected from reality... And the talk is truly a hieroglyphic for an engineer.... I understand spinors and SU(2) but this talk is so psychedelic that I could not identify what I already know with anything in the talk... Sorry....

[@Pluralist]