Hawking, via socrat44 wrote:particles have different properties of spin: 0, 1, 1/2, . . . etc. and

therefore they look different from different directions.

I specified fundamental fermions (e.g. electrons and quarks). Composite fermions, such as protons and neutrons, will exhibit a sum of spins, conserving angular momentum.

Hawking, via socrat44 wrote:A particle of spin 0 is like a dot: it looks the same from every direction. …'On the other hand, a particle of spin 1 is like an arrow: it looks different from different directions. Only if one turns it round a complete revolution (360 degrees) does the particle look the same.

"Virtual bosons" and "massless particles" model lightlike interactions, thus should be ignored as "particles". Lightlike interactions have zero-magnitude worldlines. No worldline, no particle. A better model is required for "force carriers". It is simpler to consider the force itself as an object, without inventing some other object to carry it. A ray-like (or "arrow") force object would be a component of the particle (i.e. the "thing" that is spinning), rather than yet-another sub-particle.

Hawking, via socrat44 wrote:A particles of spin 1/2 . . . looks the same ''if you turned it through two revolutions (720 degrees).

Yes! Hawking is naively referring to planar rotations, which have a spatial axis and encompass 2pi radians (i.e. spin in 3D). Chronaxial spin, which I personally attribute to "spin½" fermions, occurs in 4D (about time). It would thus, be solid angular spin encompassing 4pi steradians.

Like a Mobius strip (used to illustrate spinors), this requires two planar rotations to return the exact starting condition, exactly as observed.

The term "spin½" is absurd because there is no such thing as half a quantum! It arises because the measured spin of a fundamental fermion (such as an electron) is found to be half the expected quantum of angular momentum. Specifically, it is half of ħ. But ħ is the reduced Planck constant or h/2pi. Reducing by 2pi radians is fine for planar rotations but is incorrect for solid angular (4pi) rotations expected with chronaxial spin. With chronaxial spin, an electron should be expected to have spin = h/4pi, exactly as observed.

socrat44 wrote:Usually the form of electron seems as a ''ball''.

A ray-like force object, spinning about time, would describe a spherical (i.e. solid angular) field. The location central to that field at any moment is what we refer to as a "particle". Over time, that particle describes a worldline.

socrat44 wrote:So, it seems that Hawking is right: ''the particles DO NOT HAVE ANY WELL - DEFINED AXIS.''

Ha! Tell that to the electrons.

Jorrie wrote:FD, have you ever considered a 5th dimension for your "spin axis"? … "hyperaxial spin"

Yes. But less is more in models, and full 5D rotations would have greater* than the 4pi steradians, which so neatly fit above. "Hyperaxial spin" is rather nonspecific, as it could mean any spin in any plane greater than 2D. Chronaxial spin occurs in a 3-plane and might thus be specified as "3-spin". Rotation in a 4-plane (or an n-plane) might be specified as "4-spin" (or "n-spin").

*As "steraidians" means "square radians", try 4pi/3 "soladians" (for "solid radians")?

Jorrie wrote:BTW, what you referred to as "interval-time coordinates" looks to me the same thing as standard space-propertime coordinates in SR.

"Interval-time coordinates" does refer to proper time but not to space. This is where Epstein coordinates get in trouble. They strive for Euclidean simplicity but fail to substitute interval coordinates for space. Thus, they can’t handle light. With interval coordinates, lightlike intervals define the origin, plain and simple.

Einstein felt invariance (to all observers) to be the strongest argument for "reality". Thus, he might well have considered interval-time coordinates to be 75% more real than Minkowski's, which he was initially reluctant to accept.

"…a four-dimensional space-time… Indeed, Einstein himself was not sympathetic to this idea when he first encountered it. … The idea of space-time was not, in fact, Einstein's… It was…Herman Minkowski," - R. Penrose (forward to R. Feynman, p. xiv)