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- TL;DR Summary
- Consider 2π rotations. In SO(3) they are nontrivial loops. In SU(2) they flip a quarternion's sign. These are physically unobservable effects of a 2π rotation. Can they be mathematically avoided? If not, must we accept physical unobservables?

I will ask a mathematical and a physical-cum-philosophical question pertaining to the fact that SO(3) is not simply connected.

Classical rotations in three spatial dimensions are represented by the group SO(3), whose elements represent 3D rotations. Having said that, note that classical rotations by 2π (360°) are equivalent to rotations by 0° (no rotation at all), because the former rotation simply restores the object's original orientation. Classical systems rotated by 2π are strictly identical to their unrotated versions.

This equivalence is represented in SO(3) both algebraically and topologically. Algebraically, where SO(3) is seen as a nonabelian group, that equivalence is represented by the fact that rotations by 2π are equivalent to an infinite composition of infinitesimal rotations which results in the identity element. Topologically, where SO(3) is seen as a topological space, that equivalence is represented by the fact that rotations by 2π are closed paths (loops) in the topological space, so that it begins and ends on the same point.

What worries me is that such loops are non-trivial loops. This is to say that such loops are not equivalent (homotopic) to the identity rotation (i.e. the rotation by 0°), because they cannot be continuously deformed into each other. In fact, it is only a rotation by 4π (720°) that is equivalent to the identity rotation. In this sense, SO(3) remembers previous rotations – up to parity (even or odd number of 2π rotations). This is a form of path-dependence: although 2π rotations correspond to identity rotations, they are topologically distinct because they correspond to different topological paths. We can refer to this as

What troubles me is that the classical world has no such rotational memory (path-dependence). As I said above, classical systems rotated by 2π are strictly identical to their unrotated versions. As such, SO(3)'s path-dependence has no physical significance in classical physics. To be sure, this path-dependence is crucial for the existence of SO(3)'s double cover SU(2), which models spinorial objects such as fermions (spin-½), whose properties are physically observable, but this bears no relation to the mathematical representation of classical 3D rotations.

I am thus troubled, because as it seems it is

My mathematical question is: Can we model classical 3D rotations where 2π rotations are trivial loops?

My physical-cum-philosophical question is: Should

What do you think?

Thanks for your attention!

Cheers,

Dan

__Context__Classical rotations in three spatial dimensions are represented by the group SO(3), whose elements represent 3D rotations. Having said that, note that classical rotations by 2π (360°) are equivalent to rotations by 0° (no rotation at all), because the former rotation simply restores the object's original orientation. Classical systems rotated by 2π are strictly identical to their unrotated versions.

This equivalence is represented in SO(3) both algebraically and topologically. Algebraically, where SO(3) is seen as a nonabelian group, that equivalence is represented by the fact that rotations by 2π are equivalent to an infinite composition of infinitesimal rotations which results in the identity element. Topologically, where SO(3) is seen as a topological space, that equivalence is represented by the fact that rotations by 2π are closed paths (loops) in the topological space, so that it begins and ends on the same point.

What worries me is that such loops are non-trivial loops. This is to say that such loops are not equivalent (homotopic) to the identity rotation (i.e. the rotation by 0°), because they cannot be continuously deformed into each other. In fact, it is only a rotation by 4π (720°) that is equivalent to the identity rotation. In this sense, SO(3) remembers previous rotations – up to parity (even or odd number of 2π rotations). This is a form of path-dependence: although 2π rotations correspond to identity rotations, they are topologically distinct because they correspond to different topological paths. We can refer to this as

__rotational memory__.What troubles me is that the classical world has no such rotational memory (path-dependence). As I said above, classical systems rotated by 2π are strictly identical to their unrotated versions. As such, SO(3)'s path-dependence has no physical significance in classical physics. To be sure, this path-dependence is crucial for the existence of SO(3)'s double cover SU(2), which models spinorial objects such as fermions (spin-½), whose properties are physically observable, but this bears no relation to the mathematical representation of classical 3D rotations.

I am thus troubled, because as it seems it is

**impossible**to represent classical 3D rotations without a mathematical apparatus that introduces a physically unobservable rotational memory. If we use SO(3), we introduce rotational memory in the form of a nontrivial loop. If we use quaternions from SU(2), we introduce rotational memory in the form of negative quaternions, which are also unobservable. (A rotation by 2π transforms a quarternion**q**into its inverse –**q**. As such, it is not a closed loop, but this has no physical consequences because**q**and –**q**behave identically.)__Questions__My mathematical question is: Can we model classical 3D rotations where 2π rotations are trivial loops?

My physical-cum-philosophical question is: Should

**any**mathematical model of classical 3D rotations**necessarily**introduce rotational memory, are we forced to accept its physical reality, however unobservable it is?**My view**is that mathematics is not just a tool, not just a useful fiction. It is the abstract study of any specifiable structure. Mathematics is "unreasonably effective" in modelling reality because physical systems are structured; necessarily so, I would say. What we are seeing from our discussions of SO(3) and SU(2) is that any structure capable of 3D rotation necessarily has an inner substructure that acts as rotational memory.What do you think?

Thanks for your attention!

Cheers,

Dan