# Physical reality of nontrivial loops in SO(3)

• I
• DanCoimbra

#### DanCoimbra

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.

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?

Cheers,
Dan

Ibix
Doesn't the belt trick demonstrate that the non-nullity of a continuous sequence of rotations between 0 and ##2\pi## (paired with a return to nullity when we rotate through a further ##2\pi## in the same direction) reflects a physical reality?
Perhaps the difficulty lies in the statement "rotations by 2π are equivalent to an infinite composition of infinitesimal rotations which results in the identity element". That is an interpretation, rather than an intrinsic feature of the SO3 model. Strictly speaking, the path from North Pole to South Pole in SO3 is not a ##2\pi## rotation but a map from [0,1] to a set of rotations ranging from ##-\pi## to ##+\pi##. To say anything more than that, about equivalence or such-like, is interpretation. I would rather say that a rotation by ##2\pi## is represented by two identified antipodal points on the surface, not a loop.

andrewkirk said:
Doesn't the belt trick demonstrate that the non-nullity of a continuous sequence of rotations between 0 and ##2\pi## (...) reflects a physical reality?
Hi, Andrew! Thanks for for insightful answer.

As far as I understand, the belt trick is a pedagogic device for developing intuition. The belt trick would represent classical rotations if and only if classical objects had unobservable strings thetering them to their surroundings. In this case, a ##2\pi## rotation would indeed not restore the system's original state, due to string's so-called rotational memory. Without such strings, a ##2\pi## rotation should be null in classical reality, should it not?

That is an interpretation, rather than an intrinsic feature of the SO3 model. (...). I would rather say that a rotation by is represented by two identified antipodal points on the surface, not a loop.
The SO(3) model is an abstract object subject to multiple interpretations, so long as they are interconvertible. The different algebraic and topological descriptions are some such examples. Your description of SO(3) in terms of a solid sphere, where rotations are represented by an axis and a distance from the center, is one such interpretation!

This solid sphere has a property we must not overlook: for any given axis within the solid sphere, the antipodal points describing ##+\pi## and ##-\pi## rotations are equivalent and must be topologically identified. As such, going between antipodal points does indeed form a topological (nontrivial) loop, and one must traverse this path twice in order to obtain a trivial loop.

DanCoimbra said:
My mathematical question is: Can we model classical 3D rotations where 2π rotations are trivial loops?
Why should they be all trivial loops, and not all be non-trivial loops?

I'm asking because of another example that came to my mind:

The topology of a pac-man or asteroids level (left edge = right edge, top-edge = bottom-edge) is that of a torus. And it seems like all the loops you create there, by endlessly traveling in the same direction are non-trivial. But after each loop you arrive at states that are indistinguishable from each other.

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DanCoimbra said:
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?

[...]

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.
This is all based on a false assumption. 4π rotations are physically distinguishable from 2π rotations, in general. See my post #23 in this thread. [Edit: Oh, I see @Delta Kilo has already given a reference.]

DanCoimbra said:
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.
That depends on precisely what you mean by "classical 3D rotations". In general 3D rotation of a vector is not the same as a 3D rotation of a spinor.

A.T.
I suppose my TL;DR was misleading. The purported issue arises only for classical spatial rotations, since classical orientation is a vector. The examples given by @Delta Kilo and @strangerep are spinor rotations, which apply only to quantum states. The example given by @A.T. restates the issue: transformations in classical spaces can have topologies containing loops whose nontriviality correspond to no physical observable.

Having clarified this issue, I would like to further develop the discussion with a point that was recently brought to my attention. Topological paths through ##SO(3)##, in our application, correspond to rotation transformations occurrent in time. As such, they correspond to 4D worldlines in spacetime (of any geometry: flat, hyperbolic, or spherical). It is therefore not expected that ##SO(3)##'s topological features would corresponds to observable features in 3D events, but to observable features of 4D worldlines, if such a thing makes sense.

Consider an unmoving solid sphere in a stationary reference frame. A rigid ##2\pi## rotation about an axis crossing its center would generate a cylindrical-helix worldline for each of its constituent particles. Taken together, these worldlines fill in a 4D helical cylinder, twisted by ##2\pi## from end to end. Perhaps the question can then be reframed: can the nontriviality of ##2\pi## rotations in SO(3) correspond to some physical feature which distinguishes ##2\pi##-twisted cylindrical worldlines from untwisted cylindrical worldlines, but does not distinguish the latter from ##4\pi##-twisted cylindrical worldlines?

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