Symmetry in Higher Dimensions: Sean Carroll's Video & Physics

[Shirish]

I'm watching Sean Carroll's video on symmetry [relevant section at around 8:05]

He talks about 120 degree rotations of triangles that leave them invariant. Then he proceeds to talk about flipping them with an interesting (at least to me) remark - "there's nothing that says I'm confined to moving (the triangle) in the plane".

A natural scenario that comes to mind following this is: if there was an observer that's purely confined to the 2D plane, and if there were a triangle on that plane, then the observer could completely specify and map out the triangle by roaming around it and measuring its lengths and angles.

Also, for that observer, the notion of 120 degree rotations would be a totally natural symmetry
to think about. On the other hand, a flip would be a not-so-natural, "higher dimensional" symmetry that's not confined to the 2D plane.

I'm currently studying special relativity and in that context we usually talk about homogeneity (translational symmetry) and isotropy (rotational symmetry) as far as I know, which are symmetries in our usual 4D spacetime. So I'm curious as to whether there are "higher dimensional" symmetries in our case as well. I'm sure fields like group theory address such questions, but I'm curious about how prevalent/useful this "higher dimensional" symmetry (there's probably a proper technical term for this which I don't know, hence the quotes) idea is in Physics.

Are there any fields (in Physics) where the notion of such kind of symmetries is useful?

 

[etotheipi]

To me his 'flip' operation looks like it can be completely defined in 2D. E.g. if we orient a coordinate system so that the $y$ axis lies along the direction of the mirror line, and the $x$ axis is perpendicular to the mirror line, then the symmetry operation is $T(x) = -x$. The mini-observers in the plane drop a perpendicular to the mirror line and shift all of the points along the direction of those perpendiculars to an equal distance on the other side of the mirror line.

(Though in 3D, a reflection in the mirror plane in 3D is generally not the same as a rotation of 180 degrees. Although if we embedded his (2D) triangle in the 3D space and performed these two 3D operations (w/ the mirror plane $\bot$ the triangle), they would in this case have the same effect).

But that is all that I can tell here, I am interested to see what others have to say!

 

[strangerep]

Are there any fields (in Physics) where the notion of such kind of symmetries is useful?

Many very intelligent people have tried for 100+ years to construct useful models of physics based on higher dimensional spacetimes, i.e., higher than the usual 4D.

Afaik, all such efforts turned out to be dead ends, at least as far as physics is concerned.

 

[PeterDonis]

To me his 'flip' operation looks like it can be completely defined in 2D.

You can define it mathematically without making use of a third dimension. But you could not actually perform it in 2D with 2D objects made of 2D "atoms" that cannot interpenetrate each other, because you could not do this:

The mini-observers in the plane drop a perpendicular to the mirror line and shift all of the points along the direction of those perpendiculars to an equal distance on the other side of the mirror line.

In three dimensions, you can implement this operation because there is now "room" for the atoms to slide past each other; but in true 2D there is no "room" to do this. Or, to put it another way, in true 2D the ordering of atoms along any line cannot be changed, and the "parity" operation you describe requires the ordering of atoms along lines to be changed.

 

[Shirish]

@PeterDonis , @etotheipi : Good point - even though a "reflection" and "flip" have the same end result as you guys pointed out, one is physically impossible and the other isn't. So after your feedback, by "higher dimension" symmetries I now mean those that are physically not possible in 4D spacetime, even though they're mathematically possible.

Pretty sure group theory (or some other field's mathematicians) folks must have come up with several symmetries like that. To explain those physically, one might need to invoke an assumption that 4D spacetime is embedded in higher dimensional space (just like that 2D observer might do for a "reflection").

I'm pretty sure this is extremely speculative and possibly untestable, but since I'm a noob, I was curious whether physicists have considered this question, formulated hypotheses around it and come up with possible predictions. From what @strangerep said, any work around this question has lead to dead-ends.

 

[Ibix]

As noted, "flipping" a triangle is equivalent to a reflection, and such discrete symmetries are of interest. Physics is invariant under time-reversal, for example. Most of physics is symmetric under "parity", a spatial reflection, but anything chiral (notably the weak force) or involving rotation can be anti-symmetric. Higher dimensions are not involved.

The simplest higher dimensional theory I'm aware of is Kaluza-Klein theory, which is a 5d extension to GR. Somewhat surprisingly, electromagnetism drops out of it alongside gravity. Unfortunately so does a scalar field, a "fifth force" that is not seen in nature, so it remains mainly a curiosity. I think similar reasoning led to string theory, which is still an area of active research.

 

[vanhees71]

Interestingly physics is NOT invariant under these discrete transformations. The weak interaction violates each separately (as seen one by one in experiments). Only CPT (the "grand reflexion", i.e., the combination of charge conjugation, space reflection, and time reversal) still is not seen to be violated, strengthening the belief that there's a lot of truth in local relativistic QFTs.

 

[George Jones]

So I'm curious as to whether there are "higher dimensional" symmetries in our case as well.

This depends on what "higher dimensional" means. If I choose (because I want to ) to interpret the extra dimensions as belonging to an internal non-spacetime space, then we have the symmetry group $G \times P$, where $G$ is the the symmetry group of the internal space, and $P$ is the Poincaré group, the symmetry group of a fine (pin intended) space, Minkowski spacetime. In the standard model, $G = SU\left(3\right) \times SU\left(2\right) \times U\left(1\right)$ (quantum chromodynamics and electroweak). Physicists have tried other groups for $G$, e.g., GUTs based on $SU\left(5\right)$ (ruled out by experiment) and $Spin\left(10\right)$, which physicists sometime call $SO\left(10\right)$ (because they have isomorphic Lie algebras).