r...@zedat.fu-berlin.de (Stefan Ram) writes:
>Now a scientist comes along and says: I formally extend this world
>to a two-dimensional world with the coordinates (x,y).
To explain why I made this post, which introduces what appears
to be a redundant coordinate:
The unit circle is typically defined as a set of points in a
plane whose distance from a central point is equal to 1 (one).
Accordingly, using polar coordinates, the unit circle is the set
{(r, phi)|r = 1}.
I think of the circle as a world and the coordinate system as a
tool introduced to describe that world.
The coordinate "phi" seems redundant since the world does not
depend on it; one could omit it and just use { r | r=1 } to
describe this world.
It appears that the notion of rotational symmetry being a
symmetry operation may arise due to the redundancy present
within the coordinate system (r, phi).
After sending my previous post, I discovered some sources
expressing ideas similar to mine:
|Gauge symmetries are redundancies in the mathematical description
|of a physical system rather than properties of the system itself.
|[...] The difficulty of making these laws explicit in a natural
|and non-redundant way is the reason for "gauge symmetry".
Symmetry and Emergence (2018) - Edward Witten (1951/)
If what Witten says is true, then it is not surprising that
in some cases a physical system has no symmetry until one
considers its mathematical description!
Witten also gives an example, here it is in the words of Seiberg:
|It is often the case that a theory with a gauge symmetry is dual to
|a theory with a different gauge symmetry, or no gauge symmetry at
|all. A very simple example is Maxwell theory in 2+1 dimensions. This
|theory has a U(1) gauge symmetry, and it has a dual description in
|terms of a free massless scalar without a local gauge symmetry.
Emergent Spacetime (2005) - Nathan Seiberg (1956/).