Symmetries

[Stefan Ram]

To better understand symmetries, I have a few questions, but for
simplicity's sake I'll start with one:

Let a hypothetical one-dimensional world consist of a ray with
values x>=0. This world is completely empty except for a mass
point with unit mass 1 at x=1. This is described by a "mass
density" R(x), which is zero everywhere except for R(1)=1.

Now a scientist comes along and says: I formally extend this world
to a two-dimensional world with the coordinates (x,y). The mass
density R(x,y):=R(x) is everywhere equal to 0, except for a mass in
the form of a straight line {(1,y)|yeR}. This world is invariant with
respect to y. A translation y'=y+y0 results in the same world again.
So, there is a preserved quantity, which I call the "y-momentum".

Now, there are two reactions: One praises the "deep result". Others
say that y is just a "redundant, unphysical coordinate" that has
no meaning at all, and that the result is completely irrelevant.

So, is the y-invariance of the two-dimensional world irrelevant
or meaningful? Why?

[[Mod. note -- In order to have unit mass, doesn't your mass density
need to be a Dirac delta-function? -- jt]]

[Stefan Ram]

Post by Stefan Ram
Let a hypothetical one-dimensional world consist of a ray with
values x>=0. This world is completely empty except for a mass
point with unit mass 1 at x=1. This is described by a "mass
density" R(x), which is zero everywhere except for R(1)=1.

...

Post by Stefan Ram
[[Mod. note -- In order to have unit mass, doesn't your mass density
need to be a Dirac delta-function? -- jt]]

Yes.

To avoid delta functions, I could as well define a mass
density R(x) := 1/E if 1 <= x < 1+E, and R(x) := 0, otherwise;
for an E > 0, say, E := 0.5.

The mass density of the two-dimensional extension then
correspondingly becomes R(x,y) := 1/E if 1 <= x < 1+E
and R(x,y) := 0, otherwise.

[Stefan Ram]

Post by Stefan Ram
To better understand symmetries, I have a few questions, but for

And here's my other question:

It's about something I often encounter in texts about
symmetries, the point where I find it hard to follow.

Here's an example:

|Given a perfect circle, you can rotate it by a tiny amount
|and find that you still have the same circle.
"Why String Theory?" (2016) - Joseph Conlon (1981/)

. A kinematic "rotation", to me, is a change in the angular
position of an object. I think of a "circle" as a physical object,
a kind of very thin torus (a torus the minor radius of which
is much smaller than its major radius). A "perfect circle" is
perfectly homogeneous. It has no hair. So one cannot measure its
angular position (rotation around its center in its plane) (just
as you cannot measure the "angular position" of a Higgs boson,
i.e., in which direction it "points"). Since a perfect circle does
not have an angular position, it also cannot rotate kinematically,
contradicting the statement, "you can rotate it" from the
quotation. So, the quoted statement makes not sense to me.

Is my objection justified?

[Thomas Koenig]

Post by Stefan Ram

Post by Stefan Ram
To better understand symmetries, I have a few questions, but for

It's about something I often encounter in texts about
symmetries, the point where I find it hard to follow.
|Given a perfect circle, you can rotate it by a tiny amount
|and find that you still have the same circle.
"Why String Theory?" (2016) - Joseph Conlon (1981/)
. A kinematic "rotation", to me, is a change in the angular
position of an object. I think of a "circle" as a physical object,

A circle is a mathematical construct, which can be described in
a variety of ways. Perfect circles do not exist in the physical
reallity, where there is no such thing as an infinitely thin line
(or a fully Euclidean space, for that matter).

Likewise, rotation in this context is a mathematical transformation,
and you can, of course, show that rotating a circle by a tiny (or
huge) amount around its center does not change the set of points
that the circle is composed of.

So, whatever argument the author goes on to make, I do not think
it can be understood to be grounded in physical reality.

[Stefan Ram]

Post by Stefan Ram
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/).