Symmetry

A symmetry is a change in perspective that renders a system unchanged.

A symmetry can be a change in the system itself (like you rotate the system).

A symmetry can be a change in the way that you are perceiving the system (like you rotate yourself).

A symmetry can be a change in some structure that you have added to the system (like you have an ordered pair of things, and you decide to change which one you are calling "first" and which one you are calling "second").

The first two are true symmetries, the last is a gauge symmetry.

[TODO think about this] I want to say something like: a set of existents is said to have a $G$ symmetry if it admits a trivial $G$ action. But that isn't quite general enough. Like with my example of a can of soda, that's a single existent, not an infinite set of existents; and yet it has a U(1) symmetry.

One need only understand a part of a symmetric thing, in order to understand the entire thing. This is probably the essential reason why people like symmetric things: by virtue of being symmetric, they are easier to understand.

Examples

[TODO picture] Imagine a square S with different sides (labels on each of its sides). You can rotate it, and you get a different square S'. But if you forget about the labels (omit the measurements of the labels [TODO omission isn't forgetting]), then S and S' are the same. This thing has a natural $\mathbb {Z} _{4}$ symmetry.

Imagine a can of soda. If I rotate it, it will be arranged differently, but in some sense it will look the same. If I ignore the pattern on the surface of the can, and ignore the non-symmetric mechanism at the top of the can (omit its measurement), then the rotated can is indistinguishable from the original can. This thing has a $U(1)$ symmetry.

Imagine 4 identical balls. If I permute them, that is, if I rearrange them in such a way that they are in the same locations as they were before, then the 4 balls after the change are indistinguishable from the 4 balls before the change. This thing has an $S_{4}$ symmetry.

Imagine a pendulum, moving with very little friction so that it never really slows down. It goes left to right to left to right to left to right etc., repeating itself over a period of time $T$. It is in the same state now as it was $-T$ seconds ago, $-2T$ seconds ago, etc., and it will be in the same state again after $T$ seconds, $2T$ seconds, etc. If one were to forget all the indications that time has passed (omit measurements), then one would be unable to distinguish between the pendulum now, and the pendulum at those other times. This system has a natural $\mathbb {Z}$ symmetry.

Imagine a circular knob with 8 marks, which looks like ☀️, and which can be rotated freely. If the marks are indistinguishable, then this has a $\mathbb {Z}$ symmetry; for any $n:\mathbb {Z}$ I can rotate the circle clockwise by $n$ notches and one couldn't tell that I did anything. If the marks are distinguishable, then this has a different $\mathbb {Z}$ symmetry; for any $n:\mathbb {Z}$ I can rotate the circle clockwise by $n$ full rotations, and one couldn't tell that I did anything. Suppose I start with the latter case (distinguishable marks), and I omit the measurement of the marks; then this system can be said to admit a $\mathbb {Z} _{8}$ symmetry. [TODO I'm confused].

Suppose that I owe Bob $5. That's one way of looking at the situation. Another, completely equivalent way of looking at the situation, is to say that Bob owes me -$5. So the fact that I used the concept of an integer to describe a state of affairs, means that there is some sort of $\mathbb {Z} _{2}$ symmetry. This is the only example (so far) of a "gauge symmetry."

See also

• Group