I spent a couple hours trying to find an existing post on this, but couldn't find any. If you found one, let me know, we can link out to it and close this question out 🙂
Was reading some of Feynman's lectures on Physics and in light of recent reading I've been doing on Topology, was thinking I could formalize some of the informal statements he makes. Specifically, was interested in introducing a topological definition of symmetry and working out some of the basic consequences of that.
Feynman states:
"Weyl has given this definition of symmetry: a thing is symmetrical if one can subject it to a certain operation and it appears exactly the same after the operation."
From a topological point of view, I suggest that the definition can be phrased:
$\textit{Def}:$ Let $X$, $Y$ be topological spaces, $p: X \rightarrow Y $ be a (continuous?) function from $X$ to $Y$ and $\phi: X \rightarrow X$ be a homeomorphism on $X$. Then the subspace $p(X) \subseteq Y$ is $\textbf{symmetric}$ if $p(X) = p(\phi(X))$.
This seems to agree with the metric space definition given here (if we fix $y$ in the given metric space):
and here:
A general definition of symmetry in mathematics
if we take $\alpha = \phi$ (my $\phi$ above, not the one in the linked post) and $\beta = 1(Y)$.
My first question is: does this check out? Since I am newer to the subject, want to make sure I have my semantics correct.
Second question: without getting too deeply into the details, it seems to me that Noether's Theorem follows almost immediately, but I am much less certain on this, so I hope folks can point out where I've gone wrong. Feynman writes on this:
"[F]or each of the rules of symmetry there is a corresponding conservation law."
In physics, we phrase conservation laws as something like:
$\Delta E = 0$
We might intuitively rephrase the definition of symmetry as:
$\big[p(X) = p(\phi(X))\big] \rightarrow \big[p(X) – p(\phi(X)) = \oslash\big]$
But this looks just like the form $\Delta p = 0$ (if we aren't too pedantic about the symbols we are using). If we make the "$\rightarrow$" into a "$\leftrightarrow$" and make the suggested substitution:
$\big[p(X) = p(\phi(X))\big] \leftrightarrow \big[\Delta p = 0\big]$
But this looks like it hits the mark for the intuitive informal claim that Feynman outlines in the quote above. Conferring this with the intuitive claim here, it seems like the main key thing that I am perhaps missing is the notion of differentiablity. Assuming we can define a more general form of differentiation which conforms to the usual definition when we project down into the math of metric spaces (e.g. in the case of usual 1D calculus) and we posit that the mappings above are continuous in the topological sense, then I think we can fold the notion in. This is a rather nebulous sketch, but intuition suggests that we should be able to pursue this angle and that it conforms to basic cases in physics based on some basic arguments (such as the free particle under a spatial translation and conservation of momentum and temporal translation of a particle under an external conservative field such as gravity).
Thanks in advance for any insight!
Best Answer
"Symmetry" is an very broad term (even in physics) so I don't think trying to formalize a loose quote in the formalism you are familiar with is the best way forward to gain more insight into this topic. Instead I would try to study the actual mathematics underlying Noether's theorem. I don't know what the actual question is here, but what you have done is very far from proving Noether's theorem. First of all you should have a precise meaning of what it actually says and what the mathematical objects involved are. Below is a short introduction to just this.
The fundamental mathematical objects we use to describe physics are called fields $\phi_i(x,t)$ (in the simplest cases with classical mechanics just simple functions on time $y(t)$ describing where an object is located at any given time).
How these fields evolve and interact with other fields are described by a action functional $S(\phi_i)$, it takes in functions and outputs a real number. The fields follow the trajectories (how $\phi_i(x,t)$ evolve in space and time) that extremises the functional.
What Noether's theorem deals with is how certain symmetries of this functional implies conserved quantities along the trajectories that the fields follow. Importantly, it also tells you exactly what these conserved quantities are in terms of the fields involved.
For example if you have a particle described by $y(t)$ under the influence of a conservative force $F = -\frac{dV(y)}{dy}$ (like gravity) then the functional would be $S(y(t)) = \int[\frac{1}{2}y'(t)^2 - V(y(t))]{\rm d}t$. The trajectory that extremises the functional is in this case given by the differential equation $y''(t) = -\frac{dV(y)}{dy}$. Now the functional above is seem to be invariant under the symmetry we call time-translations: if we take $t\to t+\tau$ for a fixed $\tau$ then the functional remains the same, i.e. $S(y(t)) = S(y(t+\tau))$ or in other words the physics doesn't depend on where we place $t=0$. Noether's theorem in this case tells us that a quantity (that we call energy) is conserved along the trajectory of $y(t)$ (the one defined by the differential equation above) and that this energy is given by $E(t) = \frac{y'(t)^2}{2} + V(y(t))$. Indeed you can check that $\frac{dE(t)}{dt} = 0$, which is the definition of what it means to be a conserved quantity, using the differential equation above.
The example above is just the simplest example one can make. Noether's theorem tells us how to find conserved quantities by finding symmetries of general functionals. You can learn the basics of the underlying mathematics from studying (classical) Lagrangian mechanics and then Field theory. To get a deeper understanding of the kinds of symmetries Feynman is talking about (and which is what modern physics is built on) you should read about Lie groups.