# Conserved quantities

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## Main Question or Discussion Point

Hi,

I have a question and I was hoping for some help. The reasoning goes something like this:

There appears to be two fundamental types of coordinates

x - space
t - time

and there appears to be three types of fundamental transformations

- translations
- rotations
- boosts

If we ignore boosts for the moment, then combining these gives four combinations

- space translations
- time translations
- space rotations
- time rotations

Applying Noether's theorem to the first three gives us three fundamental laws of physics

invariance under space translations -> conservation of linear momentum
invariance under time translations -> conservation of energy
invariance under space rotations -> conservation of angular momentum

I guess my question is: If we apply Noether's theorem to invariance under time rotations, how likely is it that we will get another fundamental law of physics?

invariance under time rotations -> conservation of ???

Thanks.

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fresh_42
Mentor
Let's answer the question "How do one-dimensional rotations look like?" first.

Could we add another time dimension to Minkowski space-time to allow the rotation?

fresh_42
Mentor
Could we add another time dimension to Minkowski space-time to allow the rotation?
In mathematics, yes, in physics we will get into trouble with the principle of cause and effect.

If we just stick to the math, do you know if anyone has figured out Noether's theorem for time rotations?

fresh_42
Mentor
You can easily define and consider e.g. $O(3,2)$ which is a ten dimensional Lie group. Then you can search for differential equation systems, which are invariant under this group. But how shall we manage to find the physical system which corresponds to our differential equation system. A second time coordinate wouldn't have any correspondence in our universe.

I think you're saying that even if we come up with a conserved quantity via Noether's theorem, it "lives" in O(3,2) space-time and may not be physically defined in an O(3,1) space-time because the conserved value may dependent on two time variables, whereas O(3,1) only has one.

Does this sound right?

fresh_42
Mentor
No, the conserved quantity is just a consequence of the invariance of a differential equation system which describes a physical process. You start with physics and create the math.

You can create the mathematical environment for two time coordinates, but you cannot attach physical meaning to the quantities involved. Such a mathematical model would provide solutions for situations which cannot exist. E.g. you can consider faster than light equations, but it will remain paperwork. Another example are numbers. We deal with real or complex numbers, but the highest number I have ever heard which has been used was $10^{120}$ - and this is still a small number in comparison to infinity. String theory is similar: a mathematical model and nobody has ever found evidence. In this sense you can pretend to have a second time coordinate, and if I remember correctly, there have been such considerations. However, there is no physical evidence.

Ok, so we make it a purely mathematical exercise. We create a mathematical environment with two time coordinates and we assume that the conserved quantity doesn't have physical meaning. It seems possible that Noether's theorem might give us a mathematical expression for the conserved quantity. Something we could try and interpret. Presumably it will have units of measure. Could we try and get at it that way?

I apologize for taking up so much of your time fresh_42. Thanks for all your replies.

In case your interested, there is a really good walk through of Noether's theorem for normal space-time in

Schwichtenberg, J. : Physics from Symmetry. 2nd ed., Springer 2017

Gotta go. Cheers,

Marty4691

mfb
Mentor
Noether's theorem makes a statement about time-invariant quantities. A rotating in two time dimensions would change the time axes themselves. I also doubt that the theorem works with two time dimensions. In general you either get contradictions or trivial universes (no time evolution along at least one axis).

In special relativity boosts are similar to modified rotations in spacetime.

Google '"two time" noether' for theoretical work on this topic.

fresh_42
Mentor
I also doubt that the theorem works with two time dimensions.
Noether's theorem(s) is (are) pure mathematics (Lie theory), which means there are no restrictions on the setup, except smoothness and such things. Its application in physics is a consequence of the theorem, not the subject.

Staff Emeritus
2019 Award
Noether's theorem(s) is (are) pure mathematics (Lie theory), which means there are no restrictions on the setup, except smoothness and such things
I sort of agree with this and sort of don't. It's a mathematical theorem involving the action, or if you like, the integral over a Lagrangian density. If your theory doesn't have a well-defined action (and this can happen in ordinary 3+1 spacetime), Noether doesn't apply. It doesn't make the theorem any less true, but it makes it a lot less relevant. There's plenty of math out there that's not relevant.

mfb
fresh_42
Mentor
It's a mathematical theorem involving the action, or if you like, the integral over a Lagrangian density.
Formally they are theorems about the invariance of a differential equation under coordinate transformations. She mentions the integral form of a Lagrangian as an important example which she dealt with in the second paper. I just wanted to emphasize, that in order to apply it to a physical situation, physics has to come first. But if we ignore all physics, we still will have a valid theorem about differential equations, regardless what meaning we attach to the coordinates. The OP seemed to have - hopefully had - the opinion that each application of the theorem produces a conservation law. A second time coordinate won't affect the theorem, only the fact that it doesn't describe a physical system anymore. It's like: $A_{137}$ is a simple Lie group, but who cares?

mfb
Mentor
only the fact that it doesn't describe a physical system anymore
Well, this thread is about physics. Of course the theorem stays valid, it's a theorem. But "works with" in my comment refers to applications in physics.