Conservation of Quantity: Noether's Theorem

[Marty4691]

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 rotationsApplying 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.

 

[fresh_42]

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

 

[Marty4691]

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

 

[fresh_42]

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.

 

[Marty4691]

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

 

[fresh_42]

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.

 

[Marty4691]

I'm going to try and paraphrase your reply.

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]

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.

 

[Marty4691]

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?

 

[Marty4691]

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]

Noether's theorem makes a statement about time-invariant quantities. A rotation 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.

 

[mitchell porter]

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

 

[fresh_42]

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.

 

[Vanadium 50]

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.

 

[fresh_42]

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]

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.

 

[Marty4691]

So, it turns out that fresh_42 was right: mathematically, there is a conserved quantity due to invariance under time rotations (https://doi.org/10.3390/sym12050817). This quantity has the same units of measure as the Planck constant. For lack of a better name, perhaps we can refer to it as "angular energy". The consensus among PF mentors appears to be that this is not a physical quantity, just a mathematical quantity. This is a reasonable perspective.

Angular energy has an interesting relationship to angular momentum:

(1) The conserved quantity due to invariance under space rotations is angular momentum and has the same units of measure as the Planck constant.
(2) The conserved quantity due to invariance under time rotations is angular energy and has the same units of measure as the Planck constant.

Statement (1) underlies the Law of Conservation of Angular Momentum and is associated with the description of spin in the SM. Statement (2) seems to be encouraging us to consider time rotations in the same way we consider space rotations. This similarity in the properties of space and time might have made Hermann Minkowski smile, or perhaps, sit back and think.

Recently, a link was posted on PF/BSM to an R.A. Wilson article that seems to advocate taking a closer look at the symmetries SL(4,R) and SO(3,3) with respect to unifications. If progress is made in this multi-time direction then it's possible that angular energy may be become a "useful" quantity. Time will tell, I guess...

 

[bobob]

To reduce this to its physical consequences, if you could rotate the time coordinate, you could turn around and go backwards in time the same way you can turn around in space.

 

[Marty4691]

Nice macroscopic example. I guess I would argue that we may not understand causality at the quantum scale.

Take spin angular momentum. We know from Noether's theorem that it is associated with space rotations, but we are told that there is no actual physical rotation in space, it's an internal quantum number of a particle. Applying this reasoning (in an SO(3,3) space-time) to spin angular energy seems to imply that a "particle" can have this type of internal number without actually rotating physically through negative time.

Is this a loophole?... Hard to say.

 

[Marty4691]

Before I forget, PFers might be interested to know that SO(3,3) space-time has two classes of spin one-half particle. One class of particles has spin angular momentum, as usual. The other class of "particles" has spin angular energy and the following two properties:

(1) They are either only-left-handed or only-right-handed.
(2) They are one dimensional in space, so we shouldn't necessarily expect them to have the same mass scale as electrons.

The related math is in the linked article but as many may not be acquainted with the SO(3,3) Lie algebra I'll try and argue these points using symmetry principles:

An SO(3,3) space-time has SO(3,1) subspaces (Minkowski) and SO(1,3) subspaces (1-space, 3-time). Now SO(3,1) and SO(1,3) are isomorphic. This implies that there is a "dual" version of quantum theory and special relativity associated with an SO(1,3) subspace (just reverse the roles of space and time). So if an SO(3,1) subspace has a spinor particle with spin angular momentum then this implies that an SO(1,3) subspace has a "spinor particle" with spin angular energy.

Property (2) is easy. By definition an SO(1,3) subspace is one dimensional in space.

Property (1) is a little bit harder. Ironically, causality is our friend now. In an SO(3,1) subspace (Minkowski) a left-handed spinor particle is related to it's right-handed particle by a parity transformation. By symmetry, this implies that in an SO(1,3) subspace a left-handed "spinor particle" is related to it's right-handed "particle" by a time reversal transformation. If we break time reversal symmetry (that is, impose a causality constraint) then one of the hands is forbidden and we are left single-handed.