 # Werner Heisenberg meets Emmy Noether?

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Someone asked a question the other day that made me realise that there seems to be a parallel between the Heisenberg uncertainty principle (HUP)  and Noether's theorem.

For example, one conjugate pair of variables in the HUP is energy and time. While in Noether's theorem, the conservation of energy is related to time symmetry.

Another conjugate pair is momentum and position. And the conservation of momentum is related to spatial translation symmetry.

I assume this is not just a coincidence. So is there a deeper reason? Is Noether's theorem the reason why these are conjugate pairs?

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Huummm…
Noether's Theorem doesn't really relate position and momentum, or time and energy.
It simply states that every differentiable symmetry ( time and translation )of the action ( integral over time of the Lagrangian ) has a conserved current ( energy and momentum ).

I'd put it down to coincidence.
But if someone can show a relation, I'd be very interested in hearing about it.
( good question, Strange )

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2 hours ago, Strange said:

Someone asked a question the other day that made me realise that there seems to be a parallel between the Heisenberg uncertainty principle (HUP)  and Noether's theorem.

For example, one conjugate pair of variables in the HUP is energy and time. While in Noether's theorem, the conservation of energy is related to time symmetry.

Another conjugate pair is momentum and position. And the conservation of momentum is related to spatial translation symmetry.

I assume this is not just a coincidence. So is there a deeper reason? Is Noether's theorem the reason why these are conjugate pairs?

IIRC, the HUP is just a consequence of position and momentum being related via fourier transform.

So, global gauge invariance of the EM field gives us conservation of charge via Noether; is there a corresponding uncertainty pairing?

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7 hours ago, ydoaPs said:

IIRC, the HUP is just a consequence of position and momentum being related via fourier transform

Yes. So maybe I am asking why those specific pairs are related via Fourier transforms.

7 hours ago, ydoaPs said:

So, global gauge invariance of the EM field gives us conservation of charge via Noether; is there a corresponding uncertainty pairing?

No, there isn't. Which is interesting.

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10 hours ago, MigL said:

Huummm…
Noether's Theorem doesn't really relate position and momentum, or time and energy.
It simply states that every differentiable symmetry ( time and translation )of the action ( integral over time of the Lagrangian ) has a conserved current ( energy and momentum ).

I'd put it down to coincidence.
But if someone can show a relation, I'd be very interested in hearing about it.
( good question, Strange )

I really like that setting of Noether in a Physics context. +1

As regards to a relation have you considered Gauss flux theorem and its relation to the fundamental theorem of calculus?

Here Gauss flux = your 'current'.

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Floated this by my colleagues this morning. The quick guess was that they are related, and perhaps it can be thought of this (hand-wavy) way: Noether's theorems are classical, in that the variables are continuous and well-defined. In a quantum context, you can't say that momentum is conserved below some scale of ∆x, so you can't ensure there is symmetry on that scale. So if you violate spatial translation at a scale of ∆x, you can violate momentum conservation by ∆p.

You'd need to do some actual analysis to see how and where the h-bar shows up in the formulation. My take is that Noether's theorems are math, and the HUP is physics, and nature is what dictates the scale of the potential violations.

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6 hours ago, Strange said:

Yes. So maybe I am asking why those specific pairs are related via Fourier transforms.

No, there isn't. Which is interesting.

Then I'm inclined to think it's just a coincidence.

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1 hour ago, swansont said:

...........and well-defined................

Yes.

To go with this I like the spectroscopist's version of the HUP to explain line broadening.

Using the Δt / ΔE version

The transition take a finite time which leads to time uncertainty.

So there must be an assiciated corresponding energy uncertainty.

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