Symmetry — apokrisis

Mind if I pick you brain on the concept of symmetry.
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One kinda symmetry is geometric, we learn it in high school:

1. Reflection/mirror symmetry. The letters A C, H have them.

2. Rotational symmetry: S and O have them.
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The other kind is what I call Noether symmetry in which the past and the present, through the future are identical with respect to something. Conservation laws seem to possess this type of symmetry: You start with 10 J of energy at time t₁, you wind up with 10 J of energy at another time t₂ (law of conservation of energy).
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What's the relationship, if any, between the geometric symmetries mentioned above and Noether symmetry? They seem to be unrelated, yet I sense some connection among them. Can't put my finger on it, alas.

What's the relationship, if any, between the geometric symmetries mentioned above and Noether symmetry? — Agent Smith

Reflections - and even rotations of triangles - are discrete symmetries. You can see something has changed even if it maps back on to itself. But the rotation of sphere would be an example of a continuous symmetry.

Noether's theorem says there is a conservation law for every continuous symmetry in nature. So you have the two forms of momentum conservation - rotations and translations. Each looks the same amount of action anywhere in a Newtonian conception of space. And then you get energy conservation from actions in the time dimension.

One moment in time is considered no more special or different than any other. It is another continuous symmetry. Shift an event from last year into next year and it does nothing observable to the energy content. So time translation can't change the energy content of the Cosmos.

This is all correct in the Newtonian world, which is suitably closed and thus properly symmetric as both a spatial and temporal coordinate system.

But general relativity is not so rigid. It is an open story of spacetime dimensionality. So energy isn't conserved - although a closure can then be constructed, and symmetry restored, by talking a higher level view that balances the total inertial mass of the system against its total gravitational potential.

So the relationship is simple. Continuous symmetry means you can't see a difference just by shuffling the localised content of a system about in spacetime. Thus some general quantity of a local "stuff" is being preserved in a substantial way - such as momentum or energy.

But all this holds good only if the global container is closed and rigid enough to sustain the continuous symmetry. If spacetime evolves in some fashion, then energy conservation doesn't strictly apply.

Let me see if I got it.

1. Discrete symmetries have stages in transformation that exhibit noticeable in-between changes e.g. rotating an equilateral triangle.

2. Continuous changes are such that no stage in the transformation can be identified as different e.g. spinning a sphere/circle.

3. Noether symmetries are continuous transformations,the transformations being temporal translations.

4. In GR, there are issues with Noether symmetries.

Didn't mean non-locality alone for contextuality, although some people do claim that non-locality demonstrates contextuality.

I was thinking of more recent Bell-Wigner experiments, instead of just testing of Bell inequalities. Because the difference is that assuming non-locality might not be enough maintain an objective world, one where all observers can reconcile their recorded facts.

Modulo the potential loopholes and accepting the photons’ status as observers, the violation of inequality (2) implies that at least one of the three assumptions of free choice, locality, and observer-independent facts must fail. The related no-go theorem by Frauchiger and Renner (5) rests on different assumptions, which do not explicitly include locality. While the precise interpretation of (5) within nonlocal theories is under debate (21), it seems that abandoning free choice and locality might not resolve the contradiction (5). A compelling way to accommodate our result is then to proclaim that facts of the world can only be established by a privileged observer—e.g., one that would have access to the “global wavefunction” in the many worlds interpretation (22) or Bohmian mechanics (23). Another option is to give up observer independence completely by considering facts only relative to observers (24), or by adopting an interpretation such as QBism, where quantum mechanics is just a tool that captures an agent’s subjective prediction of future measurement outcomes (25). This choice, however, requires us to embrace the possibility that different observers irreconcilably disagree about what happened in an experiment. A further interesting question is whether the conclusions drawn from Bell or Bell-Wigner tests change under relativistic conditions with non-inertial observers (26).

https://www.science.org/doi/10.1126/sciadv.aaw9832

Noether symmetries are continuous transformations,the transformations being temporal translations. — Agent Smith

Think of it as the global Newtonian 3D spatial container and its local energetic actions. Spinning on the spot, or moving with constant motion in a straight line are both defined as inertias. Momentum is conserved by the moving mass, never being gained or lost (in the frictionless perfection of the gravitationless Newtonian frame).

So that fact reflects the continuous symmetry built into the Newtonian description of the container. Move some local rotation or translation to some other arbitrary point inside the container, and nothing is physically different as far as the amount of rotation or translation that would be observed.

Then likewise, move the action - some translation or rotation - to another moment in time, and it will still reflect the same amount of energetic effort. A one joule shove a billion years ago will look like a one joule shove a billion years from now ... if you grant the fixed symmetry of the Newtonian conception of space and time.

So it is the unchanging invariance of the whole - the spacetime stage in which the local action takes place - that ensures the conserved quantities of those local actions. The same actions would carry the same weight no matter where they happened in time and space.

We call that space-conserved property an object's inertia. And the time-conserved property its energy.

It is an accountancy trick. We know that Newton idealised the situation for the sake of simplicity. But it works at our general scale of being as observers of a now very large, very flat, very cold, and very empty universe.

Nope, didn't mean non-locality for contextuality, although some people do claim that non-locality demonstrates contextuality because ones observers' observations effect another's in such a baffling way. — Count Timothy von Icarus

I admit my views are provisional here. But for me, non-locality and contextuality seem two sides of the same coin - a dichotomy, and thus reciprocally related.

When you break the symmetry of a realist metaphysics, it breaks in these two directions, depending on whether you think you are talking about the localism of the particular interaction, or the globalism of the world that constrains the probabilities.

So like all QM's complementary variables, you can't violate both inequalities at the same time. If you are moving towards the one limit, you are moving away from the other, so to speak. And yet you also know the two are related in a deeper way.

This makes a neat fit as it promotes the two things of entanglement and contextuality to be being a more abstract commutative pair. We can see the holism of the quantum state from both its local and global angles in the one view that is united by entanglement~contextuality as its "central axis of weirdness".

I was thinking of more recent Bell-Wigner experiments, instead of just testing Bell inequalities. Because the difference is that assuming non-locality might not be enough to get rid of the lack of an objective world where all observers can reconcile their recorded facts. — Count Timothy von Icarus

Yep. But to me, this risks doing the usual thing of stumbling across the fact of a dichotomy that stands for an irreducibly triadic relation and then blindly believing it must be reducible to one or other of it two poles in a monistic fashion.

If find we have a binary choice like entanglement vs contextuality, then pick one as foundational. Do we save realism by throwing locality overboard, or by throwing out - in some arguments - the freewill of observers?

My metaphysics would say that such a dichotomy instead indicates a local~global pairs of limits. So locality almost goes overboard, but not quite. Observers go overboard, but not quite. Each would be the extreme case that could never quite be reached.

And this seems the way it works out - nonlocality as the most local view, and contextuality as the most global view, of the weirdness that makes QM non-classical.

...to complete the thought, classical realism is the place we want to get back to as the balance of what gets broken.

So QM is "weird" as it breaks realism. And folk then take one or either path and extrapolate the weirdness to infinity.

That gives rise to the different interpretational extremes. You have the Copenhagenism that offers no stopping point until it arrives at the consciousness and freewill of the human observer.

Or you head in the other no-collapse direction and have the endlessly bifurcating many worlds multiverse.

Each seems the correct interpretation - compatible with the maths. But that is because the maths doesn't contain a cut-off. Only a quantum gravity theory that absorbs all three Planck constants - the irreducible triad of c, G and h - could introduce such a cut-off to physics. And so nothing formally seems to resist the galloping off towards the infinite horizons of metaphysical irreality in one or other of its available directions.

The way to avoid the pathological metaphysics is to realise what is going on. Classical reality is emergent from the reciprocality of a pair of local~global limits. The weirdness of one is going to cancel out the weirdness of the other, at the end of the day.

Which is where we get to with thermal decoherence as a general framework uniting QM vagueness and classical crispness, or counterfactual definiteness.

And biosemiosis becomes the icing on the cake. It draws the further natural line across reality that is the epistemic cut between organisms and their environments. It shows how the Cosmos already decoheres itself, and how what human observers do is add a new level of machinery to the situation where this decoherence can be experimentally manipulated and even exploited for new technological purposes.

So at the Copenhagen end of the interpretive spectrum, you get rid of the conscious observer issue entirely. It can be left at the door of the biosemiotic epistemic cut.

And at the multiverse end of the interpretive spectrum, you can likewise rule out MWI. Decoherence says collapse is real enough due to thermal scale.

Copenhagenism is a claim about limits being taken - contextualised events becoming collapsed to a-contextual numbers. But then that Copenhagenism is just a human story. The physical reality it is based on is the nanoscale of warm water - the quasi-classical transition zone in which quantum coherence is becoming classical decoherence. Strong entanglement is giving way to strong contextuality.

Your exposition for my benefit (merci beaucoup) reminds me of the principle of the uniformity of nature, a simple statement of which is that sugar tastes sweet in Paris or in Tokyo, in 2022 or in 1927! Would you say that I, at the very least, now possess a rudimentary understanding of Noether symmetry?

Sounds like you’re done with the easy globally continuous stuff and are raring to go with the local discrete stuff. Bring on gauge symmetry and how it generates particle physics. :grin:

Sounds like you’re done with the easy globally continuous stuff and are raring to go with the local discrete stuff. Bring on gauge symmetry and how it generates particle physics. :grin: — apokrisis

Ne quid nimis. Gracias señor!

Aristotle's time puzzle:

The past doesn't exist, it's gone; the future too doesn't exist, it is yet to come; the now is an instant, it is nothing! Existence is an activity, and like all activities, requires a non-zero length of time. How can anything exist?

How can anything exist? — Agent Smith

How can everything not?

How can everything not? — 180 Proof

Did you follow the argument? It uses Aristotle's views on time, the conclusion is mine though. :smile:

Aristotle was talking out of his ass and the conclusion derived from his ass makes as much sense as a turd on a plate. :smirk:

The past is never dead. It's not even past." — William Faulkner

As I mentioned earlier in this thread, I haven't really dug into symmetry. I've been exposed to the concept in physics books, history of science, etc. but haven't really grappled with the mathematics. That said, if I recall my lectures and texts correctly, you are able to "rotate" and "flip" these matrices (e.g., Pauli matrices for spin). So, you get lines like: "An interesting property of spin 1⁄2 particles is that they must be rotated by an angle of 4π in order to return to their original configuration."

I was always told that this area of physics had some of the most obstruse, difficult mathematics in the whole field and so have always been scared away from actually working through them. I'll stick to Feynman diagrams, causal diagrams, and differential equations thank you very much :groan: .

I thought I was pretty good at math and just had gaps from going to a terrible, collapsing school system growing up because I learned a good deal of complex statistical methods and taught myself to code in several languages. Then I tried to jump into this Teaching Company course on linear algebra, without all the prerequisites (their calculus classes are accessible), and realized there are some things I'll probably just never get and have to accept on good faith.

Aristotle was talking out of his ass and the conclusion derived from his ass makes as much sense as a turd on a plate. — 180 Proof

:lol: Hell, even Aristotle made mistakes!

Any such assertions are clearly due to a lack of understanding of The Philosopher on the part of the student. Haven't you read your scholastic texts!?

That'll be 20 Hail Marys and a five-day bread fast.

The past doesn't exist, it's gone; the future too doesn't exist, it is yet to come; the now is an instant, it is nothing! Existence is an activity, and like all activities, requires a non-zero length of time. How can anything exist? — Agent Smith

It’s all relative.