Trying to understand gauge theories

Oh really? You have been lecturing us on gauge theory and yet you you don't understand these terms?

Rubbish - no group, Lie or otherwise, can possibly have a metric - how could it? Do you know the definition of a group? Of a metric?

Neither - this is nonsense

No, it is not. Measurement is the eigenvalue of an operator (a transformation) acting on a state vector

arfa brane, I tried you get you to engage in the mathematics of gauge theory, but you were not interested.
I shan't try again

 

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That's quite a leap you make there. But you have to, right? It's how your mind works.
You don't seem to have much grasp of actual physics either. I've seen little evidence of this in anything you've ever posted.

So it's not rubbish, is it? If no group has a metric then no Lie group does. That's quite logical, did you somehow skip past that because of your need to criticise and cavil?

You say neither the field of neutrons or electrons, nor the magnetic field is the gauge field. So what is the gauge field the authors refer to in the experiments, in the actual physics?
Is it the magnetic vector potential, and does that exist because there is a magnetic field?

When an electron creates a dot on a screen, that's the eigenvalue of an operator? The dot isn't there because of a transformation?
We should see mathematics, not dots?

Like I say, you seem to know very little about actual physics. I can only assume you've never been in a lab.

You're just confused; you claim something is nonsense but offer no proof. It isn't helpful, it isn't enlightening. It's just sad.

Just to underline how sad; this is what the authors (both with PhDs, and decades of research) say in that article (which I've read through many times):
"The quantum theory of magnetism provides an example of the phase-shifting properties of a gauge field. The gauge field in question is the magnetic vector potential, and it determines how electrons interact with a magnetic field."

You however, have little understanding beyond mathematics. A magnet isn't a mathematical equation, now is it?

 

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Actually what you say is demonstrably false. I assume you've heard of a permutation metric? So a permutation group is equipped with a metric, namely a distance between permutations.

An example is the symmetric group on three letters; so for instance there is a metric distance between abc and bac; indeed there is clearly a metric distance between any two permutations of abc.

So I see your rubbish and raise you some clearly observable facts. You dumbass.

 

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Changes of complex phase are an example of a gauge symmetry, but as I understand it, any kind of continuously parametrized symmetry in general can count as one.

 

Just a phase
The Aharonov–Bohm effect arises in quantum mechanics because the addition of a potential results in the introduction of a phase in the wavefunction of the electron. Normally, this phase has no effect on the observed behaviour of the electron because the measurement of a property of the electron (such as its position) determines the amplitude of the wavefunction, not its phase.

However, this phase can be detected by measuring the quantum mechanical interference between electrons that have taken two different paths from a source to a detector. If these paths travel through regions with different local values of gauge potential, then a difference in phase will alter the interference pattern measured.

This effect was proposed in 1959 by Yakir Aharonov and David Bohm and confirmed by an experiment done by Robert Chambers in 1960. Chambers sent electrons on different paths that passed next to a very long solenoid. The magnetic field outside such a solenoid is negligible (and had little effect on the electron phase) but the vector potential outside a solenoid is significant and varies in space. As a result, electrons taking the different paths around the solenoid acquire different phases.

Rich in physics
This and subsequent observations of the effect involve “Abelian” systems, in which the physics plays out in the same way when time is run forwards and backwards. In 1975 Tai-Tsun Wu and Chen-Ning Yang conceived of the non-Abelian Aharonov–Bohm effect in which the gauge fields appear differently when time runs forwards or backwards. While expected to be rich in physics, the non-Abelian version of the effect has proved very difficult to achieve.

Now, Soljacic’s team has succeeded by creating two different types of non-Abelian gauge field using fibre-optic systems – with classical light waves taking the place of the electron wavefunction. They induced the first of these fields by passing light through a specialized crystal in a strong external magnetic field. The second non-Abelian gauge field was created by modulating the light using time-varying electrical signals.

--https://physicsworld.com/a/non-abelian-aharonov-bohm-experiment-done-at-long-last/

 

Measurement isn't itself a transformation, like I say, it's a transformation because of an operator, operating on a state vector; there's an eigenvalue.
There's a dot, on a screen.

The screen is also a field (of material particles), the dot is local; measurement is local. It's also unitary, but not generally reversible. I don't know how QH thinks I got it wrong; perhaps it's just QH.

Perhaps he hasn't done any Information Theory, but I have.

Ahem: consider the question: What is the difference in information-content, between the appearance of a single dot or a small set of dots, and the appearance of interference fringes, in a 'recognizable' pattern?