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**Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?**

« **on:**06/03/2024 20:41:21 »

Hi.

I'm not entirely sure that @varsigma really needed to consider QFT for what was being discussed. The Aharonov-Bohm effect is explainable or can be described with just a non-relativistic, plain vanilla version of quantum mechanics.

If you look at the place where the wave function is being measured by the... disc shaped speedometer looking thing with the big arrow spinning around.... you'll see the magnitude of this wave function never changes. However, the portion of it that is in the imaginary axis (say running in the up-down direction of the page) compared to the portion of it that is the real axis (say running in the direction that looks like it's coming out of the page or going into the page) does change. Sometimes all of the magnitude of the value is in the imaginary axis, sometimes all of it is alog the real axis. Overall the complex valued wave function seems to spin or rotate when you view the bare complex value of the wave function. The modulus, or overall size of the wave function is what is usually important, the modulus squared is proportional to the probability of finding the particle at this place, so in this diagram that never changes and the particle is as likely to be found at place X now as it will be likely to be found there in 20 seconds or tomorrow. The fact that the complex value of the wave function is rotating in the complex plane doesn't matter in the slightest. Indeed for almost all purposes, the phase or exact way in which the magnitude of the wave function is spread between the real and imaginary axis doesn't affect any physical observation you could make. All we usually care about is the overall magnitude of the value and the complex phase of the wave function was, historically, thought to be essentially irrelevant and something that was only there as an artifiact of the mathematics.

However, the individual oscillations in the real and imaginary axis WILL matter if you brought two quantum mechanical wave functions together, i.e. made one wave packet approach another wave packet and merge. On merging, you just add the wave functions much as you would for any other sort of wave mechanics. However, it is not the modulus or magnitude of the two wave packets that you would add, you add the direct or bare complex value of the two wave functions. If the two wave packets had their rotations in the complex plane synchronised or "in phase", then the real part adds to the real part, imaginary part will add to the imaginary part, all of it gets made bigger and you end up with one really big valued wave function that still spins or rotates around the Complex plane as before (it's just got a bigger magnitude or longer spinning arrow on the speedometer).

On the other hand if the two wave packets that merge had their rotations in the complex plane precisely 180 degrees, or one half a cycle, out of phase, then when you add them you end up with the oscillation tending to cancel out: Whenever one was going big in the positive direction along the real axis, the other is going big in the negative direction of the real axis etc. When you add the two wave packets together, you get only destructive interference and 0 oscillation all together. Hence 0 probability of finding the particle(s) in this place where the wave packets merged but were so badly out of phase.

If the two split beams remained in phase, then you get a certain pattern on the screen. If the two beams went out of phase then you'll get a different pattern on the screen due to all the destructive interference as discussed just above.

Here's a diagram (the magnetic field will be explained next):

Aharanov Bohm.jpg (64.35 kB . 978x663 - viewed 205 times)

The thinking of the time was that since there wasn't any magnetic field outside the coil and in the space where the electrons were being passed, the behaviour of the split beams should be much the same regardless of which side of the coil they were passed around. In particular, the complex phase of their wave functions were in phase before they were split and they ought to just stay in phase. So, on being recombined at the other side of the coil, you'll get only the usual pattern on the screen.

As you may have guessed... you don't.... the complex phase of the beams does seem to be different depending on which side of the coil they were passed around and you end up with a pattern on the screen that suggests destructive interference has occurred when the two beams were recombined.

For quite a while this was quite a serious issue: There was no magnetic field where the electrons were travelling, what on earth was affecting one beam differently to the other? Turn off the coil and everything is back as it should be... good old pattern 1 on the screen when you recombine the electrons. Somehow the magnetic field that existed only inside the coil seemed to be influencing the electrons that were only ever outside of the coil. Sure, if some charged particle was moving and it went through a magentic field you'd expect to see some effect but you need to realise that the magnetic field is precisely 0 valued outside of the coil, there is no magnetic force of any sort to be felt where the electrons were travelling.

Let's use gravity as an example because that's usually easier for us to appreciate. Let's assume I can put two equal sized planets next to planet earth, one on the left and one on the right, equal distance from the earth, everything nice and symmetric. You would have a new gravitational force pulling to the left but also a new one pulling right, so overall no net force. You would carry on with your life as usual and expect to notice no change in anything whether these pair of planets were there or not. What the Aharanov-Bohm experiment suggests is that... actually there may be a difference. Although there is no net gravitational force acting on you, you have certainly changed your gravitational potential. There are loads of things for which we still think "potential" is fairly arbitrary and all we need is some arbitrary reference point we ca call 0 potential. In electronics, the earth line is reference 0 volts and everything works provided the +5 Volt line is at a potential 5 Volts higher. We don't try to guess what the actual potential of the earth line was and we assume it doesn't matter. If we took the box of electronics to another planet, drove an earth pylon into the ground there and used their "earth" as our electronic earth, we would still expect the box of electronics to work.

The Aharanov-Bohm experiment suggests that for magnetic potential (and possibly other potentials) things are not as we once thought. It made a difference when the coil was turned on because the magnetic potential of the space the electrons were in had changed, even if there was actually 0 magnetic field (no magnetic force if you prefer) in the space. It's maybe not that interesting these days where we flippantly add another field whenever required - but back in the day it was important to imagine that magnetism (and possibly every other force) may demand that two fields existed throughout all of space. There is a magnetic field, which can exert a force on moving particles but there is also a magnetic potential field. More importantly, the potential field may actually have physical effects rather than just being a mathematical tool. This alone is important, it means that maybe knowing that the +5V line in your box of electronics is 5 volts higher than the earth line is NOT enough, it may actually matter what the absolute potential at the earth line really is.

Unlike the electric or gravitational fields, the magnetic potential field cannot be described just as a scalar field, we require a vector valued field for magnetic potential (i.e. it has both a magnitude and also a direction). This was another reason why the notion of a magnetic potential was contested or considered to be just a mathematical artifact. An explanation for the Aharanhov-Bohm experiment is not attainable if the magnetic potential is just a scalar field. There is a perfect symmetry around the coil in the experiment , an electron passing on an equivalent path on either side of the coil should have been put into all the same potentials for all the same amounts of time, so even if it did affect the complex phase of their wave functions, they should be affected equally. On re-combining the beams at the other side of the coil, you'd still expect to have both beams remain synchronised or in phase - and get a perfectly standard pattern for constructive superposition being shown on the screen.

If we allow the magnetic potential to be vector valued instead of just a scalar, then we can break this symmetry. The potential field has direction in addition to numerical value and there can be a net circulation of the vector field around the coil: We imagine that electrons passing around one side can be travelling against the direction of the magnetic vector field, the ones travelling around the other side would be travelling in the same direction as the magentic potential vector.

That's probably all I know about the Aharanov-Bohm effect, I hope it's of some use to you.

Best Wishes.

You made some statements that I didn't understand so I asked for some clarity.I didn't originate the posts but this is what I think was meant:

What is a "matter field"?In QFT (Quantum Field Theory) it is any field for which oscillations in that field or energy in that field could be considered as a particle of matter. Usually we're thinking of fermions - typical matter particles like an electron, a quark or a composite particle like a proton.

I'm not entirely sure that @varsigma really needed to consider QFT for what was being discussed. The Aharonov-Bohm effect is explainable or can be described with just a non-relativistic, plain vanilla version of quantum mechanics.

What is the phase change you are referring to?In the standard experiment, it would be a shift in what is called the complex phase of the wave function. Basic idea: Wave functions have a value that is a complex number, they have real and imaginary parts. So a wave function might be something that looks like this:

If you look at the place where the wave function is being measured by the... disc shaped speedometer looking thing with the big arrow spinning around.... you'll see the magnitude of this wave function never changes. However, the portion of it that is in the imaginary axis (say running in the up-down direction of the page) compared to the portion of it that is the real axis (say running in the direction that looks like it's coming out of the page or going into the page) does change. Sometimes all of the magnitude of the value is in the imaginary axis, sometimes all of it is alog the real axis. Overall the complex valued wave function seems to spin or rotate when you view the bare complex value of the wave function. The modulus, or overall size of the wave function is what is usually important, the modulus squared is proportional to the probability of finding the particle at this place, so in this diagram that never changes and the particle is as likely to be found at place X now as it will be likely to be found there in 20 seconds or tomorrow. The fact that the complex value of the wave function is rotating in the complex plane doesn't matter in the slightest. Indeed for almost all purposes, the phase or exact way in which the magnitude of the wave function is spread between the real and imaginary axis doesn't affect any physical observation you could make. All we usually care about is the overall magnitude of the value and the complex phase of the wave function was, historically, thought to be essentially irrelevant and something that was only there as an artifiact of the mathematics.

However, the individual oscillations in the real and imaginary axis WILL matter if you brought two quantum mechanical wave functions together, i.e. made one wave packet approach another wave packet and merge. On merging, you just add the wave functions much as you would for any other sort of wave mechanics. However, it is not the modulus or magnitude of the two wave packets that you would add, you add the direct or bare complex value of the two wave functions. If the two wave packets had their rotations in the complex plane synchronised or "in phase", then the real part adds to the real part, imaginary part will add to the imaginary part, all of it gets made bigger and you end up with one really big valued wave function that still spins or rotates around the Complex plane as before (it's just got a bigger magnitude or longer spinning arrow on the speedometer).

On the other hand if the two wave packets that merge had their rotations in the complex plane precisely 180 degrees, or one half a cycle, out of phase, then when you add them you end up with the oscillation tending to cancel out: Whenever one was going big in the positive direction along the real axis, the other is going big in the negative direction of the real axis etc. When you add the two wave packets together, you get only destructive interference and 0 oscillation all together. Hence 0 probability of finding the particle(s) in this place where the wave packets merged but were so badly out of phase.

What electron beam? Where did that come from?What the Aharanov-Bohm experiment did was to take a beam of electrons and split it into two. Pass one of those groups of electrons around the left side of a barrier and the other electrons around the right side of the barrier. While travelling, the complex phase of any beam doesn't matter at all. Now re-combine the two beams after they get around the barrier and let the beam hit a screen that will glow on contact with an electron.

If the two split beams remained in phase, then you get a certain pattern on the screen. If the two beams went out of phase then you'll get a different pattern on the screen due to all the destructive interference as discussed just above.

Here's a diagram (the magnetic field will be explained next):

Aharanov Bohm.jpg (64.35 kB . 978x663 - viewed 205 times)

How do you shield a magnetic potential?Well you don't. What you can shield is an ordinary magnetic field (the thing that should exert force on a moving charge). The potential, it seems, you can do nothing about. What was done in the Aharabov-Bohm experiment was to use a special sort of barrier that they sent electrons around. The simplest one to use is just a very long coil. As you may know, inside a coil of wire carrying an electric current, there is a very strong magnetic field. Outside of the coil the magnetic field is really small and effectively negligble at most distances. The magnetic field wasn't really "shielded", although that could have been done, but it was just easier to use a coil and assume the magnetic field exists only inside of it and not around the outside of it.

The thinking of the time was that since there wasn't any magnetic field outside the coil and in the space where the electrons were being passed, the behaviour of the split beams should be much the same regardless of which side of the coil they were passed around. In particular, the complex phase of their wave functions were in phase before they were split and they ought to just stay in phase. So, on being recombined at the other side of the coil, you'll get only the usual pattern on the screen.

As you may have guessed... you don't.... the complex phase of the beams does seem to be different depending on which side of the coil they were passed around and you end up with a pattern on the screen that suggests destructive interference has occurred when the two beams were recombined.

For quite a while this was quite a serious issue: There was no magnetic field where the electrons were travelling, what on earth was affecting one beam differently to the other? Turn off the coil and everything is back as it should be... good old pattern 1 on the screen when you recombine the electrons. Somehow the magnetic field that existed only inside the coil seemed to be influencing the electrons that were only ever outside of the coil. Sure, if some charged particle was moving and it went through a magentic field you'd expect to see some effect but you need to realise that the magnetic field is precisely 0 valued outside of the coil, there is no magnetic force of any sort to be felt where the electrons were travelling.

Let's use gravity as an example because that's usually easier for us to appreciate. Let's assume I can put two equal sized planets next to planet earth, one on the left and one on the right, equal distance from the earth, everything nice and symmetric. You would have a new gravitational force pulling to the left but also a new one pulling right, so overall no net force. You would carry on with your life as usual and expect to notice no change in anything whether these pair of planets were there or not. What the Aharanov-Bohm experiment suggests is that... actually there may be a difference. Although there is no net gravitational force acting on you, you have certainly changed your gravitational potential. There are loads of things for which we still think "potential" is fairly arbitrary and all we need is some arbitrary reference point we ca call 0 potential. In electronics, the earth line is reference 0 volts and everything works provided the +5 Volt line is at a potential 5 Volts higher. We don't try to guess what the actual potential of the earth line was and we assume it doesn't matter. If we took the box of electronics to another planet, drove an earth pylon into the ground there and used their "earth" as our electronic earth, we would still expect the box of electronics to work.

The Aharanov-Bohm experiment suggests that for magnetic potential (and possibly other potentials) things are not as we once thought. It made a difference when the coil was turned on because the magnetic potential of the space the electrons were in had changed, even if there was actually 0 magnetic field (no magnetic force if you prefer) in the space. It's maybe not that interesting these days where we flippantly add another field whenever required - but back in the day it was important to imagine that magnetism (and possibly every other force) may demand that two fields existed throughout all of space. There is a magnetic field, which can exert a force on moving particles but there is also a magnetic potential field. More importantly, the potential field may actually have physical effects rather than just being a mathematical tool. This alone is important, it means that maybe knowing that the +5V line in your box of electronics is 5 volts higher than the earth line is NOT enough, it may actually matter what the absolute potential at the earth line really is.

Unlike the electric or gravitational fields, the magnetic potential field cannot be described just as a scalar field, we require a vector valued field for magnetic potential (i.e. it has both a magnitude and also a direction). This was another reason why the notion of a magnetic potential was contested or considered to be just a mathematical artifact. An explanation for the Aharanhov-Bohm experiment is not attainable if the magnetic potential is just a scalar field. There is a perfect symmetry around the coil in the experiment , an electron passing on an equivalent path on either side of the coil should have been put into all the same potentials for all the same amounts of time, so even if it did affect the complex phase of their wave functions, they should be affected equally. On re-combining the beams at the other side of the coil, you'd still expect to have both beams remain synchronised or in phase - and get a perfectly standard pattern for constructive superposition being shown on the screen.

If we allow the magnetic potential to be vector valued instead of just a scalar, then we can break this symmetry. The potential field has direction in addition to numerical value and there can be a net circulation of the vector field around the coil: We imagine that electrons passing around one side can be travelling against the direction of the magnetic vector field, the ones travelling around the other side would be travelling in the same direction as the magentic potential vector.

That's probably all I know about the Aharanov-Bohm effect, I hope it's of some use to you.

Best Wishes.

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