[geordief]

Is there any other way we can reproduce  that effect?

And what is "long enough"?

Two dots? Three?

I think I can give a tentative answer to your question, which is, it depends.
In experiments that demonstrate interference, it's nice to see a pattern that we can say is definitely there.
But in quantum computers, the wavefunctions of two particles can be in superposition, such that it's a form of constructive or destructive interference.
We arrange for this to happen that way, and so it must have a nonzero probability of occuring in that case.

So,is that to say that when we have just two.(or even one?) dots on the screen they can be graphically represented as an interference  pattern even though our own optical system (the eyes and the brain) do not process it  that way?

Eventually, with enough dots we do see the underlying interference pattern "with our own eyes "?

Would that be a standard interpretation?

Hopefully  not to cloud  the issue(an ironic analogy,perhaps)  ,but if optical measuring instrumentation  was sufficiently sensitive  might it ,in theory pick up  directly  the interference pattern from one ,two or three dots?

[Eternal Student]

Hi.

   I seem to have a bit of time and have tried to read some of the later posts.   I thought I'd start with some questions.

the electromagnetic field is a connection in a fiber bundle over spacetime.

    I'm willing to put my hand up and say that I don't understand that sentence.  I'm no expert and not too worried about looking ignorant - it was ambiguous and confusing to me.
    The main problem is understanding what is meant by a "connection" and the possibility that "a fibre bundle over spacetime" meant the usual one associated with general relativity.   A "connection" would then be understood as an "affine connection".  However, you probably didn't mean that but just used the word "connection" to imply some relationship or link between elements of a fibre bundle that just has 4-D spacetime as a base space.
    Overall, I'm not really sure what that sentence meant.   If the sentence was important then it needs some references or further development.

 The next sentence is equally uninformative for me:

A photon is representative of the field in that it has the same dimensions as the field.

    What dimensions does the field have?   Do you mean length, time, mass  - those sorts of dimensions    OR   the dimension of some linear space (the number of vectors in a basis etc.)     OR    a measurement of just space occupied ( 3 cm x 10 cm x 12 cm )     OR  something else?    i.d.k.

Best Wishes.

[Eternal Student]

Hi.

    Your discussion of LRC circuits and analogies is interesting and many textbooks will exhibit that analogy.   Indeed, it is commonly said that as a student of physics all you will do is study the harmonic oscillator in ever increasing levels of complexity and detail.   They are ubiquitous across most of physics.
    So rather than just go along with that I thought it would be worth taking a moment to consider a different viewpoint:   Perhaps it is mainly that human beings go looking for such simple relationships.

   

An electromagnetic force is the analog of a Newtonian mechanical force; there's a symmetry in the equations of motion for an LRC circuit and a simple pendulum.

    There is some analogy between how a LRC circuit behaves and how a pendulum or harmonic oscillator behaves but you've been very human in finding this analogy.   You've ignored the thousands of electrical circuits and situations where Current was not like velocity and/or there wasn't anything like mass or inductance.   Instead you've identified a situation where analogies can be made, assumed that would be important and run with it.    That's ok in that sometimes these analogies and ideas will lead somewhere and turn out to be very useful in very general situations.   However, sometimes they remain just happy co-incidences - an analogy that existed for that one situation only.
    Given enough time human beings will identify ways in which any two things are similar or analogous.   The way my dog eats spaghetti is going to be analogous to the way the moon orbits the earth,  I just need to find some quantities I can calculate or measure which will show a suitable correspondence (and ignore all the things I could measure that don't).   For example, the square of the orbital time period is proportional to the cube of the semi-major axis.   Given enough time and effort we will find something about the spaghetti eating that will have the same correspondence.

This only tells you that Newtonian momentum has an electromagnetic equivalent, which is inductance multiplied by the current.

     Does it?   It's an analogy you can make for an LRC circuit but not necessarily for other electrical circuits or for all things involving electromagnetism.
    For example, an electromagnetic field does contain a perfectly conventional Newtonian momentum, it has to be there and it has nothing to do with a current being observed anywhere.   If you considered current as a quantity analogous to a velocity then this means there can be a non-zero momentum where there is zero velocity  (i.e. the analogy seems to break down).
When you started this thread you ( @varsigma ) were interested in a Feynman lecture, so you might like section 27-6 of the Feynman lecture documented here:  https://www.feynmanlectures.caltech.edu/II_27.html
In that lecture it discusses precisely how much momentum is contained in the E and B fields.   In any region with non-zero E and B fields, there is a  momentum density  g  =  (1/c² ) .  E  x B .   That's all we need - just a non-zero E and B field, no current has to flow from somewhere to anywhere for momentum to exist in the electromagnetic field.   (Minor note:  It's commonly said that the momentum is "in the field" but I would prefer to say only that it is in the space permeated by those fields -  i.e. avoid suggesting that the fields could hold it, just that momentum is in the space somehow when the fields are in that space).

   It is possible to analyse and think about the LRC circuit in terms of how it changes the momentum of the space around the inductor and between the capacitor plates.  As you know, the E and B fields are changing in those components, so the momentum in the space around those components is changing (see the Feynman lecture above).  The pendulum had a mechanical component (a mass or bob) which shows a change in momentum as it swings back and forth, the LRC circuit has space (or the fields within that space) as the equivalent component showing a change in momentum.   This might take a moment to think about, so I'll say it again:   When the E and B fields change, the momentum in the space is changing,  so it is exactly like the electrical components are applying a force onto space - pulling and pushing it.
   Now, as mentioned earlier, we human beings will eventually identify a similarity or analogy between any two phenomena if we spend enough time and ignore all the things that just didn't show the correspondence we wanted.   I wouldn't exclude my own suggestion from that.   I think it's interesting to imagine an LRC circuit as being exactly like a pendulum - it just has space as the component swinging back and forth instead of the pendulum bob - but I wouldn't encourage anyone to run too far with that idea. It works nicely for this situation.

Best Wishes.

[alancalverd]

if optical measuring instrumentation  was sufficiently sensitive  might it ,in theory pick up  directly  the interference pattern from one ,two or three dots?

Easier with x-rays, but the answer is yes, we can detect individual photons, and they do indeed arrive at random with the spatial probability distribution as calculated from the continuum-wave analysis.

[alancalverd]

it is commonly said that as a student of physics all you will do is study the harmonic oscillator in ever increasing levels of complexity and detail.

Spoken like a mathematician!

Never mind the spherical cow in a vacuum, I once said "physics is a trivial particularisation of mathematics" and then spent 10 years calculating, designing and building just one particularisation of physics, based on a thermal diffusion equation, not a harmonic oscillator!

[varsigma (OP)]

Not complex. They are both real particles with charge and mass.

In field theories particles have complex probability amplitudes.

[varsigma (OP)]

When you started this thread you ( @varsigma ) were interested in a Feynman lecture, so you might like section 27-6 of the Feynman lecture documented here:  https://www.feynmanlectures.caltech.edu/II_27.html
In that lecture it discusses precisely how much momentum is contained in the E and B fields.   In any region with non-zero E and B fields, there is a  momentum density  g  =  (1/c2 ) .  E  x B .   That's all we need - just a non-zero E and B field, no current has to flow from somewhere to anywhere for momentum to exist in the electromagnetic field.   (Minor note:  It's commonly said that the momentum is "in the field" but I would prefer to say only that it is in the space permeated by those fields -  i.e. avoid suggesting that the fields could hold it, just that momentum is in the space somehow when the fields are in that space).

Yes, there you describe the free field; in an LRC circuit the evolution of the field is constrained. It's an interesting exercise finding the correspondences; although you can say there's an electromagnetic momentum which is "equivalent" to Newtonian momentum in the context of field oscillations, it's far from a unifying principle. But it's still interesting that, in order for oscillations to appear certain constraints are needed. Certain physical things need to be fixed in place.

[varsigma (OP)]

The main problem is understanding what is meant by a "connection" and the possibility that "a fibre bundle over spacetime" meant the usual one associated with general relativity.   A "connection" would then be understood as an "affine connection".  However, you probably didn't mean that but just used the word "connection" to imply some relationship or link between elements of a fibre bundle that just has 4-D spacetime as a base space.

The connection and its curvature is the field. Sean Carrol explains what that means to some extent in one of his online lectures. (it's number 15 Gauge Theory)

There's a good article in an old issue of SciAm, by Bernstein and Phillips, that discusses the geometry and the topology of a gauge field. The article starts out explaining what a fibre bundle is and how to construct one for the sphere.
Then the geometry of a gauge field is seen in neutron spin precession in a magnetic field, and the topology is seen in the Aharonov-Bohm effect. Both experiments "exploit" interference.
The first experiment is much simpler in terms of the required analysis, as the the path lifting rule is pretty obvious. Which suggests the geometry of the gauge field is less complex than its topology.

[varsigma (OP)]

What dimensions does the field have?   Do you mean length, time, mass  - those sorts of dimensions    OR   the dimension of some linear space (the number of vectors in a basis etc.)     OR    a measurement of just space occupied ( 3 cm x 10 cm x 12 cm )     OR  something else?    i.d.k.

The dimensions are the electric and magnetic fields--photons have two field dimensions; when they propagate in the vacuum they 'occupy' all three spatial dimensions. The dimensions of spacetime aren't the dimensions of the electromagnetic field.

[varsigma (OP)]

From nLab

1. Idea

The Aharonov-Bohm effect is a configuration of the electromagnetic field which has vanishing electric/magnetic field strength (vanishing Faraday tensor F=0) but is nevertheless non-trivial, in that the vector potential A is non-trivial. Since the vector potential affects the quantum mechanical phase on the wavefunction of electrons moving in an electromagnetic field, in such a configuration classical physics sees no effect, but the phase of quantum particles, which may be observed as a interference pattern on some screen, does.

More technically, a configuration of the electromagnetic field is generally given by a circle-principal connection and an Aharonov-Bohm configuration is one coming from a flat connection, whose curvature/field strength hence vanishes, but which is itself globally non-trivial. This is only possible on spaces (spacetimes) which have a non-trivial fundamental group, hence for instance it doesn't happen on Minkowski spacetime.

In practice one imagines an idealized electric current-carrying solenoid in Euclidean space. Away from the solenoid itself the magnetic field produced by it gives such a configuration.

--https://ncatlab.org/nlab/show/Aharonov-Bohm+effect

Note how it says "in practice". That means in experiments, one has to build a solenoidal coil magnet. It took a while for the first experiment to be realised because of the dimensions of the required device. Like, really small.

[alancalverd]

Not complex. They are both real particles with charge and mass.

In field theories particles have complex probability amplitudes.

But the probability is A² so real particles have real distributions.

[varsigma (OP)]

But the probability is A2 so real particles have real distributions.

Yeah. I don't know that I can explain why the square of a complex amplitude is a mass term (in a Lagrangian). Sean Carroll might be able to, or we might be able to discuss what he's talking about.

[varsigma (OP)]

How about, when we have a classical wave, the square of the amplitude is proportional to the intensity.

Intensity is a general kind of thing about waves and wave motion.
The intensity of a beam of particles and the square of an amplitude are meant to mean something.
Interference is heuristically wavefunction overlap, in a sense that some information is thereby encoded 'in' the superposition.

[alancalverd]

The meaning is simple: the square of the amplitude of a wavefunction at any point in space is the probability of finding the particle at that point. If you have lots of particles, say valence electrons in a covalent material, the square of their wave function is known as the electron density map (physics) or group orbital (chemistry) , which determines the physical and chemical properties of that material.

The intensity of a beam of particles is the number of particles passing through unit area in unit time. Spatially, it's going to look like a wave function of some sort if you wait long enough.

And once again you are in danger of confusing model with reality. We can predict  an interference pattern by superposing wave functions, but where an individual photon/electron/buckyball goes within that distribution is entirely random. It's no big deal: you can predict the outcome distribution of an infinite number of dice throws or even coin tosses very accurately, but each throw is unpredictable. Nothing is "coded in a superposition" to determine what happens next.

[varsigma (OP)]

And once again you are in danger of confusing model with reality. We can predict  an interference pattern by superposing wave functions, but where an individual photon/electron/buckyball goes within that distribution is entirely random. It's no big deal: you can predict the outcome distribution of an infinite number of dice throws or even coin tosses very accurately, but each throw is unpredictable. Nothing is "coded in a superposition" to determine what happens next.

I think the best model of an interference pattern is that it's a way to verify that your quantum computer is
"up and running". It tells you that interference is a kind of universal test, it's independent of the type of particle.
Although the dimensions, the size of the two slits and how far apart they are, are engineering requirements.

Unless there is a way to further encode known information into the beam, the interference pattern is the result of a test run, it validates . . . quantum behaviour(?)

[varsigma (OP)]

I'm going to crawl out onto a limb here, at least briefly, and try to characterize what it all means, so far. As in what's the big idea?

Feynman's big idea, or one of them, was that there is plenty of room to store and manipulate information, at the quantum field level.

Store and manipulate, as in, copy, transmit, erase, and so on. Bennett and Landauer explored the fundamental limits of computation; that is, what does it mean in terms of the energy required? What does the existence of energy mean, anyway?

My guess is it has to have a physical basis, there are forms of energy. We accept that in physics, there are two fundamental forms in any generalized "system of particles".

We can't really compare the oscillations of a simple pendulum with the free oscillations of an LRC circuit, because the fundamental gauge groups are, well, fundamentally different in ways we are still figuring out. With gauge theories there is a common approach to answering some of the questions. After all that's what physics is really, asking questions and trying to understand the answers. Philosophy tries to decide which questions are meaningful; Physics just does some experiments.

A footnote about one of the "big ideas", namely fibre bundles and quantum fields:

In short: all global structure in field theory is controled by fiber bundles, and all the more the more the field theory is quantum and gauge. The only reason why this can be ignored to some extent is because field theory is a complex subject and maybe the majority of discussions about it concerns really only a small little perturbative local aspect of it. But this is not the reality. The QCD vacuum that we inhabit is filled with a sea of non-trivial bundles and the whole quantum structure of the laws of nature are bundle-theoretic  . . .

--https://ncatlab.org/nlab/show/fiber+bundles+in+physics

[alancalverd]

What does the existence of energy mean, anyway?

Physics is the business of constructing mathematical models of what happens (or doesn't happen - the branch of physics known as civil engineering). Within those models, energy is a conserved quantity in classical physics, and remains conserved as mass-energy in relativistic physics. Nothing more or less.

[alancalverd]

Philosophy tries to decide which questions are meaningful; Physics just does some experiments.

No. Philosophers assert which questions are meaningful, then question the meaning of meaningful until they disappear through their own anal sphincters. Physics is about discovering and predicting useful and interesting stuff.

As an experimental physicist I'm quite used to the banter of engineers ("bloody scientist...") and theoretical scientists ("oily-fingered engineer...") but nobody has ever insulted me with the title of philosopher!

[varsigma (OP)]

That article in SciAm is available online here--https://www.researchgate.net/publication/250802309_Fiber_Bundles_and_Quantum_Theory

Unfortunately it isn't a good copy, if you're subscribed to the magazine online you can see a better copy.

Anyway, despite the importance of fiber bundles in gauge theories, the rather long article is a bit impenetrable. There is almost no mathematics, there are diagrams and explanations so it's up to you to map any ideas you have about topology and geometry into the mix.

Total spaces and base spaces are presented early as diagrams. A fiber is just a set of points defined over each point in the base space. They introduce the idea of combing the hair on a sphere, which is mathematically equivalent to defining a basis vector at each point so that all the basis vectors are parallel.

You can do this for a part or patch of the sphere but not the whole of the total space. The patch can be as large as a hemisphere though; the base is then the 2d projection of the hemisphere. The fiber over each point in the base is a line with a point on it identifying the angle of each basis vector. So far the ability to parallel transport along an arbitrary closed curve in the total space is not defined, something extra is needed.

This extra is the map of the gradient of each tangent space at each point on the hemisphere, relative to the flat tangent space at the "north pole". This map has to preserve the holonomy. Movement in the base is mapped to movement in the total space along the 'generalized' tangent planes, so moving from fiber to fiber is a smooth, or flat, path. The path is along the flat connection on the bundle.

Having the gradient map in the fiber bundle means having a way to lift a path from the base space to the total space. Without it all you have is basis vectors in the projection.

On a more lightfooted er, footnote, I tried discussing Bernstein and Phillip's ideas with ChatGPT. It did eventually touch on the fibrations and foliatons of the hemisphere. To foliate the hemisphere, define a set of latitude lines (circles), each leaf of the foliation has a cylinder of fibres. Each cylinder or section of the circle foliation has the same generalised gradient in the fibration. Check out how this gradient is re-configured in the diagrams on page 18 of their article.

Yeah, otherwise the bot was repetitive and not especially helpful, although it might have helped me clear up a couple of things.

[Bored chemist]

nobody has ever insulted me with the title of philosopher!

I had assumed you were a PhD.