Trying to understand gauge theories

[arfa brane]

Apparently a gauge field is really a phase-shift field.

In Yang-Mills theories there's a phase-shift 'operator' defined on particles from the Standard Model, that for instance, transform protons into neutrons and back.

But apart from that, we have fiber bundles, where a fiber can be a symmetry group, like SU(2). In an article by Bernstein and Phillips, the idea of a bundle of directions is used, such that each fibre is the interval [0,2π), but this only (obviously) makes sense if there is a reference direction. So then they present the example of a sphere (manifold), which can't have a reference direction assigned everywhere, but a hemisphere can, in fact the equator has a natural east/west direction on it.

Now, this fiber over each point on a hemisphere is what the authors call a "circle of directions", but it lies on a tangent plane so what they really mean is the tangent space of a hemisphere, where at each point one and only one plane is tangent.

Then they explain parallel transport around this space in terms of moving through the bundle of fibers over the half sphere,
which is called 'lifting a path' from the base space to the total space (the tangent space encoding 'directions' on the manifold). Then lifting a path and parallel transport are equivalent, but first the fiber bundle needs more structure, it needs an associated bundle of gradients over each point in the manifold.

So they say (I think).

 

[QuarkHead]

arfa brane: as always you are trying to run before you can crawl, let alone walk.

The mathematics of gauge theory are quite hard, and certainly beyond you from what I have seen. Fortunately it is possible to give a rough-and-ready idea about them.

So suppose a manifold. Assign to each point on your manifold a symmetry group - a Lie group. Then the particular choice of group element at each point is called a gauge. And if you have theory, say a field theory, defined at each point, that takes the same form whatever these choices are, your theory is called gauge invariant (in GR this is called general covariance).

Obviously there is a hell of a lot more to it than that - moreover rough and ready explanations cannot possibly be accurate, but I hope it helps a little

 

[arfa brane]

arfa brane: as always you are trying to run before you can crawl, let alone walk.

The mathematics of gauge theory are quite hard, and certainly beyond you from what I have seen. Fortunately it is possible to give a rough-and-ready idea about them.

. . . blah blah; blah etc.

So suppose a manifold.

Well, instead suppose I have a space I can assign a direction to at each point; oh wait, that's called a manifold because to do that I need coordinates.
Like on a hemisphere, I need a point on the equator that corresponds to zero degrees of longitude, and I need to decide where zero degrees of latitude is.

Assign to each point on your manifold a symmetry group - a Lie group.

Ok. let's make it the circle group, U(1). Then let's project the hemisphere and its coordinates onto a flat disc. Call the disc a base space and take the interval (0,2π) as the fiber over each point in the base; the collection of fibers looks like a solid cylinder; this is what the authors refer to as the bundle of directions over the hemisphere.

Now I can define parallel transport in terms of a path through the base; if I give the fiber bundle some extra structure by assigning a gradient to each fiber I can also define what the authors call a path-lifting rule.

This is all so they can explain the Aharonov-Bohm effect in terms of lifting a path; so readers get to understand the geometry of a fiber bundle and how this abstract structure exists around a shielded magnet. The authors don't really explain what the gauge is, except as a phase-shift resulting in an interference pattern.

p.s. if it's so hard, why did these two publish an article in Scientific American?; was it because hardly anyone would understand it? They thought what the hell, nobody is going to bother reading past the title?

 

[QuarkHead]

I tried tried to help you, arfa - after all it was that you that asked for help.

If it was not welcome, I am sorry.

 

[arfa brane]

I tried tried to help you, arfa - after all it was that you that asked for help.

Bullshit.

You tried to demean and condescend. You implied that I can't understand an article in Scientific American written for the lay public.
And obviously that's why these guys wrote it, so a bunch of retards could scratch their heads for decades.

 

[arfa brane]

Questions from the article:

So these two, Bernstein and Phillips actually have a profile in the mag, Bernstein, Herbert J., got his PhD at UCSD in 1967; Phillips, Anthony V., has a PhD from Princeton (1966).

But I guess their intent at posing questions in their article (which is extremely free of mathematical equations, the closest you get is some geometric diagrams, with coordinates labelled), is to give some context.

So they ask, by the third paragraph: What does it mean for a fiber bundle to have a connection?
And: How are the concepts of a connection and of a gauge field related?

What it means for a bundle to have a connection is that curvature is defined; the connection "is" the gauge field.
The second question suggests there are relations; relations can be interpreted as functional objects.

 

[arfa brane]

When the article was written it was feasible to perform an experiment with a beam of fermions (electrons) to demonstrate the reality of the AB effect.
The base space in the experiment is the projection onto a 2d annulus of the total space outside of the solenoid magnet (geometrically a truncated cone in 3d).

Remarkably, a more recent experiment also demonstrating the AB effect has been realized in a small superconducting annular ring with a shielded magnetic field whose geometry is precisely the base space from the 3d experiment.

Again from the Bernstein Phillips article: "Each of the quantum gauge fields can be understood as a connection in a fiber bundle, where the base is spacetime.
The fiber of the bundle is the set of internal symmetry transformations of particles that interact by means of the gauge field."
. . .
"The key . . . is the interpretation of the magnetic vector potential as a connection in a fiber bundle.
. . . The fiber over any point is the set of all possible phases of electrons at that point, and so the total space is made up of all possible phases of electrons at all points of three-dimensional space."

So yeah, a bundle connection is a gauge field, apparently.

 

[arfa brane]

Notice that it isn't important what these structures are called, except that an accepted nomenclature is recommended--it's less confusing, and if you want to submit a paper, you should use accepted terminology.
The important details are all about structure.

The article also compares the AB experiment with a spin-precession experiment. Spin precession/rotation in fermions has the structure of a point moving along the edge of a Mobius strip. Here the base space is a circle (no interior) and the total space is a circle wrapped once around itself, called the double cover of the base space. So a pair of points is the fiber over each point in the base, one point is "up" the other is "down" in terms of spin direction (here, apart from the direction of rotation, spin direction is an arbitrary choice).

p.s. this thread may be composted without damaging the environment.

 

[arfa brane]

More from the authors:
"The study of fiber bundles is part of the branch of mathematics called topology, but bundles have also been investigated in differential geometry because of their relations to the geometric concept of curvature.

The idea of a connection in a fiber bundle grew out of attempts to generalize the notion of curvature of a two-dimensional surface . . . to the curvature of a space with three or more dimensions. Hence another way of expressing the mathematical difference between the two experiments . . . is to note that the neutron-rotation experiment concerns the topology of a fiber bundle, whereas the electron beam experiment concerns the geometry of a fiber bundle.
. . .
The outcome of the neutron-rotation experiment shows in a sense that fiber bundles exist . . . and can be observed."

So, just to clarify, the precession of spin in fermions, in the author's case neutrons, maps a 1-dimensional space to a 1-dimensional space; both are closed curves, but one of them is embedded in three dimensions (the total space), the other is embedded in two dimensions (the base space).

The electron beam (Aharonov-Bohm) experiment is "more general". Things are less simple. In the first topological bundle there is an unambiguous rule for any path through the base--no "lifting rule" is needed.

Finally, fiber bundles being observable is a bit tongue-in-cheek; The experiments actually observe particle counts in a changing external magnetic field, or an interference pattern in a fixed and shielded magnetic field, respectively for particles propagating through the field or its potential.

 

[arfa brane]

After a bit of googling "gauge theory", it seems the consensus is that a gauge field (and its associated theory/theories) is a quantum field.

But I think that's an historical mistake; Herman Weyl first coined the term "gauge invariance" in a theory that attempted to unite gravity and electromagnetism.
It seems that gauge fields can have global or local symmetries; I've seen noted authors such as G. t'Hooft state that Maxwell's theory is a classical gauge theory, so presumably so is Einstein's.

Wikipedia has this to say:

Since any kind of invariance under a field transformation is considered a symmetry, gauge invariance is sometimes called gauge symmetry. Generally, any theory that has the property of gauge invariance is considered a gauge theory.
For example, in electromagnetism the electric and magnetic fields, E and B are observable, while the potentials V ("voltage") and A (the vector potential) are not.[4] Under a gauge transformation in which a constant is added to V, no observable change occurs in E or B.

--https://en.wikipedia.org/wiki/Introduction_to_gauge_theory

It seems that a choice of gauge is arbitrary; it can be an angle or a distance (i.e. as in distance- or angle-preserving Lorentz transformations), or a spin direction or an electric or magnetic field strength, but it represents a kind of fixed measurement, and so, it must have physical units with definite values . . .

 

[arfa brane]

So when you choose, for your gauge, the phase-difference between two beams of fermions then use it in experiments that can change this phase gap, by acting on half the beam after dividing it you have this gauge field. You introduce an observable shift in the phase difference, after the dividing part.

In both the experiments in the SciAm article, a magnetic field is the cause of the effect, but the way a beam of neutrons is halved is different to the way a beam of electrons is; this essentially synchronizes the spins of a pair of beams, but what is meant really is two halves of a matter-wave, or a region where the phase difference is zero between the two halves, so they interfere with zero phase-difference, if the phase in one half is changed by introducing a magnetic field the interference pattern will be changed.

So getting a classical output from either experiment has to do with how a neutron or electron interacts with magnetic fields, the electrons are much lighter, an electron beam doesn't stay coherent for as long as a neutron beam. Pretty much, both experiments had to wait for certain technology to arrive.

 

[arfa brane]

This is a tale of two experiments. Both experiments are in a certain class, which is nominally the class of interferometric experiments.

One experiment is neutron interferometry; the other is electron interferometry. In the first case, a real magnetic field acts on one half of a pair of beams so the neutrons travel further than a reference beam, so when they recombine they are out-of-phase. In the second case an electron beam is divided, one half goes one way, the other half the opposite way, around a curved region. Each half of the beam interacts with a magnetic potential that rotates an abstract spin vector, but in equal and opposite directions. When the two halves are recombined there's a shift in the interference pattern compared to when the electromagnet is off.

But working backwards, it seems the experiment in the superconducting ring is physically equivalent to the abstract mathematical space of an annulus which is a geometric projection of a truncated cone.

The fiber over each point of a cone, truncated or not, looks like the fiber over a hemisphere but all the fibers have the same gradient, so over an annulus you have a solid cylinder with its centre removed. The fibers are all identical.

So its about the map from a path in the base to a path in the bundle of fibers over the base, and, the fiber bundle unambiguously defines a path along a gradient.

To see it from the point of view of a curved region 'acting' on a path, so its about the curvature in a connection (a path through the bundle), the diagrams show where all the curvature is. The beams themselves are doing the transporting (of abstract spin vectors).

 

[arfa brane]

Finally found a decent online copy of the Bernstein and Phillips article:

The last diagram:


So we get to see what a connection in a fiber bundle looks like. When it was written the base space shown was an analytical slice through the solenoidal electromagnet; the total space for this slice is a three-dimensional fiber bundle that looks like a set of ramps. But with the advent of the AB experiment in a 2d ring, the total space here does describe the connection and its curvature.

In other words, whether you do a 3d or a 2d experiment the shift in phase is an invariant.

 

[arfa brane]

This geometry, represented in a diagram, is where the spherically-capped, truncated cone over the solenoidal current is laid out.

So although this article is free of formulas you can still think of the diagrams as a kind of notation. If you want to you can use your own notation or one that you know about.

But here is how you need to put the physics back in. The straight line along the bottom is a geodesic when it's mapped to the cone. The trick is to start at the midpoint, the point with a Q on it, with P at the left and R at the right. So then you map a path for a pair of points moving away from Q, each taking a direction with them so it stays parallel in the plane. The plane in the diagram is 'wrapped around' the cone, so everything is also rotated, if you fix the cone.

The sphere inside the cone intersects it at a line of latitude, so this is a curve that lies on a circle in the plane; parallel transport of a direction along this curve also introduces a phase difference, This region is excluded from the experiment because it lies inside the solenoid, the electrons go around, through a region with a constant potential gradient.
The physical solenoid is also lying on one of the planes the cone is tangent to. So the cone is at an angle to the incoming beam and most of it extends forward of the solenoid.

 

[arfa brane]

Some comments:
A beam of particles, and in particular a coherent beam, is a very unusual thing. It's unlikely they would exist if humans hadn't evolved and gotten curious enough about certain things. But these days, coherent beams of abstract 'particles' with DeBroglie wavelengths is as common as a high school education.

A laser pointer is a device that generates a beam of 'material particles' which is a quantum field; if you can split the beam and recombine it, and if you can introduce a delay in one path before recombination, you have a device that controls the difference in phase between two 'subfields' with a constant wavelength. This is a continuous function, acting on path lengths by curving the space the matter-waves propagate through.

I say a beam of light or of laser light in particular, is material. This is because it can transport information from place to place, that's all I need.

Which immediately casts the status of space and time into the non-material; neither can transport any information or be a resource. To have a resource I need to show it can be a store of information; although I can store space and time information, that's because I can transform certain material fields in a reliable or controlled way. I certainly can't store this information about space or time in a spacetime with nothing else in it.

 

[arfa brane]

Some more comments:
The diagram in post 13 is a flat connection, because it's in a thickened cylinder which if you cut open at some fiber can be laid flat. In that case the two paths from the centre are straight, one goes down-gradient, the other goes up. Restoring the curvature of the bundle by identifying opposite sides restores the symmetry of antipodal fibers having equal but opposite gradients.

 

[arfa brane]

Why is this magnetic vector potential about a geometry that includes spheres and cones?

If you use cones to explore spherical symmetry, some rules or relations appear off the bat.

For regular cones:
If the apex of a cone is beyond the sphere or 'above' it, then the cone intersects a circle on the sphere which is not a geodesic.
If however the apex is inside the sphere or at its centre, then the cone intersects a circle on the sphere; this circle or perimeter of the cone is a geodesic when the cone is a flat disc with apex the centre of the sphere.

So if you have a disc (flat cone) with perimeter an equator, the bundle of lines tangent to this line on the sphere is a cylinder. Every disc parallel to this equatorial one has a perimeter with tangent bundle a cone with apex outside the sphere; as the discs get smaller inside the sphere the apex of the cone gets closer, eventually there's a flat cone at the pole.
I saw that one in an online lecture, so it must be true.

 

[arfa brane]

Another comment about the fibers the authors discuss. If you indeed are taking a circle of directions over each point in a manifold as the fiber, then it doesn't matter if you represent each point as a circle or as an interval which is a straight line. Topologically the interval on the real line equivalent to the distance around a circle is the same as the circle.

Because if you cut the circle open and straighten it, that shows the equivalence. Another way to connect the two is to imagine rolling the circle along the real line so there is a 1-1 map between points. So since the circle rotates a full turn, it can also be seen as a transformation that takes angles to angles.

Similarly rotate a cone along a plane so there's a 1-1 map between tangents, and a full rotation of the cone.

But for instance, again the authors point this one out, for say a hemisphere if you choose the "circle of directions over each point" fiber, then you need to give coordinates to each fiber. If you use spherical coordinates that's a pair of angles.

If instead you take this "associated" gradient at each point of the sphere, increasing from flat at the pole towards vertical at the equator, to the bundle of directions, you don't need the coordinates, because any path through the total space is given by the connection (between fibers). The gradient shouldn't be parallel to the fibers, one way to avoid this is to divide the actual gradient by a constant, so in the total space it's never greater than 45 deg. say.

 

[arfa brane]

I think I can say I now have some understanding of what someone meant when they said the useful gauge theories are those with local symmetry.
In constructing a theory with this property, apparently you figure out what happens when you make a global symmetry into a local one.

That's pretty vague, but for instance when you divide a beam of particles into two halves, you can then apply a "local" transformation to one half, so changing a phase in this matter field, or you can apply the same transformation to both halves, i.e. see a global symmetry transformation in the field.

So this example given in the article above, of the global and local symmetries of a sphere are about being able with the fiber bundle model, of finding a flat connection for a path along the manifold. The manifold is the sphere which is considered a kind of canonical example, like a circle in one less dimension is.

So obviously, globally or from a distance, the sphere is nowhere flat, but up close it is locally. Mathematically it's easy to show there is only one plane surface at a given point.
So you take this set of locally plane surfaces and find a way to connect them in an abstract topological space. This "total space" lets you define the angles in a circle as an interval; a topological line.

To map the gradients of each locally flat, plane surface to such an object, you lift the surface above the sphere and rotate it. So when all the fiber gradient surfaces are aligned along a path from say, the North pole to the equator, the gradients are all transverse to the path which is flat as you move from fiber to fiber. The flat surfaces that give this lifted path, are all rotated the same in the bundle over the hemisphere; it's a global symmetry, but "made local" to each fiber.

Here is a diagram that explains it.




You can see that, in the total space for each path on the sphere, the gradient on the sphere is transformed (reduced and rotated 90° on each fiber, the plane surfaces shown are greatly exaggerated). Those flat surfaces are the tangent bundle of the sphere, after transforming. The bundle connections for each path on the sphere as shown, are flat, upwards movement in the bundle is the same as rotating as you move along a flat surface.

Or wander through the desert of the mind.

 

[arfa brane]

Some more stuff:

The terms "gauge field", "gauge group", and "gauge transformation", can seem a bit hard to pin down.

And there's the phrase: "gauging the symmetry". That one appears to be about the fact that none of the symmetry groups, indeed no Lie groups, in modern physics (i.e. U(1), SU(2), SU(3)), are equipped with a metric.

There isn't just one field in either of the experiments; there is a matter field of fermions, and there is a magnetic field, so which one is the gauge field? I'm pretty sure it's the magnetic field because it's what "gauges the symmetry" of the neutron or electron fields. The "gauge transformation" of either field means an observable shift; you need to know stuff about how to detect this shift, but that's not really involved in the physics of the transformations, although measurement is itself a transformation.

Heh.