Local and Global properties of physical systems

[arfa brane]

Apparently the big deal in physics is gauge theories.

Gerardus t'Hooft tries to explain what these are in an old SciAm article, about which I can remember he says that gauge theories are derived from the operation of making a global symmetry a local one. This is a bit hard to abstract, but there are plenty of explanations about how, for instance, the Standard Model itself has the structure of such a theory.

Anyways, some more stuff I found about what this all is and why it's important in physics:

A global property is something which is inherent to a system as a whole. A local property is something attached to a particular point in space and time.

Assume for the moment that the earth would be a perfect sphere, which it is to a rather good approximation. Then the rate at which the earth's surface [curves] under one's feet is a global property, because it is the same on the whole planet. On the other hand, whether there is water and land under the feet is a local property, and depends on where on the earth one stands.

So far, this is a static situation, which permits to divide between global and local properties.

Yeah, ok . . . so now what?

 

[arfa brane]

Even more important in physics is the difference between local and global changes. A local change modifies something at a given place. E. g., the property whether there is land or water below one's feet is changed locally by the tides. A local change is not limited to a certain point, but it can affect many (or all) points at the same time, but something different may go on at every point. The tides all over the world are an example of a local change, which let the water rise at some point and removes it at another point. A global change is then a special case of a local change in that it makes the same change at each and every point. For example covering the earth's surface everywhere by a meter of sand would be a global change.

This leads back to symmetries. It is now possible to divide between a global and a local symmetry. A global symmetry is something inherent to the system as a whole. A global symmetry transformation would then be a symmetry transformation applied to every point which leaves the system unchanged.

A local symmetry transformation is much more complicated

Oh, hell no!

 

[arfa brane]

It is indeed, complicated. Again, I recall t'Hooft saying that in order to "make" a global symmetry into a local one, the [gauge] theory has to have a force included, which is although a force isn't that complicated or hard to understand, not an easy thing to do or understand how to.

In the SciAm article, t'Hooft discusses an abstract internal arrow inside protons and neutrons, such that an internal rotation transforms neutrons into protons or protons into neutrons. Doing this to all the protons and neutrons in a physical system would be a global change that leaves the system unchanged--i.e. a symmetry.

 

[James R]

Is there a question? Something you'd like to discuss with the group?

 

[arfa brane]

Of course there is. But I expect there won't be any actual discussion. Or that much interest, except perhaps from you, and that won't be about actual physics. It might be about bottles though.

The questions: what is a force, and why do gauge theories need them? Why are there force carriers in modern physics?
Lastly, how are forces transmitted in Newtonian mechanics, what is the force carrier?

 

[arfa brane]

The global symmetries are found to be associated with properties of particles, e. g., whether they are matter or antimatter, whether they carry electric charge, and so on.

Local symmetries are found to be associated with forces. In fact, all the fundamental forces of nature are associated with very special local symmetries. For example, the weak force is actually associated in a very intricate way with local rotations of a four-dimensional sphere.

Maybe the global/local symmetries are connected to measurements. So do questions about forces devolve to questions about measurements?
Or what?

p.s. the quotes are from a blog by a dude called Axel Maas. links are available on this page:
--https://axelmaas.blogspot.com/2010/08/global-and-local-symmetries.html

 

[James R]

The questions: what is a force, and why do gauge theories need them?

I think it works the other way around. The gauge theories are supposed to describe/explain the forces. Theories are meant to explain real-world observations.

Why are there force carriers in modern physics?

Because our most accurate theories in modern physics are quantum field theories. Excitations of the fields correspond to particles.

Lastly, how are forces transmitted in Newtonian mechanics, what is the force carrier?

In Newtonian physics, the only known forces are electromagnetism and gravity. Both are treated as action-at-a-distance. There are no force carriers.

Hmm, maybe that's not quite right. The classical fields themselves can transmit forces via classical waves (the prime example being electromagnetic waves, which carry momentum). Newtonian gravitational waves are also a thing, but it turns out they aren't a correct description of such waves. Our best classical theory of gravity is, of course, general relativity, which also predicts gravitational waves. All gravitational forces are explained in general relativity as relative reference frame effects, however. There are no force carriers, as such.

 

[arfa brane]

I think it works the other way around. The gauge theories are supposed to describe/explain the forces. Theories are meant to explain real-world observations.

However, gauge theories are constructive theories. A "gauge particle" is constructed which has to obey all the symmetries of the field in question. That is to say, the force-carrier in a gauge theory is an object with the same degrees of freedom as the field, for instance.

To obey the physics the proposed field particle must not break any of the field symmetries. CPT symmetry however, doesn't seem to be about a particular field but rather about the SM itself. CPT is about global, not local, symmetries. (\note to self)

 

[arfa brane]

About those CPT symmetries. It's useful to think about an abstract mirror, say a matter-antimatter mirror. Or a time mirror.

Obviously, measurement is going to need the latter abstraction in some theory of measurement. Isn't that what measurement is, since we decide what the physical response in some system, actually represents? Or rather, what the symbols and the logic we use to construct "measurement" devices, mean? Or what meaning in measurements, is?

You measure your weight accurately on a device, designed for the purpose. Your weight, is responsible for that pull you can feel, towards the centre of the planet. Are you, just by being aware of this weight you have, with or without standing on some scales, measuring the earth's gravity? Why or why not?

 

[QuarkHead]

Well, the simplest (but not the most accurate) way to think of it is this: suppose you have a theory. Further suppose there exists a 4-manifold (spacetime, if you like), at every point of which you have a group of continuous symmetries (a Lie group). Then if, as you "carry" your theory continuously over the manifold and encounter a possibly different Lie group element at every point, then if your theory takes the same form at every point, you have a gauge theory.

Needless to say, you needa ton of other things like a connection, a curvature measure and a way of doing meaningful calculus on your manifold,

Interestingly, you can, very loosely speaking, think of General Relativity as a gauge theory

 

[arfa brane]

More stuff from stackexchange:

"Making the global symmetry local" is for some reason a popular pedogogy, but it doesn't actually make any physical sense as a motivation. See this answer of mine or this questions and its answers for better motivations for gauge theory.

You seem to think that quantum gauge theories are somehow completely different from classical gauge theories because the "from global to local" motivation doesn't work in classical electrodynamics. This is not the case, a quantum gauge theory is the quantization of a classical gauge theory like any other quantum field theory is the quantization of its corresponding classical field theory.

The transformation Aμ↦A′μ=Aμ+∂μα is a local transformation. The definition of a local transformation in this case is simply that it is a transformation in which the transformation parameter α depends on spacetime.

(The transformation is also "classical". In fact, this transformation only makes sense for the classical field, not for the quantum operator-valued field, since α is not a dynamical field of the classical theory and therefore quantization does not turn it into an operator-valued field.)

A bit to unpack there . . .

The global transformations of a U(1) gauge theory are those where α does not depend on spacetime, and indeed those are the transformations under which the gauge field is invariant.

It is crucial to note that it is the latter and not the former property that generalizes to the notion of "global" transformations in non-Abelian gauge theories. The gauge symmetry is, in a very real sense, "unphysical" in that it encodes a redundancy in our mathematical description, not a property of the physical system, see e.g. this question and its answers or this answer of mine. Its global part - in the meaning of the transformations which leave the gauge field invariant - is a true property of the system, not of the description, for instance, it is the global symmetry, not the gauge symmetry, that is broken in the Higgs mechanicism, see this excellent answer by Dominik Else.

--https://physics.stackexchange.com/q...-u1-transformation-properties-of-gauge-fields
Uh huh!

 

[arfa brane]

From a poster at stackexchange:

A non-quantum motivation of gauge theory can also be given: If you write down the Lagrangian of free electromagnetism, motivated because its equations of motion are the Maxwell equations, not because we like gauge symmetry, then you find it comes naturally with a U(1) gauge symmetry, corresponding to the well-known fact that adding a gradient to the vector 4-potential is physically irrelevant.

Now, if you want to couple other fields to this free electromagnetism, you need to make the additional terms gauge invariant, too, else the theory is no longer "electromagnetism coupled to something else" in any meaningful sense since suddenly adding gradients can change the physics. Once again, gauge symmetry is something one discovers after physically motivating the Lagrangian from something else, not some sort of a priori assumption we put in.

I think the comment about adding a gradient being physically irrelevant says something about redundant degrees of freedom; this idea that the gauge in a gauge theory is redundant information is intriguing. As if it's a thing the theory needs, but the physics doesn't, until as stated, you want to couple your redundant d.o.f. to say, a matter-field.

So the oft-used example of doing experiments in a lab which is raised to a certain voltage potential, and this doesn't affect any experiments is just an example of adding a gradient which is physically irrelevant (in a classical gauge theory).

 

[arfa brane]

Wikipedia has an entire page about the mathematical side of gauge theories, and explicitly tables the differences between the mathematical and physical terminology. One interesting paragraph:

Gauge theory has been most intensively studied in four dimensions. Here the mathematical study of gauge theory overlaps significantly with its physical origins, as the standard model of particle physics can be thought of as a quantum field theory on a four-dimensional spacetime.

The study of gauge theory problems in four dimensions naturally leads to the study of topological quantum field theory. Such theories are physical gauge theories that are insensitive to changes in the Riemannian metric of the underlying four-manifold, and therefore can be used to define topological (or smooth structure) invariants of the manifold.

--https://en.wikipedia.org/wiki/Gauge_theory_(mathematics)#Gauge_transformations

That table on the wiki page says that a mathematical connection is physically the gauge potential or gauge field. That's why you see phrases like "the curvature of the connection". The Aharonov-Bohm effect (a phase change in a matter-field of electrons) is exactly because of this curvature in the gauge fieid.

 

[arfa brane]

I also recall now, that in that SciAm article t'Hooft talks about how an airplane's altitude is given as above sea level.
It's "physically irrelevant" if instead the altitude is given as above ground level, and then equally irrelevant would be introducing something extra into the "theory of measurement" that restores the symmetry.



The extra thing is a force acting on the airplane, "correcting" the path. All irrelevant to the actual physics, but nonetheless a gauge field, with a gauge symmetry.

Notes: the altitude of the airplane is a local measurement; the path is a global measurement.

 

[arfa brane]

So the global symmetry--the whole path of the airplane through spacetime--is "made local" w.r.t. the local altitude. A choice.
This motivates the introduction of a force that "makes" the global constant altitude asl, into a constant altitude locally; but you need a Lagrangian you can take with you.

 

[arfa brane]

So, the quoted stuff in the OP, from Axel Maas, relates to the screenshot or diagram of the way a path changes when you change the coordinates as I see it, in this way:

Globally, sealevel varies depending on tides, storm surges and so on, it isn't the same everywhere, so the airline industry uses a standard. Modern aircraft have terrain following radar, GPS, and lots of other kit. A global sealevel value is something that's calibrated into an instrument, an altimeter which actually reads air pressure (right?).

The diagram is really just a simple visualisation of what a comoving observer would see if they moved along the surface of the mountain range; the plane appears to move correspondingly or covariantly. In physics you can put observers wherever you like.

Such that a locally comoving observer sees forces acting along the path because of the coordinate change(s), these disappear for an observer moving parallel to the globally fixed sealevel. It's just physics.

And of course the diagram says something about measurements.

 

[arfa brane]

Notice how this:

"Making the global symmetry local" is for some reason a popular pedogogy, but it doesn't actually make any physical sense as a motivation.

from post 11, is clearly an opinion.

There's a good pedagogical example in that diagram, from a youtube vid about symmetries in physics. In this case the symmetry is that the laws of physics don't change when coordinates change.

The rubric--make a global symmetry local--isn't hard to see when the global property is a fixed sealevel or zero altitude, everywhere.
Try to make this a local property, and observe the mathematical consequences; futz with differential equations, yada yada.

 

[arfa brane]

The diagram is simple--obviously you can analyse a physical situation by focusing on certain details and ignoring a lot of other real-world stuff.
You consider an ideal situation.

An observer moving along the surface of the mountains, in the diagram would see the level flight path change, the local altitude would change, there are gradients because of the changes in coordinates. But the observer isn't a necessary part of the physics, it's all down to a local parameter, the altitude, and how it transforms.

You can see how the global symmetry is broken; locally the altitude "measurements" describe a path which is not smooth.
Something about measurements and symmetry-breaking goes after this. (viz. post 11).

 

[arfa brane]

Just a comment about the mathematical part; let's say this lives in an abstract space. This space has a lot of room in it, there are literally pages of equations and explanations of them, that pertain to that diagram and many other kinds of simple looking diagrams that describe motion of objects.

Although you can say the extra redundant stuff you put in, because of a 'simple' change of coordinates is a 'gauge field', that is a really long way away from showing that laws of motion describe a gauge symmetry in general.

Just sayin'