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What exactly does unification mean in physics?

Posted: Thu Sep 14, 2017 12:27 am UTC
by Pfhorrest
I understand that the photon that mediates the electromagnetic force is an excitation of the electromagnetic field.

I understand as well that the W and Z bosons that mediate the weak nuclear force are excitations of another field (or fields maybe... I'm a little unclear on how one field can have excitations that manifest as two different kinds of particles, and maybe just clearing that up will answer my ultimate question I'm leading up to here. For that matter, more basically, I'm unclear on how the effects of two different kinds of particles constitute a single force).

I understand that electroweak unification means that at high enough energies, photons interconvert with W (and Z?) bosons freely, in some kind of equilibrium state, so if you've got one then you'll immediately have some of the others and then vice versa.

What I'm wondering is if that means that there is really just one single electroweak field, that can be excited in different ways, ways that can transform between each other at high enough energies but that can only stay one way or another at lower energies... and if so, how exactly that works, and what exactly that means. I'm picturing an excitation of a field as the classical sinusoidal wave shape. Are there like... different shapes of waves possible, or something?

And I'm assuming whatever the answer to that is, it would also hold if full grand unification of the electronuclear forces should turn out to be the case; that gluons would just turn out to be yet another different kind of excitation of that one field?

Re: What exactly does unification mean in physics?

Posted: Thu Sep 14, 2017 2:41 am UTC
The reason you have multiple particles corresponding to a single force is that the field transforms under a gauge group larger than U(1) (just plain phase changes).

All of these theories are Yang-Mills theories which generalise the Electro-Magnetic field Fμν to a Yang-Mills field Gaμν where a indexes over the generators of the gauge group (in the same way the two spacetime indices index over the generators of translations (the basis vectors).

The gauge groups of electromagnetism and electroweak theory are U(1) and U(1)xSU(2) respectively. U(1) is just the group corresponding to phase changes, i.e. e which has a 1-dimensional fundamental rep (that given here). SU(2)'s fundamental rep is generated by the 3 Pauli matrices so it is a 3-dimensional algebra with a representation in 2-by-2 matrices.

As with the case of electromagnetism, where we have 6 independent possible excitations (a photon travelling in any of three directions, in one of two polarisations) corresponding to the number of independent components, so too do we have 6 excitations for each independent combination of a and b (so in the electroweak case, there is a single U(1) boson (Y), and 3 SU(2) bosons (corresponding to each of the three Pauli matrices which generate SU(2), W+, W- and W0).

At the moment, the SU(2) bosons all have the same spectrum so there isn't any way to distinguish them (although you would still be able to detect their existance from branching ratios in radioactive decays). "Luckily" the Higgs field couples to all these bosons and has a typical "Mexican hat" potential; this causes the Higgs to spontaneously break symmetry at low energies and, because (as with so much of the standard model), things couple to each other in ways that don't quite line up, we end up mixing the Y and W0 bosons giving us the observed γ, Z0, and W± bosons with these three having different spectra, and the two W bosons having identical spectra but being distinguished by charge.

Re: What exactly does unification mean in physics?

Posted: Thu Sep 14, 2017 2:48 am UTC
by Pfhorrest
Thank you for that very detailed explanation, but almost all of it was somewhere over my head, I'm not even sure enough how far to say one specific part I need explained to me further. Is it possible to dumb it down a little bit and see if I can grasp it then?

Actually, I guess I can maybe name some parts I do and don't get already:

I'm not clear what a gauge group is, what a generator of one is specifically, or what that U(1)/SU(2) notation means. I think this is probably the root of my difficulty.

I'm guessing that the phi in your e^i*phi stands in for where I would usually expect a theta, as in Euler's formula, and gauge groups have something to do with transformations, and U(1) somehow or another means something to do with rotation in the complex plane, from which I can see the connection to phases of cyclical graphs or waves, which has some obviously connections to physics, but I'm not clear exactly how it relates to my specific question.

I don't know what "fundamental rep" means.

I don't completely follow the bit about photons moving in different directions being different possible excitations of the electromagnetic field, or why there are only three directions. (Guessing some connection to three dimensions, but photons don't just travel positively along the three axes of a given coordinate system, so I still don't really follow).

I think I understand something about the mixing and Higgs field coupling you talk about, though I'm very uncertain of my understanding there, and I'm not sure I understand the word coupling correctly. As I understand it, particles "mix" via interaction with the Higgs field by extremely rapidly transforming into each other every time they interact with the Higgs field, which is basically continuously as the Higgs field is nonzero everywhere so you can't do anything without immediately interacting with it. So e.g. what would be a massless left-handed or right-handed electron continuously interacts with the Higgs field, switching its spin state every time, acquiring rest mass in the process of that constant interaction, and mixing up its spin state instead of having just one constant one. But, even if I understand that correctly, I don't completely follow what that means for mixing bosons as you describe.

Thanks again!

Re: What exactly does unification mean in physics?

Posted: Thu Sep 14, 2017 1:49 pm UTC
The gauge group is a symmetry of the field. In electromagnetism the gauge symmetry was discovered after the formulation whereas it was fundamental to the development of Yang-Mills fields from electromagnetism.

In electrostatics, I'm sure you're familiar with the fact that you can choose the zero of your potential however you want. In the general case though, it's a bit more complicated and we instead find that all 4-potentials Aμ+dμϕ (for arbitrary scalar ϕ) have the same dynamics and so produce the same field tensor Fμν. This gives an infinite class of symmetric potentials related to each other via a continuous transformation. As with any symmetry, this has a group structure which here can be shown to be that of 1-by-1 unitary matrices U(1) (the U means unitary and the 1 means they are 1-by-1) the elements of which are the elements of the unit circle in the complex plane eiφ (I use φ here because in quantum mechanics, you have to also do a phase change of the wavefunction when you transform the gauge of the electromagnetic potential, but any other variable could be used).

In the 50s, people looked at this gauge structure and decided to try and generalise it by picking other gauge groups. It turns out that the only groups that work well are those of the special unitary groups SU(n), that is, the group of n-by-n unitary matrices of determinant 1 (U(1) is the same as SU(1) and so it is usually written without the S). The fundamental representation is just the representation implied by the definition of the groups I've given above, the fundamental rep of SU(n) is just the set of n-by-n unitary matrices of determinant 1.

Now, these groups are Lie groups, which means that they also have a manifold structure and because we're usually interested in infinitesimal transformations, we're interested in the tangent space at the identity which is the definition of the Lie algebra. As a tangent space, we can find an orthonormal set of basis vectors, which here we call the generators of the gauge group Ta and, with some potential topological complications, we can recover any element of the gauge group as exp(xaTa).

We find that the gauge potential Aμ=AaμTa must transform under the action of a member G of the gauge group as A'μ=GAμG-1 + (i/g)(dμG)G-1 (where i/g is a coupling constant). This reduces to the EM case when the gauge group is U(1).

The most useful basis for the SU(2) Lie algebra are the Pauli matrices but instead of the σ1, σ2, σ3 basis, it makes more sense to use σ+, σ-, and σ3 (using the standard ladder operator style construction) which we'll call T+, T-, and T0.

Now, the whole point of the gauge symmetry is that whatever gauge you choose yields the same Fμν but it still has the gauge index from the gauge potential so in YM theories, we have a field tensor Gaμν. We know that physically Fμν gives us one photon, so because of the extra gauge index, which in SU(2) can take one of 3 values, we get 3 bosons which we'll call W+, W-, and W0 because they correspond to each of the three T generators and, (although currently massless, the W± bosons are those we expect).

If we also have a U(1) theory as well, it behaves very similarly to the W0 but coupling to different things. If we coupling this U(1) to "hypercharge" we can posit a "weak mixing angle" to rotate the U(1) B field into the W0 field and vice versa to get a photon and a Z boson. This allows to relate the photon, W, and Z bonsons' coupling constants which is great! So far though everything's entirely equivalent though and there isn't really any reason to choose this basis over the B, field and the three W's. The thing is, the Higgs doesn't couple to the photon, but does to the Z (and the W's) which gives us a reason for choosing this basis. To 0th order this predicts that mW=mZ cos(weak mixing angle) which isn't too inaccurate, but there are corrections due to the mass of virtual fermions and virtual higgs particles.

Does that help things a bit? It's been a few years since I first looked over this but I have my old notes so it's good getting me to read over it :)

Re: What exactly does unification mean in physics?

Posted: Thu Sep 14, 2017 10:38 pm UTC
by Tchebu
Both of eSOANEM's responses are entirely correct, but if they're over your head perhaps I can have a stab at explaining this without the technical jargon. This will be long however because I want to start real simple and apologies if I overdo it and start way under(?) your head...

You're actually pretty much on the right track in your last two paragraphs and you're also right that your basic confusion stems from not wrapping your head around how we can have multiple particles from the same field. The short answer is that it's basically a semantic ambiguity, but like many semantic ambiguities, it captures something important, so let's unpack.

One might think that when we talk about "one" field, we're talking about just one number per point in space. Of course you already know this to be false, because, say, the electric field already requires three numbers per point, i.e. the components along some arbitrarily chosen axes. However, there's nothing really preventing me from saying

"nono, you see those are actually three DIFFERENT fields!! This collection of 3 numbers that you call the electric field is actually 3 different fields"

At this point any reasonable physics undergraduate will instantly tell me that what those 3 numbers are depends on my choice of coordinate axes and that it's better to regard the entire triplet as a single field. There's simply no fact of the matter as to which direction in the universe is "the x-axis", and for that reason rotating my coordinate axes will produce a change in my "three different fields" without reflecting an actual change in any of the physics. It also means that I can't make any physically meaningful statements about the directions of the electric field unless I was talking about the RELATIVE direction of, say, the fields coming from two different sources or the RELATIVE polarization of two electromagnetic waves. In other words, if you give me two differently polarized electromagnetic waves, I wouldn't be able to tell you that one is polarized along the x-direction and the other along the y-direction, but I would be able to tell you that they're not the same, and are in fact orthogonal. So there are still several distinguishable excitations of this "one" electric field.

Now imagine there was a background uniform electric field permeating all of space for whatever reason (maybe the observable universe is located between two enormous capacitor plates, who knows..) In that case there would be "absolute" statements we could make regarding the direction electric fields from other sources. We can now unambiguously distinguish the component along this background field from the components that are orthogonal to it. So now if I choose my coordinates so the x-axis is along this background field, then the "x-component" of the field actually acquires a physical meaning! I can actually now measure it in a lab as the answer to the question "what is the component of this field that's parallel to the 'universal background field'? ". The y and z components remain rotatable among themselves and there's still no fact of the matter as to "what the y-component is".

Notice what happened. In the absence of the background field the individual components are physically meaningless and it's better to assemble them together into one object and call it by one name: "the electric field". Once we introduce some background field, suddenly it's better to split off the component along this direction, because it actually carries physical significance, and perhaps give it a name of its own (i dunno, say, the "longitudinal field"), but the other two components remain interchangeable and we should perhaps still combine them together into a 2-component "transverse field" or something.

So now the million dollar question: is there just one "electric" field, two fields (a one-component "longitudinal" and a two-component "transverse" fields) or three fields (x, y, z- components)? Hopefully by now it's obvious that the answer is that it's basically just a naming scheme and which one is better depends on context.

Ok, so now it's time to generalize...
In the example with the electric field these three different numbers corresponded to some arbitrarily chosen directions in physical space. But we can just have several types of fields, each described by a single number, whose physics might be such that, there's no fact of the matter as to "which number means which field" just like earlier there was no fact of the matter as to "which number means which direction" and the only physically relevant quantities are always made up of some combinations of these numbers that don't actually depend on "which number means what field". For example it could be that the only things that manifest themselves physically are sums or averages over these fields.

In this case these fields basically behave like the "components" in our earlier example, having no physical meaning of their own. So we just lump them together and call them a single (multi-component) field, but because it's still described by several numbers, there are several different types of distinguishable excitations of this field. This is why you can have several particles as excitations of the same field.
As before, in order for this to be the case, there must be some transformation rules that allow us to "rotate" the different fields into each other, but now this rotation is no longer a rotation of our coordinate axes, but a "rotation" in some auxiliary abstract "field space".

Some terminology: The set of such possible transformations for a given set of fields is called the "gauge group" (and these are typically some combination of the Special Unitary groups that eSOANEM referred to) and the fact that these transformations don't change the physics of our system is known as "gauge symmetry" or "gauge invariance". Theories that have this feature are called "gauge theories".

Note that at this point, you still can't tell which particle is which (because there's no fact of the matter), but if I give you two different ones you'll be able to tell me that they're not the same, or more realistically you'd be able to do experiments that will tell you how many different types of them there have to be.

Now suppose that the dynamics of these fields are such that one of the fields likes to take a specific non-zero value everywhere across the universe. Just for whatever reason, that's the lowest energy configuration it can take, so as the universe cooled down that's the value it took. In that case we'll be in a situation somewhat analogous to the background electric field case (although not exactly, treat this as a conceptually correct but technically inaccurate analogy). This now breaks some of the gauge symmetry, i.e. the freedom we had to "relabel" our fields, because now some subset of the fields becomes distinguishable from the rest due to how they relate to this background field. This is analogous to our "longitudinal" and "transverse" separation for the electric field components.

In fact because all these fields also interact with each other (unlike the electric field which doesn't interact with itself, which is what makes the analogy inaccurate), it's not just that some of the fields are "along that direction" in our abstract field space and others aren't, but their interactions with this background field will actually make them behave differently from the rest!! The fields will naturally split up into subsets that behave qualitatively different! This is basically the Higgs mechanism in a nutshell.

Notice that in this example we started with a big set of fields that were all "mutually inter-rotatable" and that it only split up into unambiguously distinguishable subsets because of dynamics that broke some of the gauge symmetry. Of course us mortals, when we delve into the mysteries of particle physics will just see two categories of particles that behave differently and won't immediately realize that secretly they're part of this bigger unified theory that has this much bigger symmetry that's not apparent because of how the dynamics of some of these fields played out. That's the basic idea behind unification! We're looking for that bigger symmetry group! If it exists, then at some high energies, the culprit fields that broke the symmetry will no longer need to take their non-zero value (remember they only took it because that was their lowest energy configuration, and now we have energy galore). Then the full symmetry will be restored and all the different particles will again become "mutually inter-rotatable".

Re: What exactly does unification mean in physics?

Posted: Fri Sep 15, 2017 12:26 am UTC
by Pfhorrest
Thanks Tchebu, that helps a lot! I still probably need to re-read it a few times to really let it sink in, but it's in a language I speak at least so I can actually do that now.

Re: What exactly does unification mean in physics?

Posted: Fri Sep 15, 2017 8:56 pm UTC
by Xanthir
Ah, yeah, that helped a *ton*; I was laughing to myself at how eSOANEM’s "simpler" second explanation flew right over my head as well. ^_^