Finally a reward for the hard work we did in the last few videos! (Not that hard work isn’t its own reward.) This week we talk about Gauge Theory, explaining how the forces of nature arise because of local symmetries in quantum fields.
The Biggest Ideas in the Universe | 15. Gauge Theory
And here is the Q&A video, where we go into more specifics about the Higgs mechanism, how it gives mass to particles, and how that plays out in the Standard Model of particle physics.
The Biggest Ideas in the Universe | Q&A 15 - Gauge Theory
Sean – thanks for doing these wonderful videos. I would like to question your comments (starting about :51) about how a charged particle (electron) can emit photons with no energy. Photons do have energy (E=hf) and momentum (p=hf/c). So emitting photons of any frequency would still violate energy and conservation laws. This has bothered me since I first started looking closely at the Standard Model. The “virtual” photons that theory requires do not seem to be photons at all but something completely different. If they are completely different why are they considered photons?
Thanks for the lecture, great as usual! I wonder if you could go into more detail on the actual form of the gauge theory for electromagnetism, like explaining polarization, and what the actual components of the electron and photon fields are.
Weak theory was an SU (2) group, meaning that it has dimensionality (2*2)-1 = 3. Is that why there are three weak theory bosons (W+, W-, and Z).
Bravo! Wonderful talk. I have read all your books and probably listened to a large proportion of your talks and even read a few papers (pretty much impenetrable to me). But to be able to go through this subject with a minimum number of equations (and explain the ones you used), was a tour de force… and somewhat improvised at times, I think It reminded me of one of my favorite jazz guitarists Larry Coryell. … high praise. I’m going to raise a nice glass of red wine to you tonight Professor Carroll. You have a unique skill. Especially valuable in these challenging times, where Enlightenment values are under attack. … Questions on this talk can wait for later. Bravo!
First of all, what a wonderful payoff video! It’s remarkable how much of physics can be explained from just geometry. Do you think it’s inevitable that a theory of everything will be geometrical too? I’d also like to hear more about GUTs and if we should believe their predictions, and how the electroweak interaction fits into the standard model.
Great talk, thanks!
What, if anything, is the relationship between the U(1) gauge symmetry of QED and the invariance of Maxwell’s equations under choices of Lorenz gauge?
Thanks!
Ontology:
Taking SU(3) as an example, I understand that the symmetries and conservation laws follow from the group properties, and I understand that the observables match the calculations derived from that group. But I don’t understand why the universe likes that particular group – why not SU(14) or SU(1,2,3,4)?
For that matter, all these involve functions of x, which is position – but why is there a position to have a function of?
Which role do Goldstone bosons have in gauge theories?
Electrons and positrons annihilate producing photons but what happens when a quark meets its antiparticle? I’d guess that they produce an excitation in their connecting field, i e gluons, and if that’s right then I must have learnt something.
There must be quantum fluctuations of the Higgs field around its ground state v value, so the mass of other particles as the W and Z bosons should fluctuate also, and all these fluctuations should be correlated. Is this effect real and is it detectable ? I expect it will be very weak, because if it is not the case it would break all the spatial and temporal symmetries and most notably the momentum conservation law…
Thanks for another great video!
I understand that making a symmetry local requires the introduction of a connection. (That’s just math.) But I don’t understand why this connection necessarily must have curvature. (Which is what gives rise to the force fields.)
In other words, could there be a world in which we have a local U(1) symmetry and the corresponding gauge field, but no E and B fields?
You said something like the proposed symmetry group for spacetime was SO(3,1) such that the spacial dimensions do not rotate into the time dimension. Yet I’ve heard that within a black hole space and time swap roles. Sounds like a super-duper-symmetry to me.
Thank you again for the excellent video. I have three questions:
1. I am unsure if it is just me who missed something, but the whole discussion centered around rotations of field vectors. You mention using rotations on the color fields… But what I do not get is why would one want to rotate anything? What is the physical significance in rotating fields? You mentioned that you can pick any rotation, then parallel transport the vector then compare to another field. But if you can arbitrarily rotate the field what is the point of the rotation and comparison?
2. You show how a product between a complex value and its complex conjugate produces an answer that is interpreted as mass. But why? I know a complex number multiplied by its complex conjugate is a real number, and that can look like a mass (scalar) but surely it can also be anything else such as charge? There is a lot of math that you equate to physical properties like mass, momentum etc. However, how do you know the answers relate to those specific physical properties of the fields?
3. I get that you are simplifying a LOT and skipping over a ton of detail to make this material accessible, however one thing that bugs me is that you start with mathematical formalism (geometry, topology, symmetry, complex numbers) and postulate many things such as “Let’s try gamma squared” as the mass term. How do you know to try gamma squared, and not some other function? Why is the square of the photon field the mass? The deeper question – it feels like QCD is a theoretical model and then reality is supposed to “fit with the theory” – unlike the classical approach where we investigate reality and come up with math that describes it. The statistical likelihood that a theory based on abstract topology, geometry and symmetry happens to be exactly what reality is based on seems to me almost zero. Surely reality drove the discovery of QCD, not the other way round?
Several of my questions are very similar to those of Waldo Nell.
1. What is the physical significance in rotating fields?
2. You show how a product between a complex value and its complex conjugate produces an answer that is interpreted as mass. But why?
3. Why is the square of the photon field the mass?
4. Could you comment on the interplay in the history of particle physics between theory and experiment? I generally think of a scientific theory making “predictions’ but it seems that ,in general , particle physics theory accounts for experimental results (except perhaps “looking for the Higgs”). I’m not even sure that Dirac’s prediction of the “anti-electron” made the experimentalists “look” for it. Even though it was a great prediction!
5. Gravity… I know you are going to devote a talk (or several) to gravity. But, could you briefly reiterate why QM/GR can’t be a gauge theory? You said something like, ” the geometry of spacetime… you can derive the “metric”… With quarks there’s just a connection, an internal symmetry. You’re not rotating directions of spacetime. It’s he internal “fictitious space” (??) of R/G/B. There’s no metric (??).”
I think you can take next week off if you like. There’s tremendous amount to absorb in this video, at least for me. Thanks again for being so good at doing this.
One more question:
You said that the electron field is a complex field that is symmetric under a U(1) phase rotation. That’s fine. But there is also a wavefunction that assigns complex numbers to the electron-field configurations. So it seems that there are two U(1) symmetries associated with the electron (complex electron field & complex wavefunction). Could you please elaborate on these two U(1) symmetries? How are they related and how do they relate to the gauge field?
arghous said:
> You said something like the proposed symmetry group for spacetime was SO(3,1)
> such that the spacial dimensions do not rotate into the time dimension. Yet I’ve
> heard that within a black hole space and time swap roles.
> Sounds like a super-duper-symmetry to me.
You have it backwards. The fact that you can even say “time and space swap roles” shows you that there is no symmetry that exchanges time and space. If there were, the statement would be meaningless, or trivial, depending on your point of view.
For comparison, it wouldn’t make much sense to say “in this region of space the x and and y axes swap roles”. This is because the x and y axes are completely arbitrary, and you can exchange them with a rotation. Not so for the t (time) and r (radial) axes centered on a black hole.
Waldo Nell said:
> I am unsure if it is just me who missed something, but the whole
> discussion centered around rotations of field vectors. You mention
> using rotations on the color fields… But what I do not get is why
> would one want to rotate anything? What is the physical significance
> in rotating fields? You mentioned that you can pick any rotation,
> then parallel transport the vector then compare to another
> field. But if you can arbitrarily rotate the field what is the point
> of the rotation and comparison?
I can try to answer this one.
Physicists like to work “in coordinates”. That means they put down some axes, then write their equations in terms of the corresponding coordinates. Unfortunately, axes have no physical meaning, so one has to separately figure out what happens to equations if you make a different choice of coordinates.
To organise this, physicists keep track of all the ways to relate a system of coordinates to another one (locally at every point). Such a thing is called a gauge (transformation). To understand the significance of rotations and why they come up so much, imagine we are talking about actual axes in 3d space. Then every choice of (positive orthonormal) axes is related to any other one by a rotation. So the fact that you don’t want to assign physical meaning to a particular choice of x, y and z axes translates into using the group SO(3) for the underlying symmetries.
In the example of QCD, things are more abstract and complicated, but ultimately it is a similar situation. The particular choice of colours r, g, b has no physical meaning. And not just the order of the three labels, but also what particular three independent mixtures we take as “primitive”. This translates into being able to “rotate” (in the complex sense) any choice of r,g,b into any other choice.
Mathematicians tend to have a different approach, which I find clearer, but it does require a bit more “technology”. In math we usually prefer to work with “intrinsic” objects (such as manifolds), and build our equations in terms of constructions that make absolute sense within these objects, without an arbitrary choice of coordinates.
In the specific examples of gauge theories, the object of study is a so-called “principal G-bundle” over the manifold M of interest, where G is the desired symmetry group. One way to understand principal G-bundles is to think of them as collections of G-torsors, one for every point in M. A G-torsor is a copy of G where one has forgotten “where the unit is”.
Because we don’t know where the unit of the group is at every point, we don’t know how to relate elements of the group living over different points. This is why the ideas of connection and parallel transport come up. There are very nice coordinate-independent formulations of connections on principal (or even more general) bundles (Ehresmann connections), and one can then recover the ones that physicists use when working in coordinates.
This idea of working with torsors is really powerful, but it takes a while to get used to. Because of the insistence on coordinates, physicists have naturally settled on formulations of theories that are based on these ideas (although not many physicists talk about them in these terms).
To get comfortable with this way of thinking, it is useful to consider simpler examples. In linear algebra, the usual approach is to give an abstract definition of “vector space” in terms of algebraic operations (addition and scaling). Then one can define new notions and prove theorems without any reference to coordinates.
The physicist-style torsor approach, instead, would start with ℝ^n as the “only” vector space. Notions would be defined in coordinates, then separately proved “covariant” with respect to the symmetry group, which in this case is GL(n) (invertible n × n matrices). When working with inner products and such, again ℝ^n is the “only” space, but now covariance is considered with respect to O(n).
Mathematically, what is really going on is that in the second approach we are not using the usual definition of vector space. Rather, we are identifying vector spaces with GL(n)-torsors. The actual vector space in the usual sense underlying a GL(n)-torsor X is the “twisted product” X ×_GL(n) ℝ^n. That explains why every constrution can be performed on ℝ^n, but has to be separately proved covariant. Another way of thinking about it is that we work on a GL(n)-principal bundle over a single point.
I hope this helps.
I’m not going to try to answer for Waldo but for myself. First, I very much appreciate your explanation. When you say, “So the fact that you don’t want to assign physical meaning to a particular choice of x, y and z axes translates into using the group SO(3) for the underlying symmetries.” I understand the abstraction but I believe the question was about the “physical reality of rotations” which is a question I have too. “Physical reality” might not be exactly the right phrase, but reality in terms of the fundamental forces of physics might. I don’t know. Again, thanks for the explanation though.
As far as I understand, rotations have no “physical reality”. You just need to take them into account to compensate for the fact that you introduced another “physically unreal” object into the picture, namely axes/coordinates. It’s a consequence of the choice of formalism. You can’t work with coordinates without mentioning how things transform under change of coordinates.
Note that I’m basically equating the word “real” with what physicists might call “covariant” or “intrinsic” or “well-defined”. For example, in Newtonian mechanics velocity is “real” in this sense, but its x,y,z components are not. If we insist on working with components rather than the abstract notion of velocity (using bundles), we get to work with very concrete entities (real numbers!) but pay a price by having to consider its behavior under rotations (and Galilean boosts, in this case!).
Thanks so much for the clarification. Funny how much of this comes down to a language. I “understand” covariant… but at the end of the day, only the math is the “real” language. I think Maxwell wanted to make a mechanical picture of his equations by a system of “spinning gears”.
So, you have a Higgs field that has symmetry, but it has settled into a configuration that isn’t symmetric because it was seeking its lowest energy configuration. You call that “spontaneous symmetry breaking.”
My question is whether there can be any non spontaneous symmetry breaking. For example, can we force the U(1) symmetric electron field into a non symmetric configuration, and thereby induce an interaction in which charge isn’t conserved?
I’m not sure how you could break the symmetry of a circle, and maybe that’s the point. However, it seems like you could multiply psi by e^i(e^i(theta)), just doing the same transformation twice, but maybe not leaving psi invariant. Anyhow, that is the question: can we, as an experiment, break symmetries on purpose?
Not sure how to formulate this as a question, but it seems like a local gauge symmetry isn’t exactly a symmetry in the traditional sense. When you rearrange your relevant coordinates at different points in space, you must necessarily change the values of the corresponding gauge field in between those points. So a local gauge symmetry transformation changes the configuration of the gauge field. I think of symmetries as transformations that don’t change anything physical. The gauge transformation then seems more like just a dynamical relationship between the matter field and gauge field..?
Why is it that in order to have a mass term in the lagrange density, a gauge particle necessarily has to interact with itself? Or, to put it another way, why does the particle have to appear twice in a term in order for it to be the mass-term of that particle?
Regarding the discussion between Waldo and Numizo:
There is a very useful analogy between gauge theory and finance that helped me a lot to get an intuitive understanding of abstract ideas such as “color rotations”, “connection”, “curvature”.
The gist is as follows:
color of quark at different points in space currencies of different countries.
local SU(3) color rotation currency re-scaling in a particular country (e.g. ,1,000 Pesos becomes 1 Austral in Argentina).
connection exchange rate of currencies between neighboring countries.
curvature ability to make money by exchanging money in a loop (arbitrage) .
All this is explained much better in this freely available paper by Juan Maldacena:
https://arxiv.org/pdf/1410.6753.pdf
Hi doctor Sean! Thanks for the awesome video! Here’s a few questions:
1. I get that we need a new field to transpose the vector… But why that must gave rise to a new particle? I mean, why would we want to transpose the vector in the first place?
2. The spacetime is a field, like the electron field? And graviton is its particle? It feels weird to think that spacetime is just a field like the others…
3. We assume spacetime is a field that all the other fields relate to… But couldn’t we think that the electron field, for example, is the one field that all the others relate to?
4. (This question is one that always bothered me) isnt the folding in spacetime enough to account for gravity and the relativistic effects? Why do we need the graviton?
5. Ontologic speaking, are all those fields really existant or are they just some different manifestation of a greater field? It seems inelegant that nature would make so many different fields, hehehehe!
6. If the electromagnetic force and the weak force are unified in the eletroweak force, why is there a field for the weak force and one to the eletromagnetic force?
Thanks!