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
Would it be possible (at this point) to say something about how the electric and weak fields were combined?
Also what must be true if there is a GUT ?
Brilliant set of talks.
Clarifying the definition of “gauge” in railroads: this is the distance BETWEEN parallel tracks, not the distance ALONG a trackway. It doesn’t much matter what gauge a railway system uses, as long as the wheel-spacing gauge in the cars matches the gauge of the tracks.
I feel like the last lectures and this one, on Symmetry and Gauge, need more time. Please consider doing a fork/expansion from that point. I say that because both felt rushed given the volume and depth of the subject matter and it was clear there was more you wanted to say. By itself, the entire topic of what the SU rotations can really “mean” in each case for the different domains deserves its own lecture. I would love for you to expound upon the notion of “curvature” in each symmetry case as it relates to the i dimension – does the imaginary plane do the same qualitative “work” in 3/2/1 or are there variation/flavors/shades? There is just so much juice to squeeze here!
Thank you for doing this series. It is the a most beautiful kindness. You make my mind happy.
I agree Alec. Each one of these “segments” could be expanded upon. My comment was that he should take a week off! But your comment is more to the point. Perhaps the Q&A will show how deep each area is. But, what a wonderful talk!
If I could rephrase my earlier question: What quantities exactly need to be invariant (/covariant?) under a gauge transformation? It seems like the matter field itself and the gauge field will both be transformed. But the connection derived from the gauge field is invariant? (actually covariant?) And therefore the energy in the gauge field is invariant? Or is it just the total action of both which needs to be invariant?
Is this equivalent to asking what the ‘real’ or ‘observable’ quantities are?
Thanks!
Joao Victor Sant Anna Silva,
Good questions. I am not a physicist, but let me “think aloud” about them:
1. We are trying to develop a theory of *physical dynamics*, so in order to say things like “travel at constant velocity” we need to compare (velocity) vectors along that path/trajectory. Otherwise, you just have a bunch of vectors parameterized by time and zero ability to talk about how those vectors relate to each other, i.e. how they evolve over time etc.
2. Agreed. I believe the term “spacetime” is getting overloaded. Technically, a field is just some function on a manifold (i.e. the underlying points of space) which attaches some object (e.g. a scalar, vector, spinor, etc.) to every point in the manifold. However, in order to describe the dynamics of gravity General Relativity needs the metric which happens to be a field, so perhaps the “field of spacetime” really means the “metric field of GR”.
3. So all our fields still exist “relative to” the underlying manifold. It sounds like the metric field just happens to be special.
4. General Relativity seems to work just fine as a classical (i.e. non-quantum) theory. However, GR also has fields. Since we have gotten so much profit out of quantizing every other field, why not go ahead and see what happens when we quantize the metric? I am guessing that this is where the graviton comes from. Quantizing gravity in this way seems like a pure extension/generalization of GR, and once you have a quantum theory of gravity then you can start doing calculations about how the quantized metric field (i.e. “gravity”) interacts with all the other quantized fields lying around.
5. This is something that popped into my mind as well. For n fields, it’s a trivial matter to wrapp them together with an n-tuple. Explicitly, given n fields f_i : M -> B_i, we can form a “greater field” f : M -> B where B = B_1 X B_2 X … X B_n by taking f : x |-> (f_1(x),f_2(x),…,f_n(x)). However, unless we impose some further structure on B, I’m afraid this probably doen’t really do much for us, physical theory-wise.
6. Looking at the Wikipedia page on the Standard Model, it looks like SU(3) corresponds to QCD but U(1) X SU(2) together correspond to electroweak interactions. The Lagrangian given for the Electroweak Sector claims that U(1) corresponds to something called “weak hypercharge” and SU(2) corresponds to something called “isospin”. So, apparently, the underlying fields are a bit more exotic, with the electromagnetic and weak fields some derived entities.
I have a question from the beginning of the eposide where we talk about the quark “color” gauge symmetry. I have a problem understanding, if the quark is invariant under the rotations of the r,g,b axes then why even assume those axes “are there” at all? If their orientation at any point doesn’t matter than it seem’s a little pointless for the “colors” to even be there at all. What is the physical underlying meaning of the color?
Passing my quarantine in the pampas of Argentina, this is amazing. Thank you so much for your generosity in putting this together. This video in particular was incredibly helpful in pulling together what is going on. A couple of questions, If the strong coupling constant decreases at higher energies, and I think the weak coupling also does, and EM fine structure constant increases, is there an energy where they are the same? Is that a point of unification of forces or further symmetry? Is this where electroweak unification occurs?
A second question is does the extra degree of freedom in the “inner spaces”, the phase for the electron, the SU3 color position, carry any real information about the particles state, or is it only the differences from other particle states that matter.
So we take our incredibly tiny tweezers and pull two quarks apart. Why does that energy form as a quark/anti-quark pair and not a Z or W boson?
Dr. Carroll,
This series, and this video in particular, have been the only explanation of the Standard Model that successfully communicates its *beauty*. Thank you so much for your time and efforts reaching out to the public in this way.
Question: Not sure I follow the discussion about quark confinement. In particular, we have a decreasing relationship between the structure constant, \alpha_s, and the energy cutoff, E_\star. I believe you mention that this implies QCD confinement; however, I fail to see the connection with the rubber band analogy. Would you mind going into a bit more detail for us?
Thanks to Nomizu for attempting to explain my question. I am starting to understand. The one thing I still do not understand is when would you perform a transformation? If axis are arbitrary, when would you be prompted when working with fields to perform a rotation/other transformation and to what target space would you transform to? I think what I am missing is an example. I am used to be taught a concept followed by a physical example in my undergrad physics classes, so the lack of seeing when a transformation is used is perhaps holding me back.
In the QA video at 43:54 I got a little confused. If the potential is Phi = e^(-mr) and the force is a derivative dPhi/dr. Then isn’t the derivation F = dPhi/dr = e^(-mr)*(-m) = -me^(-mr)? How does the 1/r factor come into the equation?
Also please do not stop giving even more than more information than we need. Thank you for all the (lectures) videos!
Arrgh, I got it wrong again! It’s not that the derivative is different from the potential, it’s that the potential itself actually is e^(-mr)/r. Thanks for catching this, Tom.
In order to supplement some of the concepts here, is anyone familiar with the three volume set of books by Feynman (https://www.feynmanlectures.caltech.edu/)? Is it any good and would anyone recommend it? I just looked at one of my undergrad physics book (Halliday, Resnick and Walkers 1995 edition) and there is no mention of Hamiltonians or Principle of least Action or Lagrangians or topology, making it a bit harder to remember all the new core concepts needed to keep up.
Hi Sean, I’m a (semi-retired) computer programmer by profession but now mainly play with fractals and fractal art. My math ended at A level as my cousin put me off doing a math degree, he did a Math degree at Oxford and told me (when I was deciding what to do at Uni) that math at degree level was more like philosophy – not something I was enamoured of at the time. Consequently I have a whole lot of higher math missing, at least formally…..especially newer stuff like group theory 😉
One of the things I’ve been trying to do relating to fractals is investigate higher dimensional math such as hypercomplex/bicomplex/quaternions etc. hopefully finding something >2 dimensions that constitutes a true ID zero mathematical field i.e. commutative (unlike quaternions) and with division algebra (unlike bicomplex) etc.
So far no luck but one of the forms I’ve come up with I’ve called “Multiplex” which obeys most requirements for an ID zero field and interestingly produces proper 4D+ fractals with bulbs (like the White/Nylander Mandelbulb) and is infinitely extensible to any 2^n dimensions.
It’s incredibly simple to describe, using values a, b, c…. etc as complex numbers and a as (ax,ay) where ay is imaginary then the form is as below:
ax, ay (2D standard complex)
bx,by cx, cy (2 complex unit vectors)
In cartesian the 4d is (ax*bx, ax*by, -ay*cy, ay*cx)
Add a new row of 4 complex unit vectors to give 8D cartesian etc.
My thoughts only:
1. If at some point in such a tree a complex is entirely real or entirely imaginary then the branch below the null part still exists but is no longer valued in the parent….gives wave/particle duality?
2. The binary tree form makes entanglement obvious.
3. Maybe dimension/location are interchangeable (imagine the multiplex tree is infinite and the nodes are “points” of existence)