I keep reading that about 30 years ago we got a good theory that links electromagnetism and the weak force so convincingly that they're called the electroweak force, and somebody got a Nobel Prize for it, but that's about as far as my understanding goes, mumble, mumble, strings, gammon & spinach.
James? I thought we must have something akin to Maxwell's equations or some kind of inverse-square law for the strong and weak forces, but that it was always left out of the popularized explanations that make their way to me. Is that not right? Do we really not know more than that the strong force holds neutrons and protons together and the weak force has something or other to do with the decay of neutrons? Do we not have predictive equations of some kind?
Yes, E/M and the weak interaction are described by the same equations these days. The E/M part (the part that has the massless photon mediating the interactions) is nice and simple. The weak interaction part has a couple of massive particles mediating the interactions, so it is short-range, and there's a technical complication having to do with helicity, but experiment matches the equations very well--the unification works. If you ignore the weak interaction, Maxwell's equations describe E/M very well, and the weak interaction only matters at short distances. So we understand these.
The strong force is stronger, and short range, but appears to have a massless particle mediating it. That doesn't seem to make sense unless there's something else baked into it that keeps it short-range. So we have a theory that has the short-ranged-ness built in, that has a much more complicated symmetry (SU(3), if you're curious--I have no idea how to describe that in terms of everyday rotations), and which is strong.
The relevance of that last bit of emphasis needs some explanation. The equations are typically really really hard to solve exactly. So the usual approach is to use a power series, like the Taylor series. For small angle "x" a sine function is approximately x-x^3/6+x^5/(5!) - x^7/(7!) and so on. If "x" is small enough you can say sin(x) =approx x. If "x" is a little bigger, then x-x^3/(3!). (BTW, I'm using radians not degrees) Check it for yourself. If x=.01, my calculator gives sin(.01)= 0.009999833334, while .01-.01^3/6 = 0.009999833333. Pretty close.
OK, in electroweak theory the "x" in the expansion is small, so the first term gives a fair result, the second terms give a much better one, and the third terms gives values that we don't have experiments accurate enough to distinguish.
However, for the strong force the "x" is large, and the simple expansion doesn't work--it doesn't converge nicely. Try plugging x=2 into that expansion for the sine and you'll get an idea of what I mean. (That formula for sine _will_ eventually converge, but not all expansions do: e.g. 1/(1-x) = 1 + x + x^2 + x^3 + ... : it's only good if |x|<1)
So calculations of the strong force (Quantum ChromoDynamics) are fiendishly hard, and there are a number of clever people trying to figure ways to do them. Hence the "mumble mumble mumble" We're pretty sure it is OK, but ...
On the other hand, we know that the universe at small scales is described with quantum mechanics. Our theory of gravity works like a charm for big objects (though people have proposed "MOND" theories to modify it to avoid having to posit dark matter). But for technical reasons it just isn't quantum-mechanical--the two don't fit smoothly.
That's the big reason for the popularity of string theory--it promised to provide a framework that would allow one to construct a theory that unified gravity and quantum mechanics. Notice that I didn't say that it _was_ such a theory, just that it would provide a framework to make one.
Yours truly would need to spend several years studying to get up to speed on string theory. So I'm perhaps not the best person to evaluate it. But I have a bias towards simplicity ("I don't know, but..."). I'll buy the 21 dimensions if the thing works, but it feels artificial and so far they haven't gotten it to work. I think the paradigm is wrong somehow.
So no, we don't understand gravity--except for large objects.
And we don't know how dark matter interacts either. Except that gravity works...
Sorry, the "mumble mumble" for weak interaction has to do with the technical details, and the "um" for the strong because we have so much trouble actually doing the calculations. I should check back with the original from time to time :-)
I have always loved that the simpler equations are tied to the more easily explained forces, and yet those more easily explain forces are the ones that are ultimately less well understood.
I'm still waiting for the discovery of magnetic monopoles and gravitons.
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I keep reading that about 30 years ago we got a good theory that links electromagnetism and the weak force so convincingly that they're called the electroweak force, and somebody got a Nobel Prize for it, but that's about as far as my understanding goes, mumble, mumble, strings, gammon & spinach.
James? I thought we must have something akin to Maxwell's equations or some kind of inverse-square law for the strong and weak forces, but that it was always left out of the popularized explanations that make their way to me. Is that not right? Do we really not know more than that the strong force holds neutrons and protons together and the weak force has something or other to do with the decay of neutrons? Do we not have predictive equations of some kind?
I do love that XKCD.
It's a great strip, yes.
Yes, E/M and the weak interaction are described by the same equations these days. The E/M part (the part that has the massless photon mediating the interactions) is nice and simple. The weak interaction part has a couple of massive particles mediating the interactions, so it is short-range, and there's a technical complication having to do with helicity, but experiment matches the equations very well--the unification works. If you ignore the weak interaction, Maxwell's equations describe E/M very well, and the weak interaction only matters at short distances. So we understand these.
The strong force is stronger, and short range, but appears to have a massless particle mediating it. That doesn't seem to make sense unless there's something else baked into it that keeps it short-range. So we have a theory that has the short-ranged-ness built in, that has a much more complicated symmetry (SU(3), if you're curious--I have no idea how to describe that in terms of everyday rotations), and which is strong.
The relevance of that last bit of emphasis needs some explanation. The equations are typically really really hard to solve exactly. So the usual approach is to use a power series, like the Taylor series. For small angle "x" a sine function is approximately x-x^3/6+x^5/(5!) - x^7/(7!) and so on. If "x" is small enough you can say sin(x) =approx x. If "x" is a little bigger, then x-x^3/(3!). (BTW, I'm using radians not degrees) Check it for yourself. If x=.01, my calculator gives sin(.01)= 0.009999833334, while .01-.01^3/6 = 0.009999833333. Pretty close.
OK, in electroweak theory the "x" in the expansion is small, so the first term gives a fair result, the second terms give a much better one, and the third terms gives values that we don't have experiments accurate enough to distinguish.
However, for the strong force the "x" is large, and the simple expansion doesn't work--it doesn't converge nicely. Try plugging x=2 into that expansion for the sine and you'll get an idea of what I mean. (That formula for sine _will_ eventually converge, but not all expansions do: e.g. 1/(1-x) = 1 + x + x^2 + x^3 + ... : it's only good if |x|<1)
So calculations of the strong force (Quantum ChromoDynamics) are fiendishly hard, and there are a number of clever people trying to figure ways to do them. Hence the "mumble mumble mumble" We're pretty sure it is OK, but ...
On the other hand, we know that the universe at small scales is described with quantum mechanics. Our theory of gravity works like a charm for big objects (though people have proposed "MOND" theories to modify it to avoid having to posit dark matter). But for technical reasons it just isn't quantum-mechanical--the two don't fit smoothly.
That's the big reason for the popularity of string theory--it promised to provide a framework that would allow one to construct a theory that unified gravity and quantum mechanics. Notice that I didn't say that it _was_ such a theory, just that it would provide a framework to make one.
Yours truly would need to spend several years studying to get up to speed on string theory. So I'm perhaps not the best person to evaluate it. But I have a bias towards simplicity ("I don't know, but..."). I'll buy the 21 dimensions if the thing works, but it feels artificial and so far they haven't gotten it to work. I think the paradigm is wrong somehow.
So no, we don't understand gravity--except for large objects.
And we don't know how dark matter interacts either. Except that gravity works...
Sorry, the "mumble mumble" for weak interaction has to do with the technical details, and the "um" for the strong because we have so much trouble actually doing the calculations. I should check back with the original from time to time :-)
I have always loved that the simpler equations are tied to the more easily explained forces, and yet those more easily explain forces are the ones that are ultimately less well understood.
I'm still waiting for the discovery of magnetic monopoles and gravitons.
We're still looking for monopoles. So far just "limits."
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