Well, my question could be viewed as about the math. ie: “does the math of the standard model have the property that if you removed any explicit mention of electromagnetism, strong force, or weak force and just kept the quark and lepton fields + the math of the symmetries for those, would that be sufficient for it to effectively already contain EM, strong, and weak forces?”
And as far as gravity being force or geometry, uh… there’s plenty of math associated with that. I mean, how would one even begin to talk about the meaning of the Einstein field equation without interpreting it in terms of geometry?
Perhaps there is a deeper underlying principle that gives rise to it, but the Einstein field equation is an equation about how matter shapes the geometry of spacetime. There’s no way really (that I know of) to reasonably interpret it as a force equation, although one can effectively solve it and eventually get behaviors that Newtonian gravity approximates (at low energies/etc...)
(EDIT: to clarify, I’m trying to figure out how to semivisualize this. ie, with gravity and curvature, I can sorta “see” and get the idea of everything’s just moving in geodesics and the behavior of stuff is due to how matter affects the geometry. (though I still can only semi “grasp” what precisely G is. I get the idea of curvature (the R tensor), I get the idea of metric, but the I currently only have a semigrasp on what G actually means. (Although I think I now have a bit of a better notion than I used to). Anyways, loosely similar, am trying to understand if the fundamental forces arise similarly, rather than being “forces”, they’re more an effect of what sorts of symmetries there are, what bits of configuration space count as equivalent to other bits, etc...)
I guess I’m not enough of a theorist to answer your question: I do not know whether the symmetries alone are sufficient to produce the observed particles. My intuition says not, for the following reason: First, SU(3) symmetry is broken in the quarks; second, the Standard Model contains parameters which must be hand-tuned, including the electromagnetic/weak separation phase that gives you the massless photon and the very massive weak-force carriers. Theories which spring purely from symmetry ought not to behave like that! But this is hand-waving.
As an aside, I seem to recall that GR does not produce our universe from symmetries alone, either; there are many solutions to the equations, and you have to figure out from observation which one you’re in.
If you like, I can quote our exchange and ask some local theorists if they’d like to comment?
But GR explains (or explains away, depending on how you look at it) the force of gravity in terms of geometry. I meant “does the standard model do something similar with the gauge bosons via symmetry?”
May still leave some tunable parameters, not sure. But does the basic structure of the interactions pop straight out of the symmetries?
And yeah, I’d like that, thanks. It’s nothing urgent, just am unclear if I have the basic idea or if I have severe misconceptions.
The basic fact about quantum field theory is field-particle duality. Quantum field states can be thought of either as a wavefunction over classical field configuration space, or as a wavefunction over a space of multi-particle states. You can build the particle states out of the field states (out of energy levels of the Fourier modes), so the field description is probably fundamental. But whenever there is a quantum field, there are also particles, the quanta of the field.
In classical general relativity, particles follow geodesics, they are guided by the local curvature of space. This geometry is actually an objective, coordinate-independent property of space, though the way you represent it (e.g. the metric you use) will depend on the coordinate system. Something similar applies to the gauge fields which produce all the other forces in the Standard Model. Geometrically, they are “connections” describing “parallel transport” properties, and these connections are not solely an artefact of a coordinate system. See first paragraph here.
You will see it said that the equations of motion in a gauge field theory are obtained by taking a global symmetry and making it local. These global symmetries apply to the matter particle (which is usually a complex-valued vector): if the value of the matter vector is transformed in the same way at every point (e.g. multiplied by a unit complex number), it makes no difference to the equation of motion of the “free field”, the field not yet interacting with anything. Introducing a connection field allows you to compare different transformations at different points (though the comparison is path-dependent, depending on the path between them that you take), so now you can leave one particle’s state vector unmodified, and transform a distant particle’s vector however you want, and so long as the intervening gauge connection transforms in a compensatory fashion, you will be talking about the same physical situation. However, as the link above states, the gauge connection is not solely a bookkeeping device; there are topologically distinct gauge field states which are not equivalent under some continuously varying transformation.
It all sounds horrifically abstract, but if you follow the link above you’ll see some simple examples which may help. Anyway, the bottom line is that classical gauge field configurations do contain a coordinate-independent geometric content just as gravity does, so you can’t completely do without them; and whenever you quantize a field, you have particles.
Well, my question could be viewed as about the math. ie: “does the math of the standard model have the property that if you removed any explicit mention of electromagnetism, strong force, or weak force and just kept the quark and lepton fields + the math of the symmetries for those, would that be sufficient for it to effectively already contain EM, strong, and weak forces?”
And as far as gravity being force or geometry, uh… there’s plenty of math associated with that. I mean, how would one even begin to talk about the meaning of the Einstein field equation without interpreting it in terms of geometry?
Perhaps there is a deeper underlying principle that gives rise to it, but the Einstein field equation is an equation about how matter shapes the geometry of spacetime. There’s no way really (that I know of) to reasonably interpret it as a force equation, although one can effectively solve it and eventually get behaviors that Newtonian gravity approximates (at low energies/etc...)
(EDIT: to clarify, I’m trying to figure out how to semivisualize this. ie, with gravity and curvature, I can sorta “see” and get the idea of everything’s just moving in geodesics and the behavior of stuff is due to how matter affects the geometry. (though I still can only semi “grasp” what precisely G is. I get the idea of curvature (the R tensor), I get the idea of metric, but the I currently only have a semigrasp on what G actually means. (Although I think I now have a bit of a better notion than I used to). Anyways, loosely similar, am trying to understand if the fundamental forces arise similarly, rather than being “forces”, they’re more an effect of what sorts of symmetries there are, what bits of configuration space count as equivalent to other bits, etc...)
I guess I’m not enough of a theorist to answer your question: I do not know whether the symmetries alone are sufficient to produce the observed particles. My intuition says not, for the following reason: First, SU(3) symmetry is broken in the quarks; second, the Standard Model contains parameters which must be hand-tuned, including the electromagnetic/weak separation phase that gives you the massless photon and the very massive weak-force carriers. Theories which spring purely from symmetry ought not to behave like that! But this is hand-waving.
As an aside, I seem to recall that GR does not produce our universe from symmetries alone, either; there are many solutions to the equations, and you have to figure out from observation which one you’re in.
If you like, I can quote our exchange and ask some local theorists if they’d like to comment?
But GR explains (or explains away, depending on how you look at it) the force of gravity in terms of geometry. I meant “does the standard model do something similar with the gauge bosons via symmetry?”
May still leave some tunable parameters, not sure. But does the basic structure of the interactions pop straight out of the symmetries?
And yeah, I’d like that, thanks. It’s nothing urgent, just am unclear if I have the basic idea or if I have severe misconceptions.
It is a while since I thought about this. But…
The basic fact about quantum field theory is field-particle duality. Quantum field states can be thought of either as a wavefunction over classical field configuration space, or as a wavefunction over a space of multi-particle states. You can build the particle states out of the field states (out of energy levels of the Fourier modes), so the field description is probably fundamental. But whenever there is a quantum field, there are also particles, the quanta of the field.
In classical general relativity, particles follow geodesics, they are guided by the local curvature of space. This geometry is actually an objective, coordinate-independent property of space, though the way you represent it (e.g. the metric you use) will depend on the coordinate system. Something similar applies to the gauge fields which produce all the other forces in the Standard Model. Geometrically, they are “connections” describing “parallel transport” properties, and these connections are not solely an artefact of a coordinate system. See first paragraph here.
You will see it said that the equations of motion in a gauge field theory are obtained by taking a global symmetry and making it local. These global symmetries apply to the matter particle (which is usually a complex-valued vector): if the value of the matter vector is transformed in the same way at every point (e.g. multiplied by a unit complex number), it makes no difference to the equation of motion of the “free field”, the field not yet interacting with anything. Introducing a connection field allows you to compare different transformations at different points (though the comparison is path-dependent, depending on the path between them that you take), so now you can leave one particle’s state vector unmodified, and transform a distant particle’s vector however you want, and so long as the intervening gauge connection transforms in a compensatory fashion, you will be talking about the same physical situation. However, as the link above states, the gauge connection is not solely a bookkeeping device; there are topologically distinct gauge field states which are not equivalent under some continuously varying transformation.
It all sounds horrifically abstract, but if you follow the link above you’ll see some simple examples which may help. Anyway, the bottom line is that classical gauge field configurations do contain a coordinate-independent geometric content just as gravity does, so you can’t completely do without them; and whenever you quantize a field, you have particles.