The thing I was looking at recently was talking about hadrons. The weird thing about hadrons is that they are defined by their quarks, but quarks are fermion-sized dots while hadrons are big lumbering planetesimals by comparison. How can 3 point-like particles form a huge object like a proton?
The theory is that the gluons, strong-force bosons that bind the quarks together, carve out a hole in spacetime that accounts for the mass of the hadron. But, beyond just that, inside the region of the hadron, virtual quark/antiquark pairs are being created by the gluon interaction and then destroying each other in short order.
Not only is empty space a very busy place, but non-empty space is as well. Protons and neutrons are not just sitting there (flying through space, because nothing is motionless in this here universe) but are a constant seething roil of activity.
Noether’s theorem is connected to charge conservation from these symmetries, yes.
But the big value is how the symmetries relate to interactions happening at all, and then separately the need for anomaly cancellation leading to connections between the charges.
The Standard Model is an example of a “gauge theory” in which a fundamentally interaction-free theory gains interactions through the insistence that the theory respect a particular gauge symmetry. In brief, it goes like this.
Imagine a simple quantum field theory that has one type of particle – represented by fields – with no interactions between fields. Fields in QFT are necessarily “complex” valued, meaning the value of a field at any point in space is a complex number* of the form a+ib, or more tidily Ae^{i\theta}. If this notation is completely foreign, the rest of this post will be a challenge.
The only physics that can take place in this theory is whatever a simple mass and its motion (kinetic energy) allows, i.e. not much interesting. But you can write out the math of this theory all the same.
In quantum mechanics, observable quantities can be calculated in a theory by taking combinations of three things and integrating the result in some fashion: thing 1 = a field (or a wavefunction in non-relativistic QM), thing 2 = the complex conjugate (i.e., taking i\rightarrow -i) of another field (possibly the same field), and thing 3 = the relevant measurement operator, usually involving derivatives of one of the fields.
If you arbitrarily decide to multiply every field in the theory by, say, -1, no observable quantities will change because observables always involve the product of two fields, as above, and no measurement operator can prevent that cancellation of negatives. This is an example of a symmetry of the system, but not a very interesting one.
If you arbitrarily decide to multiply every field in the theory by e^{i\alpha} for some constant \alpha, the measurements still are unaffected since the first field gains e^{i\alpha}, the second complex conjugated field gains e^{-i\alpha}, and the product of these two expressions is also exactly 1. Another symmetry, and it remains not too interesting. This is called a global gauge symmetry (“global” since the field transformation we are applying is the same everywhere since \alpha is constant through all of space and time.)
If you decide to multiply every field in the theory by e^{i\alpha(x)} for some function \alpha(x) that is not the same everywhere in space (and maybe time), then now the operators which involve derivatives of one of the fields will pick up extra factors that have no counterpart from the other field to compensate. This transformation is apparently not a symmetry of the system.
If you insist on making this local gauge transformation a good symmetry of the theory anyway, you need to do something else to compensate for the extra pieces that show up. One introduces a new type of derivative that can handle this type of transformation, and that new type of derivative happens to look the same as if you added a brand new field to the theory that has certain key properties.
Among these properties are that the field transforms in a way that is tied to the local gauge transformation (a very special property indeed); that the new field is massless; and that the new field leads to interactions in the theory mediated by this new gauge field, i.e. interactions between the original fields we started with. This first property is an input, while the second and third properties are consequences.
Bringing it back around: So, the symmetries we’re talking about are these local gauge symmetries. They involve the presence of special “gauge fields”. The photon is an example of a gauge field. Yes, these are rather more esoteric than ordinary spatial symmetries.
The local transformation above (e^{i\alpha(x)}) is the simplest gauge transformation you can do, and it happens to be of the form that leads to quantum electrodynamics. More complicated versions with higher-dimensional sorts of complex number “rotations” lead to the other forces.
And then if you want all these seemingly separate gauge symmetries to play nice together, you find that the charges across all the particles in the full model have to be related in a very specific way, as noted in the previous post.
* It doesn’t have to be a simple number but could be a vector, etc., but it will always be a complex-valued field.
Note that this “weirdness” is not special to hadrons. A simple hydrogen atom – a proton nucleus with its orbiting electron – is about one angstrom (10-10 m) in size, yet that lumbering size has nothing to do with the “size” of a single electron, which is as point-like as your quarks. The space-taking nature of the electron bound by electrical forces to the nucleus is the same as the space-taking nature of the quarks bound together by strong forces to form that nucleus.
Honestly, the majority of you (any given solid object) is electrical fields. The object (you) appears to be a vast amount of “empty” space, but the bulk of the object is a binding of electrical fields.
its very unattainability is the reason why it’s the best…
Until you try to put it all back into the closet you just took it out of.
Stuff is like a virus. It keeps breeding more stuff, space tries to get out of its way, and so expands the universe. Now if only my car could do that.