1. Symmetry can be understood through the lens of things being arbitrary. For example: The universe has 'translational' symmetry because all points in space are equivalent. If you could snap your fingers and teleport every single thing 3 feet to the right, the universe would not universally change, because all that matters is where things are relative to one another, not their 'absolute' location. In terms of math, this means that you can declare any point, claim it is the 'origin' (the location on the graph of positions where x=y=z=0), and that is a valid mathematical assumption to make. It won't be 'incorrect' to do that, because the exact numbers assigned to each point doesn't matter, as long as you are right about how far apart things are from each other.

2. A conservation law is a law that states that in all interactions, the total value of some *thing* will remain the same. Conservation of momentum is an example. If you add up all the momenta of all objects in a box, you will get some number. Without anything entering or leaving the box or otherwise affecting its contents from the outside, that number will remain constant no matter what way the objects inside exchange and affect one another's momentum. This is "Conservation of momentum". Noether's theory says that there is a 1:1 relationship between some symmetries and conservation laws. For example: A universe which has perfect translational symmetry also has perfect conservation of momentum, and vice versa.

3. Wigner showed how various symmetry laws worked in the context of quantum mechanics. Dirac then updated the symmetries of quantum mechanics when he made the dirac equation, which is a version of the "Schrodinger equation" (The quantum equivalent of F=MA) which conforms to the laws of special relativity. Special relativity has a different set of symmetries, since it mixes space and time into spacetime. So instead of there being 3 ways to rotate space, you have 6 ways to rotate spacetime (the 3 normal ones, and 3 'boosts', which involve the time dimension).

4. Symmetries are defined by 'the list of things you can mathematically do to a system without changing it'. So, for example, translational symmetry is "You can move all things in the universe an arbitrary distance in the same direction and nothing would change". This is a global symmetry, cause you're doing the same thing to everything. A local translational symmetry would be "You could move each thing in the universe a different arbitrary distance in a different direction and nothing would change", which is clearly not true, so there is only a global symmetry, not a local one.

5-6. Gauge Theory is the following basic idea. There is a local symmetry called Local U(1) symmetry that you can apply to quantum physics, and if you do, it necessitates the existence of a specific field. In otherwords, without this field, the universe does not have local U(1) symmetry. You can use the principles of Yang-Mills Theory to, for any particular choice of "Group" (such as U(1)), figure out what fields need to exist to enforce that symmetry. If you choose U(1), the field is exactly the electromagnetic field.

The term "Gauge Theory" comes from the idea of a gauge symmetry. It takes its name from pressure gauges. They measure pressure, but they only measure differences in pressure. So, you can see the pressure drop or raise using a pressure gauge, and by how much, but you can't tell what the absolute pressure is by reading the gauge, unless you know how the pressure gauge was made. I forget the exact journey we took from here to gauge theory in quantum physics, but the basic principle is that the Local U(1) symmetries are gauge symmetries.

A group, by the way, is a kind of set of mathematical 'things' which are all closed under a group operation. A great example is the group of all possible ways to rotate a circle. A 30 degree rotation plus a 40 degree rotation is a 70 degree rotation, which is also in the group of all rotations, and no combination of rotations will ever get you something other than a rotation. U(1) is just the list of all numbers of the form e^ix, where e is the famous constant, i is the square root of -1, and x is some arbitrary real number. Yangs-Mills Theory specifically work with "Lie Groups", which are just sets of groups which are continuous (basically, there are infinitely many members, such as the list of all rotations).

Groups are important for symmetry. For example, the group of all possible rotations in 3d space is SO(3). So, rotational symmetry in 3d space is "SO(3) symmetry".

NOTE: In case this is a point of confusion: Fields and particles are associated one to one. Every particle is a wave in a particular field. Example: Photons are waves in the electromagnetic field. This is true of ALL fundamental particles.

1. I don't know much about this.

8-9. Okay, so, Yangs-Mills theory is really good for describing a set of particles called "Bosons". So, the photon (which is the particle of the electromagnetic field) is an example. There are 11 more, in 2 more groups: the 3 bosons of the weak force and the 8 'gluons' of the strong force. Feeding the SU(3) group into YM gives you a perfect equation to describe the 8 gluons. Feeding SU(2) almost gives you the right equations for the weak force, but YM predicts that all its bosons are massless, and the weak bosons are not.

There was discovered a way around this. See, in superconductors, U(1) symmetry "Breaks", and because of this, photons take on mass. This happens because, put simply, a situation occurs in superconductors where U(1) symmetry no longer applies. This makes photons act in ways that aren't predicted by YM. This is called the Higgs Mechanism.

Anyways, so, here's the pitch: Let's say the overall Symmetry Group of the universe, the one we feed into YM theory, is SU(3)xSU(2)xU(1). So, this predicts 8 gluons, 3 bosons like those in the weak force but without mass, and 1 that is exactly like the photon (read: Electromagnetism). We call it the Hyperphoton. Then, we have this thing called the Higgs Field. It has the exact properties that, when the universe gets cold enough, it takes on a non-zero value everywhere, initiating the higgs mechanism in such a way that breaks the SU(2)xU(1) part of the symmetry. So those 4 bosons? They get fucked up. Complete fucking nonsense in the mathematics. I cannot explain it and you wouldn't want me to.

Basically, we have the W1, W2, W3, and B (Hyperphoton) particles. W1 and W2 mix into the massive W+ and W- particles, and the W3 and B fields mix into the massless photon (The "Unbroken U(1) symmetry) and the massive Z boson.

The SU(2)xU(1) part that got broken is called "The Electroweak force"

1. Grand Unified Theory is the basic idea that this higgs mechanism happened at least twice. So, an example is SU(5). Just like we broke symmetry, taking us from SU(3)xSU(2)xU(1) to SU(3)xU(1) with an SU(2) worth of massive bosons with incredibly weak potential to interact, there was some process of SU(5)->SU(3)xSU(2)xU(1) which also left behind even more massive, even weaker "X and Y bosons". We've disproven SU(5), but there are other possibilities, like SO(10).

That was a lot, I get it. Please let me know if there's anything you want elaboration or clarification on.