How could all four fundamental forces be one single force at the beginning of the universe? Did this single force then behave like all four at the same time? How do we know that?
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The main idea behind the unification of fundamental forces is that ultimately each force is a manifestation of the same underlying field^1, they just act differently because they operate under different symmetries. A symmetry in physics is just a transformation on something that leaves it like it was before – a geometric example is rotating a perfect sphere. Sure, you technically rotated it, but it looks identical to how it did before you did so – like a sphere. Another example is if you painted the entire sphere another color. It’s still still has all of looks of a sphere, so nothing of note really changed.
This constrains how much we can change the object though. If changing the sphere’s color somehow made it no longer look like a sphere, we would have another **degree of freedom** that we can play with. It would be a dial we could turn to see real results, rather than turning a dial and seeing the same exact thing. The symmetries that each force operates under each have their own degrees of freedom that allows the forces to act differently. Visually speaking, rotating a sphere 10 degrees changes nothing but rotating a cube 10 degrees changes a lot.
Now to the unifications. At certain energy levels the forces will unify because the different symmetries they obey, which can be represented as **mathematical groups** [(wiki link if interested)](https://en.wikipedia.org/wiki/Group_theory), are really just components of a more general symmetry (i.e. subgroups of this larger group). A group in physics is, in short, just a set of transformations. To return to our sphere, we can have a group to describe all of the possible rotations you can make on it.
Once these energy levels are hit, the subgroup symmetries begin to become less important while the greater symmetry group begins to dominate our descriptions of things. A close analogy is melting an ice cube: when it’s frozen the water molecules are forced into a highly ordered lattice structure but when it melts the molecules can flow freely as a liquid, no longer burdened with needing to stay specific positions. The degrees of freedom for movement increase.
In the same way, the higher energy allows the greater symmetry group to take over and free the forces from their subgroups. New dials to play with become available. But if they all share the same dials, since they’re all operating under the same group, they’re all effectively the same force.
As for how we know, the answer is twofold: one, we have rigorous math that has worked for us before (and so we have no reason to expect to just stop working) and force strength projections telling us they should meet in strength and unify and two, we have physical evidence. Amazingly, we have observed the electroweak force, the merging of the weak nuclear and electromagnetic forces, at CERN before!
Footnote 1: This is an oversimplification but the nitty gritty of it doesn’t matter so much.
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