U(1) rotates 2D complex values. SU(2) rotates real 3D vectors. SU(3) rotates complex 3D vectors. SU(3) combines 2D phase rotations internal to each component with 3D rotations that mix components. What, if anything, makes U(1) × SU(2) distinct from SU(3)?
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What's the connection between these symmetry groups and fundamental particles/fields? I always see people say that U(1) and SU(2) are intrinsic or closely tied to the standard model but I've never been able to understand what they're talking about
Fields are essentially maps from spacetime to some space of values (e.g. the complex plane or R3). The next step is to figure out how they behave. You do that by guessing an action, and extremizing to get the equations of motion (Euler-Lagrange)...
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In guessing the action, it turns out to be really helpful to include symmetries. Some obvious ones are translation and rotation. But you can also have “rotations” in the space of values (which the field gets sent to). And local means you can rotate them differently at different..
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Points in spacetime. By assuming this symmetry you get accompanying gauge fields. In short, the group acts on the “matter field” (e.g. electron field) by rotating its value. The gauge field keeps track of the different rotations at different points. And they describe reality.
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