system
Given the following post, respond as if you're commenting on Reddit:
human
How does the Higgs field work with Lorentz transformation, I'll settle for someone explaining it in Galilean transformation.

Wouldn't this field provide an frame of reference that was "better" than another one where the average velocity of the Higgs particle is zero?

I am in my Senior year of a duel degree in physics and engineering. So I know some stuff but I can't wrap my mind around this.
gpt
The Higgs field is a scalar field, meaning at every point in space and time it has a numerical value (as opposed to, say, a vector field, whose value is a vector). That value doesn't change depending on your coordinate system or your velocity - that's what Lorentz invariance means. (An example of a "scalar field" which is *not* Lorentz invariant is the time, since that changes between observers depending on their velocity.)

Lorentz invariance also means that there's no preferred rest frame, physics is the same no matter how you rotate or what your speed is. But, as you notice, if the Higgs varies throughout space and time, it should define a preferred frame and violate Lorentz invariance.

And, in principle, you're right. *And that's totally fine.* The Higgs doesn't actually do that (more on that in a sec), but it could in theory, indeed it has before, and that's no problem. The laws governing the Higgs are still Lorentz invariant (i.e., if the LHC were moving on a spaceship zipping past the Earth, it would have gotten the exact same Higgs results it did here), but the solution doesn't have to be. After all, we're living in a part of the Universe which is very, very different from other parts - stars, planets, government shutdowns, and so on. Definitely not Lorentz invariant at the level of the solution, as we'd say. But the laws of physics still are, so it's alright.

As for the Higgs itself, today it's uniform (on average) everywhere and has been for billions of years, so it doesn't define a preferred frame. Back in the very early Universe it was likely different, because of all the incredibly high energy interactions that were happening, so then it would have had some time evolution and hence a preferred frame (likely the same as the cosmic rest frame which today is the rest frame of the CMB), but that's long since not been the case.