#### ChrisVer

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If we have the lagrangian:

[itex]L= (D_{μ}Φ)(D^{μ}Φ^{*}) - \frac{1}{4} F_{μν}F^{μν}+ V(|Φ|^{2}) [/itex]

with V being the Higg's Potential we can try to put as a solution of the dynamic system the relation:

[itex]Φ(x)= ρ(r) e^{iθn2π} [/itex]

This solution causes "problems" in the vacuum, because each winding of Φ stores up some energy.

One question I have about it, is why is this the case? I understand that in the covariant derivatives, terms of [itex]∇_{θ}[/itex] will give additional energy proportional to the winding number n. But isn't the Φ field U(1) invariant? So I can make U(1) transformations which will cancel out the exponential factor giving me the Goldstone boson dof as longitudial dof of the gauge bosons?

Also, any good book from which I can look up for Domain walls, these cosmic strings and GUT monoples would be appreciated :)

Thanks