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## Main Question or Discussion Point

Hello! I am going to talk here about a real scalar field for simplicity. If we have a Higgs-like potential for a real scalar field, the graph of the potential looks like a section of a "Mexican hat", with a bump at 0 and two absolute minima at, say, ##\pm a##. This is the plot I see in any book talking about spontaneous symmetry breaking. Although I understand how the symmetry is broken I am a bit confused about the graph itself (the actual picture of it). The graph has on the y-axis the potential ##V(\phi)## and on the x-axis the field ##\phi##. I am not sure how to think about the x-axis, as the field is not an independent variable, it depends on the space-time variables i.e. ##\phi = \phi(x,y,z,t)##. So if I read on the graph that ##V(a)## is a minimum, this means means that when the value of the field is ##\phi(x,y,z,t)=a##, the potential reaches a minimum. But what does it mean that ##\phi(x,y,z,t)=a##? The field doesn't take a single value. One can have ##\phi(2,7,\pi,12.4)=a## but ##\phi(e,22,0,12.4)=7*a+25##. Do we assume for this plot that the field is constant everywhere? And if so, why do we do this? The field has a kinetic term, so it can have change in time. Do we do it because we want to see the actual minimum value of the vacuum and hence, as we need minimum energy, we set it constant just to discard the kinetic term? Thank you!