As we saw previously, a wave solution is possible - the dispersion relation for the wave simply becomes

\(E^2 = p^2 c^c + m^2 c^4 + g_0 \hbar^2 c^2\).

Now comes the fun part - why bother having a mass at all? If the mass and \(g_0\) are both constants, then anything I can do with a mass term I can do with \(g_0\). This is the essence of the Higgs mechanism: I can generate a mass for particles by coupling the particle to this extra "Higgs field" and setting the empty space value of the field to some constant \(g_0\). The different masses for photons, electrons, muons, etc. are determined by how strongly each particle/field couples to the Higgs field, which allows each species of particle to essentially feel the effect of a non-zero \(g_0\) differently and hence have different masses.

How does one generate a non-zero \(g_0\)? This is done via the dynamics of the Higgs field itself. The Higgs field interacts with itself, but how it interacts is dependent on who you talk to (it's a matter of debate). In one common example, the Higgs field satisfies an equation of the form

\( (\frac{1}{c^2} \frac {\partial^2}{\partial t^2} - \frac {\partial^2}{\partial x^2} - A + B g^2(t,x)) g(t,x)=0\)

where the last two terms are the result of the Higgs field interacting with itself. The lowest energy solution to this equation is when the Higgs field is a constant \(g_0\) everywhere, and as we know, nature likes to settle into states of lowest energy (hence this constant value is the natural value we expect the Higgs field to be in in our universe). If the Higgs field is such a \(g_0\) everywhere, what is \(g_0\) in terms of A and B?

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