Question

Quantum Mechanics: Breaking SO(3) symmetry.

Probably a dumb question, but I'm stuck. I'm looking for input on how to expand a state, in matrix form, around a minimum. (More in comment).

Answer 1

\[State:\left(\begin{matrix}\phi_1 \\ \phi_2\\ \phi_3\end{matrix}\right)\]
\[Minimum: \left(\begin{matrix} \phi_o \\ 0 \\ 0\end{matrix}\right)\]
Basically, how would I go about expanding the state around the minimum? Once I do that, I should be able to break the symmetry and find the resulting interactions of the hamiltonian on my own.

It's probably just the change in notation that's confusing me, but any advice is greatly appreciated.

Answer 2

The lagrangian density in question is:
\[l = \delta_\mu \phi^* \delta^\mu \phi - \frac{m^2}{2 \phi_o^2}[\phi^* \phi - \phi_o^2]^2\]
Where:
\[\phi_o = Constant\]
In case that's helpful.

Answer 3

Figured it out, I think :)