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The Lagrangian for pairs of fermions $\psi_+$ and $\psi_-$ is $${\cal L}~=~i(\gamma^\mu\partial_\mu\bar\psi\gamma^\nu\partial_\nu\psi|_+~+~\gamma^\mu\partial_\mu\bar\psi\gamma^\nu\psi|_-)~-~m(\psi_+\psi_-~+~\bar\psi_+\bar\psi_-)$$ This is a form of the Dirac equation. We now assume that this mass is due to the Higgs field. The mechanism of symmetry ...

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1) NO 2) The usual way of doing it is to first solve the instanton solution in euclidean time, which is equivalent to obtaining soliton solution of a given potential. Since you have read the book, I am not going to explain how it is done for this case. Then, plugin your instanton solution to the euclidean action and evaluate it. Since $$<q_f|e^{-iHt/\... 1 There is a typo in Rajaraman's book. The correct expression for the \Delta time delay should be$$ \frac{\Delta}{2}=\frac{\sqrt{1-u^2}}{u}\ln(u) $$You can check this in Rajaraman's 1975 review article (R. Rajaraman; Physics Reports, Volume 21, Issue 5, p. 227-313. (1975)). With this, the soliton-antisoliton solution can be written as$$ \displaystyle\...

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I suspect you took an unfortunate left turn. Your hamiltonian density is fine, and for a stationary solution it is just the Bogomol'nyi trick,  {\cal H} = \tfrac{1}{2}\left ( (\partial_x \phi)^2 + \frac{4a}{b} \sin ^2 \frac{b \phi}{2} \right )= \tfrac{1}{2}\left (\partial_x \phi-2\sqrt{\frac{a}{b}}\sin \frac{b\phi}{2}\right )^2 - \frac{4a^{1/2}}{b^{3/2}}\...

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Since you are just asking about language, I will start the exercise for you but will not solve it to the complete, elegant, punchline. You are probably encouraged to read up on coherent states and their displacement operators. To start with, dimensionally, your extra interaction term makes no sense, and you need to normalize x in your extra term to yield a ...

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