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This is spontaneous symmetry breaking rather than explicit symmetry breaking. The Lagrangian respects all the symmetries in question, but the vacuum state does not.

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Ryder explains spontaneous symmetry breaking in this book [1]. Just about any book on QFT and particle physics will say something.

 

 

References

 

[1] L. Ryder. "Quantum Field Theory", Cambridge University Press; 2 edition (June 13, 1996).

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No problem. Ryder presents two nice analogies, one mechanical and the other from statistical mechanics. The essential feature of spontaneous symmetry breaking is a degenerate vacuum. The Lagrangian or Hamiltonian describing the system remains invariant under the symmetry in question, the physical system is not invariant as one of the vacua has to be chosen.

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Quite generally the Lagrangian approach works well, especially when quantisation is required. However, one can have theories that are not governed by a Lagrangian and those whose Hamiltonian formalism is not equivalent.

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AJB ...... For the equivalent Hamiltonian Model; I have The Perturbed Dirac's Model ..... have written (down) it since years.

 

[latex]H_{ew}=H_{Dirac}+H_{zo}+H_{w+}+H_{w-}+LT[/latex]

 

Do you have any idea, about this?

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Gauge theories are quite complicated to deal with in the Hamiltonian formalism. Technically we have constraints to deal with, but this can be done.

 

Most particle theorists like to use path integrals and this is best formulated in terms of a Lagrangian.

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But, you will have to apply LAP on the Lagrangian which is very complicated, especially when the Lagrangian is long and unabbreviatable. Are there sufficient approximations and neglectance in Electroweak?

Or, there is simpler manipulation.

 

[Latex] H_{w+}=1/2W^{ij}_{t}W^{k}_{ij} epsi^{t}[/Latex]

Could you confirm, AJB?

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But, you will have to apply LAP on the Lagrangian which is very complicated, especially when the Lagrangian is long and unabbreviatable.

 

What is LAP?

 

[Latex] H_{w+}=1/2W^{ij}_{t}W^{k}_{ij} epsi^{t}[/Latex]

Could you confirm, AJB?

 

I am not really familiar with the Hamiltonian formulation of the electroweak theory. All the references I have use path integrals in the Lagrangian formulation.

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LAP: Least Action Principle.

 

Ok, so you use this to get at the classical equations of motion. These are important in quantum field theory, but as I am sure you now the path integral approach takes into account all configurations.

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