What does off-diagonal term mean?

[physlad]

when we apply Higgs mechanism to the lagrangian of a complex scalar field... we get terms for a Glodstone boson, a massive scalar and a massive vector field in the new lagrangian..

we also get other terms.. one of them takes following form

$$A_\mu \partial^\mu \xi$$

this term is said to be off-diagonal in the fields... what does this mean? and what does it imply?

I mean how can we know just by looking at such a term that our lagrangian does not describe physical fields? and that we need to rewrite the lagranigan in terms of real fields... until we get the lagrangian which contains a massive vector and a massive scalar (Higgs particle)..

 

[azntoon]

The term you mentioned is off-diagonal because it involves two distinct fields, the vector field A_μ and the scalar field ξ. This implies that the Lagrangian does not contain physical fields describing a massive vector and massive scalar (Higgs particle), and must be rewritten in terms of real fields. This is done by introducing a unitary transformation on the scalar fields, which allows us to write the original complex scalar field as a combination of two real scalar fields. The same transformation can then be applied to the vector field, resulting in a massive vector and a massive scalar (the Higgs particle).

 

[Entropia]

The off-diagonal term in this context refers to a term in the Lagrangian that involves two different fields, in this case the vector field A and the Goldstone boson field \xi. This term is called off-diagonal because it does not involve just one field, but a combination of two fields.

The presence of this off-diagonal term implies that the Lagrangian does not describe physical fields in a straightforward way. This is because the Lagrangian should only involve terms that correspond to physical particles that can be observed in experiments. The off-diagonal term indicates that the fields A and \xi are not the physical fields, and that we need to rewrite the Lagrangian in terms of other fields that correspond to observable particles.

In this case, the Higgs mechanism is used to rewrite the Lagrangian in terms of real fields, which correspond to the massive scalar and vector fields that can be observed in experiments. This is why the off-diagonal term is important - it allows us to identify the fields that are not physical and to rewrite the Lagrangian in terms of the physical fields.