I am studying about Lorentz symmetry now, and I’m curious about what does a term ‘symmetry’ exactly means for the field.
Consider a Lorentz transform $x^{\mu}\rightarrow x’^{\mu}=\Lambda^{\mu}_{\ \nu}x^{\nu}$, and there is a scalar field satisfying an equation $F(\phi )=0$. I know that scalar field is always Lorentz-invariant, so which of these conditions represents the invariance condition?
- $\phi’=\phi$
- $\phi’(x’)=\phi(x)$
- $F(\phi )=0 \rightarrow F(\phi’ )=0$
How about vector or tensor field? Also, is there any other difference with gauge symmetry, or etc?
