It is known that if we use the constraint $R_{\mu\nu}(P^{a})=0$ , i.e. the curvature of local translations vanishes, then we can modify general coordinate transformations (gct) to a covariant generalisation thereof (cgct) in such a way that these cgct include local translations. Rephrasing this, local translations can be thought of covariant diffeos.
If
a. LOCAL Lorentz covariance corresponds to the statement that there is no preferred inertial reference frame on curved space.
b. GLOBAL translation invariance corresponds to the statement that a NON-GRAVITATIONAL theory does not distinguish different points in spacetime, and
c. diffeomorphism invariance corresponds to the statement that the coordinate system chosen does not matter
THEN my questions are
Is it safe to assume that LOCAL translation invariance is just the curved space generalisation of the GLOBAL translation case, such that they still correspond to not having a preferred origin, but in curved manifold parlance? Or do they mean something completely different?
What does it mean to say that LOCAL translations are eaten up by covariant gct? I assume that the theory is still invariant under them, in the sense that there is no preferred origin. But somehow now this statement comes from diffeos?
In a way I have always been confused about this, as I fail to clearly see the difference between diffeo invariance and the "no preferred reference frame" of local Lorentz. Now I am even more confused because diffeo invariance seem to contain both 'no preferred origin' and "no preferred reference frame".
Notation:
$R_{\mu\nu}(P^{a})=\partial_{[\mu}e_{\nu]}^{a}+2\omega_{[\mu}^{ab}e_{\nu]b}-\frac{1}{2}\overline{\psi}_{\mu}\gamma^{a}\psi_{\nu}$
$P^{a}$ is the generator of local translations.
References:
- D.Z. Freedman & A. Van Proeyen, SUGRA, 2012; p. 225
