General relativity as a gauge theory of the Poincaré algebra

Question

Let the Poincaré algebra be given without any factors of i as

$[P_\mu,P_\nu]=0$,

$[M_{\rho \sigma},P_\mu]=\eta_{\sigma\mu}P_\rho-\eta_{\rho\mu}P_\sigma$,

$[M_{\mu\nu},M_{\rho\sigma}]=\eta_{\nu\rho}M_{\mu\sigma}+\eta_{\mu\sigma}M_{\nu\rho}-\eta_{\mu\rho}M_{\nu\sigma}-\eta_{\mu\sigma}M_{\nu\rho}$,

where $P$ are the generators of the translational symmetries of the Poincaré group, $M$ are the generators of the Lorentz rotations and boosts, and $\eta$ is the spacetime metric 1.

The author of this thesis 2 writes on pages 20 and 21, that infinitesimal transformations take the form,

$\delta \bullet = \xi^A P_A$

where recall that the $P_A$ translation generator is given by $\partial_\mu$. This holds for the bullet denoting an arbitrary field.

When discussing the first step in gauging the Poincare algebra — assigning a gauge field to each generator — the author states that the transformation rules for these gauge fields can be derived from the above expression and the structure constants of the algebra.

From here he states that we can "define a connection $A_\mu$ that takes values in the adjoint of the gauge group."

$A_\mu = E_\mu^A P_A +\frac{1}{2} \Omega_\mu{}^{AB} M_{AB}$

where $E_\mu^A$ and $\Omega_\mu{}^{AB}$ will eventually be realized as the vielbein and spin connection.

Where did this come from? Can anyone provide me some more clear motivation for constructing such an object? My only idea is comparing it to the following expression for the global Poincaré transformation acting on a scalar field,

$\delta(a,\lambda)\phi(x)=[a^\mu \partial_\mu - \frac{1}{2} \lambda^{\mu\nu}M_{\mu\nu}]\phi(x)=[a^\mu P_\mu - \frac{1}{2} \lambda^{\mu\nu}M_{\mu\nu}]\phi(x)$

where it seems the construction of $A_\mu$ was done by considering the "vielbein and spin connection" as parameters for the Poincaré and Lorentz symmetry transformations.

I have addition questions after this is resolved, but I will either edit the question later to include more, or post another question.

1 D. Freedman and A. Van Proeyen, Supergravity, Textbook (2012).

2 T. Zojer, Non-relativistic supergravity in three space-time dimensions, Van Swinderen Institute for Particle Physics and Gravity at Rijksuniversiteit Groningen, PhD Thesis (2016).

Answer 1

Main idea to introduce gauge field for every generator, is to provide invariance under some group of transformation, in your case under group of diffeomorphism and local Lorentz transformations (local version of global Poincaré group). This logic is very similar to gauge invariance, where we introduce gauge field and covariant derivative to provide invariance under local gauge transformations. But now gauge group is Poincaré group.

1) Gauge symmetry with generators $T^t$:

$$ \nabla_m = \partial_m - i A_m^t T^t $$

Poincaré group with generators $P_a$ and $M_{ab}$:

$$ \nabla_m = \partial_m -i e_m^{\;a}P_a -\frac{i}{2}\omega_m^{\;\;\;cd}M_{cd} $$

2) Now we can directly calculate commutator:

$$ [\nabla_m, \nabla_n] = -i R_{mn}^{\;\;\;a}P_a -\frac{i}{2}R_{mn}^{\;\;\;ab}M_{ab} $$

$R_{mn}^{\;\;\;a}$ is curvature (or field strength), that corresponds to $P_a$, $R_{mn}^{\;\;\;ab}$ curvature for $M_{ab}$. They expressed only in terms $e_m^{\;\;a}$ and $\omega_m^{\;\;cd}$. They will be identified with tetrad and spin connection.

3) Gauge transformations for $e_m^{\;\;a}$ and $\omega_m^{\;\;cd}$ follow from requirement:

$$ \phi^\prime = h \phi $$ $$ (\nabla_m\phi)^\prime = \nabla_m^\prime\phi^\prime = h \nabla_m\phi \;\;\;\;\;\Rightarrow \;\;\;\;\; \nabla_m^\prime = h \nabla_m h^{-1} $$ $$ h = e^{i a^a P_a + i\omega^{ab}M_{ab}} $$ Here $\phi$ is some field, $h$ is element of Poincaré group.

From here you can found laws of transformation for $e_m^{\;\;a}$ and $\omega_m^{\;\;cd}$. From this transformations you can argue, that this fields transform as tetrad and spin connection.

Note, that curvatures transforms homogeneously $R^\prime = hRh^{-1}$.

4) In gravity, spin connection is expressed in terms of tetrad. So we need covariantly constraint geometry: $$ R_{mn}^{\;\;\;a} =0 $$

From this equation one can express $\omega_m^{\;\;\;cd}$ in terms of $e_m^{\;a}$.

One can also check vielbein postulate: $$ \nabla_n e_m^{\;\;a} = 0 $$

5) From tetrads one can construct metric:

$$ g_{mn}(x) = e_m^{\;a}(x) e_n^{\;b}(x) \eta_{ab}$$

Indeces $a,b, \dots$ are Lorentz indeces, and transforms onle under Lorentz transformations.

Indeces $m,n, \dots$ are world indeces, and transforms only under diffeomorphism transformations.

Using tetrad one can convert Lorentz indeces to curved indeces:

$$ \xi_m = e_m^{\;\;a}\xi_a $$

From Riemannian tensor $R_{mn}^{\;\;\;ab}$ one can construct standard gravity actions.

Please, Ask questions, I will try to clarify answer..!