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I will briefly explain my understanding on the subject.

In the following explanation i refer to the Poincarè group meaning the group:

$$\mathcal{P}_{1,3} = \mathbb{R}^{1,3} \rtimes Spin^+(1,3)$$

The standard model

To build the standard model we need 2 main ingredients:

  • a global Poincarè group to build our fields as representations of such a group, scalars vectors 2-tensors spinors and all that.
  • a gauge group, namely $SU(3)_C×SU(2)_L×U(1)_Y$, needed to obtain the interactions between the fields and have the charges.

I know that we also have a Higgs field which is not associated to a gauge group but is necessary to break the $SU(2)_L×U(1)_Y$ part and gain the masses through the Yukawa couplings.

We then build the fields using representations of a global Poincarè and the local gauge groups and we build the usual standard model action with kinetic terms (with covariant derivatives), theta terms, Yukawa terms and the Higgs potential.


Gravity in a gauge theory form

The classical General Relativity is built on two key arguments:

  1. Locally it is always possible to choose a locally inertial frame (LIF)

  2. The connection is the torsion free one$^1$

From such assumptions you can built the usual gravitational action and the coupling to matter via the covariant derivative and the metric.

It is also possible to build gravity as a gauge theory.

Taking as gauge group the Poincarè group, you build a connection on the principal bundle which is composed of a translational gauge field, the tetrad field $e$ and a rotational gauge field, the spin connection $\omega$.

Each of this two fields has its own curvature tensor: $R^{(e)}$ and $R^{(\omega)}$. Imposing the choice of a torsion free connection give rise to the so called soldering equation, which states that $R^{(e)}=0$ in vacuum, and makes possible to obtain the spin connection out of the tetrad, as in classical GR you obtain the Levi-Civita connection from the metric.

After setting the translational curvature tensor to zero one recovers the action of classical general relativity but with the new variables.

Principle (1) tells us that locally the metric is always Minkowsky, so combining this with the fact that now this is a gauge theory, one uses the Poincarè connection to build the covariant derivate for matter fields. We have a (classical) theory which encompass all the interactions using the gauge formalism machinery.

It becomes QFT on curved spacetime once one fixes the classical Poincarè solution to a specific $(e_0,\omega_0)$ and quantize the remaining fields.


Question:

Since we already know that the standard model is built upon Poincarè representations, why can't we incorporate into the standard model the gravitational interaction gauging the Poincarè group, having then a $$G = \mathcal{P}_{1,3} × SU(3)_C×SU(2)_L ×U(1)_Y$$

gauge group?

I already know that gravity is probably not UV-complete, but neither is the standard model, so what is the reason to not use such a theory as our most complete effective theory up to some scale?

I know that at low energies gravitational interactions are negligible, and that we cannot do high energy perturbation theory, but perturbation theory is not a necessity of natural laws but rather a useful tool for us when we are not able to exactly solve equations.

So what is the reason to not say that the standard model + Poincarè gauge theory is a (UV-incomplete) quantum theory of all interactions?


$^1$ Here i'm talking about the standard formulation of general relativity where the connection is the torsion free Levi-Civita connection derived from the metric.

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    $\begingroup$ I clarified my definition of Poincarè group according to the suggestion of @Moguntius $\endgroup$
    – LolloBoldo
    Commented Sep 2, 2023 at 15:15

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Two things here:

1.) When you are constructing "gravity" (whether it be GR, or Einstein-Cartan theory, or any extension that may include torsion but not non-metricity) and want to include spinor/fermionic fields it is advisable to pass to the double cover of the Poincaré group, i.e. the "Poincaré-Pin-group" or whatever you want to call it, that is the semi-direct product of the translations and the Pin group of the signature with which you are working.(In fact it suffices to use the connected component of the identity of the Pin group (so $SL(2, \mathbb{C})$ when you are working in 1+3 dimensions) because that is sufficient for gauge theories as all other components can be "reached" by multiplying with a "constant" group element).

2.) If you want to reproduce "ordinary" GR rather than starting with gauging the whole Poincaré-Pin-group and then "artificially" imposing the condition for the torsion to vanish you can simply start with the orthochronous Spin group (or the proper, orthochronous Lorentz group if you do not want/need to pass to the double cover) and then you will never find a connexion with torsion as a solution of your field equations.

As to answer your question: While you are correct that neither gravity (in the sense of GR being quantised) nor the standard model of particle physics are UV-complete, there is a difference between them: the standard model of particle physics is perturbatively renormalisable while gravity is not$^1$. Furthermore, I suspect that the expected corrections to say scattering angles one would receive by adding a form of quantised gravity are negligible for the energies at which we can probe physics at the moment.


$^1$ That is at first glance. There are some people that think that gravity might by asymptotically safe and therefore one would only need to fix a finite amount of couplings to "determine" how gravity works at high energies. While there is some evidence that this might be true for GR it has thus far not been rigourously proven.

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    $\begingroup$ I edit my definition of Poincare group to include the double covering $\endgroup$
    – LolloBoldo
    Commented Sep 2, 2023 at 15:18
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    $\begingroup$ The reason i wanted to use the Poincarè instead of the Lorentz group (or the double covers) is that in the SM you need the Poincare one to fond the representations of massless fields, see the Weinberg definition of massless spin 1 states $\endgroup$
    – LolloBoldo
    Commented Sep 2, 2023 at 15:20
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    $\begingroup$ I edit the $\mathcal{P}$ definition. Yes i saw the answer, i'm reasoning over it a bit before accepting :) $\endgroup$
    – LolloBoldo
    Commented Sep 2, 2023 at 16:59
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    $\begingroup$ Very nice answer. I would only add that one can really use quantized GR as an EFT with a tower of UV cutoffs and it will never run into an inability to predict anything at current and near-future experimental capability/accuracy. Quantum gravity is not a problem for essentially any observational or experimental area of physics. It is just that people are trying to repeat the same story with this EFT as they did with Fermi's EFT of weak interactions (see chapter 1 of Hořejší's excellent electroweak textbook for the history, arxiv.org/pdf/2210.04526). $\endgroup$
    – Void
    Commented Jun 10 at 12:11
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    $\begingroup$ @Moguntius Yes, start with the assumption that quantized gravity is an EFT that was obtained as a low-energy limit of some renormalizable theory and try to find that theory, possibly unifying gravity with other interactions along the way. This is the route that took one from Fermi's four-fermion theory to the Weinberg-Salam theory and what people tried to repeat with string theory etc. $\endgroup$
    – Void
    Commented Jun 14 at 10:38

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