# Why are particles described with Poincaré symmetry even though space seems inhomogeneous?

Poincaré transformation consists of translation, rotation, and boosting. And by assuming the physical quantities are invariant and equations are covariant under the transformations, we build the models on particles. The invariance and covariance make sense if space is the same under the transformations. Space has symmetry in the case.

But from Einstein's field equation,

$$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$$

space is curved when energy is placed on it.

So the very existence of particles would make the space irregular by making dents randomly. And with this process, space would not keep the symmetry anymore. The earth itself would distort the space by a large amount. But the particles are described with the Poincaré symmetry and the experiments on earth verify the theories constructed with the symmetry.

Why does the Poincaré symmetry work to describe the particles even though space doesn't seem homogeneous?

P.S. The several answers point that the symmetry holds approximately. But I feel difficulty with the approximation view. For example, there is the spin-statistics theorem. The theorem relies on the exchange symmetry:

$$\phi(x)\phi(y) = \pm \phi(y)\phi(x)$$

The equation represents equality between the left and the right side and there is no room for approximation. If the particles are approximate entities, I guess the spin-statistics theorem would not work.

• Concerning your statement "the earth itself would distort the space by a large amount": to within a few orders of magnitude, the deviation from flatness of an object of mass $M$ and radius $R$ is $\mathcal{O}(GM/R c^2)$. For the Earth, this works out to be $\approx 10^{-9}$. So it's not surprising that flat-spacetime approximations still work pretty well on Earth. Jul 26, 2020 at 19:43
• Just a food for thought: due to equivalence principle, the structures defined locally need to retain the flat spacetime symmetries. If you have some vector equation it needs to retain symmetry under Poincare group acting in the tangent space, as this is where the vector lives. Thus it seems to me, that invariance and covariance still makes sense even in curved spacetime. The problem I guess arises only with nonlocal structures, as is the exchange symmetry. Jul 27, 2020 at 6:02

## 3 Answers

The quick and easy answer is that the amount of spacetime curvature created by particles is so small compared to the effects of the electromagnetic, strong, and weak interactions that gravity can be ignored when studying particle interactions in the lab.

By analogy, one can consider Euclidean planar geometry as a useful model approximating the geometry of your table, which is intermediate between the microscopic scale that sees the irregularities in the wood and the larger scale that sees the curvature of the non-flat earth.

Poincare symmetry should probably be thought similarly of as a useful model approximating symmetries seen at a particular scale, certainly intermediate between the very small (where spacetime might not make sense and/or the spacetime might not be like $$R^4$$) and the very large (where significant spacetime curvature at extreme-astrophysical or cosmological scales might arise and/or the spacetime might not be like $$R^4$$).
(It may be that "the very-very small and smaller" or "the very-very large and larger" shows Poincare symmetry... but, for at least the intermediate range of scales that we work with right now, there are lower and upper bounds.)

As others have noted, the effect of gravitation (curvature of spacetime) is relatively small at typical particle-physics scales (in space and time).

The homogeneous and isotropic property of the space holds in appoximate sense, or when looking at large enough distances. Definitely, the world around us is not a homogeneous mass, we have stars, planets, galaxies, or on shorter distances - mountains, lakes, trees.

At these scales, the existence of localized density clusters breaks the translational and rotatitonal invariance. However, when investigating the properties of universe on cosmological scales, larger than the galaxy clusters, the matter would be distibuted almost uniformly and isotropically (however, the CMB has an anisotropy of $$\sim 10^{-5}$$ ).