# Why reflection is generator of Minkowski space, if we know that only CPT symmetry is true?

In particular, we know that $$P$$ symmetry (parity transformation, which is basically a reflection), is not the symmetry of the universe, a whole combination of $$C$$, $$P$$, and $$T$$ is. Because of this, it's not clear to me why Minkowski space preserves reflections (universe doesn't). Is this property really necessary to model spacetime?

Related question: Is there a similar mathematical construct that preserves only rotations and translations, but not necessarily reflections?

Let me try to clear things out. You are mixing the symmetries of spacetime with the symmetries of the interactions of the Standard Model.

The first, are symmetries of spacetime to which according to special relativity (in relation to the Standard Model) is the Poincaré group of symmetries. So spacetime itself is translationally invariant, isotropic, parity symmetric, time symmetric (this actually just a case of translation in the time coordinate).

A different set of symmetries are those associated to the fields and the interactions themselves. Namely, to the models we build on spacetime. For this we use fields, they correspond to forces and to matter, but they have different internal symmetries, technically they are associated to irreducible representations depending on the types of charges they carry.

Now here is what is more specific. $$C, P$$ and $$T$$ are symmetries that speak about the fields and how they transform within a model. Sadly $$P$$ and $$T$$ already appear in the spacetime symmetries therefore the confusion, but what people mean is their action on the fields and the interactions of the theory.

So they are two different things under study (with a similar name). The symmetries of a theory do not have to be the same as the symmetries of the spacetime it is built on.

P.D. As soon as you assume rotations and translations your spacetime is isotropic and homogeneous so will also be symmetric under reflection.

• Thank you for an answer. I know they are different symmetries, but it would be meaningless to define symmetries of spacetime without considering fields and particles within them, and they are governed by Standard Model and subject of CPT symmetry. Isn't it reasonable to assume that CPT is not only symmetry of Standard Model fields, but also spacetime? – sheerun Nov 16 '20 at 23:27
• @sheerun Symmetry is a property of a mathematical structure. Minkowski space is a mathematical structure, and it has its own symmetry. The Standard Model is a more elaborate mathematical structure, and it has its own symmetry. Minkowski space is part of that structure, and some of the Standard Model's symmetries act trivially on Minkowski space. For example, as far as Minkowski space is concerned, CPT is identical to PT, because the C doesn't do anything to spacetime. We use mathematical structures to model reality, but they are models, and "symmetry" is a property of the model. – Chiral Anomaly Nov 17 '20 at 0:22
• Hmm, PT actually sounds like a reflection, thanks! I wonder if Theory of Everything will use spacetime-something structure that incorporates charge and C symmetry already within it (more dimensions?) – sheerun Nov 17 '20 at 1:08