$T$-odd vs $T$-violation I am a bit confused by the difference between $T$-odd and $T$-violation. For example, I read that the existence of a fundamental particle EDM is a violation of time symmetry. However, placing an electric dipole in an electric field, would produce a hamiltonian (non-relativistically, which is usually the region of interest for e.g. atomic experiments): $H = -d\cdot E$, where $d$ is the electric dipole and $E$ is the electric field acting on the (say) electron. The dot product between d and E is odd under time reversal. But I am not sure I understand where the T-violation comes from. I thought that T-odd means just that the system changes sign under T operator, but it is still an eigenstate of it, which means that $T$ and $H$ commute. However, T-violation, I imagined, it means that T and H don't commute. Can someone help me clarify what odd and violation mean in this case?
 A: If you blinded yourself to everything about the particle except for the electric dipole moment, then it would indeed look like not much has changed. However, fundamental particles have other properties besides an electric dipole moment, such as the spin (an intrinsic magnetic dipole moment).
If all properties of a fundamental particle were T-even, or all properties of a fundamental particle were T-odd, then time reversal wouldn't do anything, as they would either all stay the same or all flip together. But if one property is T-odd and another is T-even, then only one of those properties changes upon time reversal, and therefore a difference can be detected.
For example, the electric dipole moment is T-even and the magnetic dipole moment (the spin) is T-odd. This means that if we start with a particle whose electric and magnetic dipole moments are aligned, and we reverse time, then the spin flips but the electric dipole moment doesn't. Time reversal gives us a particle whose spin and electric dipole moment are antialigned rather than aligned.
