I am actually a fan of bra-ket notation. But in my mind, <i|j> is the amplitude for going from state i -> j. My point is, if you just have one state that you wa

I am actually a fan of bra-ket notation. But in my mind, <i|j> is the amplitude for going from state i -> j. My point is, if you just have one state that you want to talk about, which one do you write down <a| or |a>? It doesn't matter. You only write the brakets when you want to calculate transition amplitudes/inner products, or cross products. I agree about just keeping them the way they are originally facing and nothing will go wrong. The nice thing is that you can mechanically rearrange things almost like regular algebra, i.e. draw out a common factor, and don't have to think what these objects are (a vector? a dual vector? a functional?). Same with four-vector notation / Einstein summation. As long as you are following the rules for raising and lowering indices, you cannot form "wrong" products or mess up the transformation properties. Said in another way: as a (former) experimentalist, I don't care if the vector is p^mu or p_mu. All I care about is the product p^mu p_mu.