What are kets?

Abstract

According to Dirac's bra-ket notation, in an inner-product space, the inner product $⟨x|y⟩$ of vectors $x,y$ can be viewed as an application of the bra $⟨x|$ to the ket $|y⟩$. Here $⟨x|$ is the linear functional $|y⟩\mapsto ⟨x|y⟩$ and $|y⟩$ is the vector $y$. But often -- though not always -- there are advantages in seeing $|y⟩$ as the function $a\mapsto a\cdot y$ where $a$ ranges over the scalars. For example, the outer product $|y⟩⟨x|$ becomes simply the composition $|y⟩\circ ⟨x|$. It would be most convenient to view kets sometimes as vectors and sometimes as functions, depending on the context. This turns out to be possible. While the bra-ket notation arose in quantum mechanics, this note presupposes no familiarity with quantum mechanics.

Bulletin of the EATCS 141 October 2023