0.4 Daggers

Although \mathcal{H} and \mathcal{H}^\star are not identical spaces — the former is inhabited by kets, and the latter by bras — they are closely related. There is a bijective map from one to the other given by |v\rangle\leftrightarrow \langle v|, and denoted by a dagger:11 \begin{aligned} \langle v| &= (|v\rangle)^\dagger \\|v\rangle &= (\langle v|)^\dagger. \end{aligned} We usually omit the parentheses when it is obvious what the dagger operation applies to.

The dagger operation, also known as Hermitian conjugation, is antilinear: \begin{aligned} (c_1|v_1\rangle+c_2|v_2\rangle)^\dagger &= c_1^\star\langle v_1| + c_2^\star\langle v_2| \\(c_1\langle v_1|+c_2\langle v_2|)^\dagger &= c_1^\star|v_1\rangle + c_2^\star|v_2\rangle. \end{aligned} Also, when applied twice, the dagger operation is the identity map.

You might already be familiar with Hermitian conjugation under another name: the conjugate transpose12 of an (n\times m) matrix A is an (m\times n) matrix A^\dagger, obtained by interchanging the rows and columns of A and taking complex conjugates of each entry in A, i.e. A^\dagger_{ij}=A^\star_{ji}. In particular then, |v\rangle = \begin{bmatrix}v_1\\v_2\\\vdots\\v_n\end{bmatrix} \overset{\dagger}{\longleftrightarrow} \langle v| = \begin{bmatrix}v_1^\star,&v_2^\star,&\ldots,&v_n^\star\end{bmatrix}. We will come back to this \dagger operation on matrices in Section 0.6.

  1. “Is this a \dagger which I see before me…”↩︎

  2. In mathematics texts this operation is often denoted by {}^\star rather than {}^\dagger, but we reserve the former for complex conjugation without matrix transposition. Note, however, that scalars can be thought of as (1\times1) matrices, and in this special case we have that \dagger=\star.↩︎