## 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, |v\rangle\leftrightarrow \langle v|, denoted by a dagger:4 \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. In the matrix representation,5 |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}.

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

2. Recall that the conjugate transpose, or the Hermitian conjugate, 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 mathematics texts it is often denoted by {}^\star rather than {}^\dagger.↩︎