# Why is harmonic oscillator creation operator called “a-dagger”, $a^\dagger$?

As is well known, Dirac introduced the creation and annihilation operators for the quantum harmonic oscillator ($\hat{a}^\dagger$ and $\hat{a}$ respectively), which are now part of every first course in quantum mechanics due to their far-reaching importance and simplicity.

$$\hat{a}^\dagger=\sqrt{\frac{m\omega}{2\hbar}}\left[\hat{x}-\frac{i}{m\omega}\hat{p}\right],\,\,\,\hat{a}=\sqrt{\frac{m\omega}{2\hbar}}\left[\hat{x}+\frac{i}{m\omega}\hat{p}\right]$$

However, in English we say "a-dagger" for the creation operator, and the word "dagger" is certainly more associated with destruction (killing, stabbing, etc.) than it is with "creation". This is almost always an unspoken source of confusion for native English-speaking students of physics.

Given that this was originally done by an Englishman, how come we haven't simply swapped the definitions for $\hat{a}^\dagger$ and $\hat{a}$, and then called $\hat{a}^\dagger$ the annihilation operator and $\hat{a}$ the creation operator, a redefinition that would have no effect on the physics but have a substantial effect on a student's ability to remember these operators? Is there a reason why Dirac chose this language-awkward definition?

• The name "dagger" is used for the typographical symbol † because of its appearance. This goes back way before Dirac. Here, it denotes the complex-conjugate. – Gerald Edgar May 9 '16 at 13:38
• For a discussion about the origin of the dagger, one can check Why is a cross † used as footnote marker for people? – Laurent Duval May 12 '16 at 19:34

$\dagger$ is the transpose conjugate of a linear operator; once $a$ is defined as the annihilation operator (presumably because annihilation starts with the letter a), Dirac showed that the corresponding creation operator corresponds to it's transpose conjugate, hence the creation operator is $a^{\dagger}$.

Thus the a comes first, and then the $\dagger$.

I wouldn't assume that Dirac is responsible for the $\dagger$ notation for transpose conjugate; it is not the only way it is written in mathematical texts, though it is certainly the favorite among physicists today.