Creation and annihilation operators can be defined in several different ways, some more general than others. We usually choose to denote by $a$ the annihilation operator and by $a^\dagger$ the creation operator.

It can be seen that the two operators must be Hermitian adjoint of each other. But is there a reason why we make this particular choice (i.e., which operator doesn't have the dagger)?

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    $\begingroup$ because the dagger looks like a plus sign so one can remember that it adds a particle. This would be my reasoning ;-) $\endgroup$ – Noldig Oct 17 '14 at 13:16

It boils down to a matter of convention. Nothing stops you from choosing the annihilation operator to be $a^\dagger$.

Still, in quantum field theory, you decompose e.g. a scalar quantity in plane waves $$ \Phi(x) = \int \frac{d^4 k}{4 \pi} \left( a(k) e^{ik\cdot x} + a^*(k) e^{-ik\cdot x} \right)$$ where obviousely the second part is the c.c. of the first part. Here, it is usual to combine $a$ with $e^{ikx}$ which annihilates a particle from your Fock state.

But again, this is just convention and you are free to choose which operator is the "real thing" and which is "just the adjoint" any way you like.

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