# Defining creation and annihilation operators

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)?

• because the dagger looks like a plus sign so one can remember that it adds a particle. This would be my reasoning ;-) – 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.