Given the following pair of operators $a$ and $a^{\dagger}$ that satisfy the usual bosonic CCR:  
$$[a,a]=[a^{\dagger},a^{\dagger}] = 0;\ [a,a^{\dagger}] = 1$$  
For what values of $\alpha \in\mathbb C$ are the following expressions well defined?  
$$a^{\alpha} \ \text{and}\  (a^{\dagger})^{\alpha}$$  
For integer $\alpha$, I know that these expressions are well defined, but I am interested in knowing what kind of constraints are imposed on $\alpha$, if such an operator must be appropriately definable. If the above expressions can be defined in precise manner, then please also provide the appropriate construction of these operators.