What does the colons in $:\hat{P}:$ mean? When going through some past papers I came across the following exercise:

For the momentum operator 
  $$\hat{P}^\mu =\frac{1}{2}\int \frac{d^3 k'}{(2\pi^3)2E(\textbf{k'})}k'^\mu (\hat{a}^\dagger(\textbf{k'})\hat{a}(\textbf{k')} + \hat{a}(\textbf{k'}) \hat{a}^\dagger(\textbf{k}'))$$
  prove the commutator relation 
  $$[:\hat{P}^\mu:, \hat{a}^\dagger(\textbf{k})]=k^\mu\hat{a}^\dagger(\textbf{k})$$

Does the ": :" symbol around the momentum operator represent anything? If so would it affect the calculations or would I just prove the commutation by doing:
$$\hat{P}^\mu\hat{a}^\dagger(\textbf{k})-\hat{a}^\dagger(\textbf{k})\hat{P}^\mu$$
 A: The "::" symbols represent normal ordering, a very common operation in Quantum Field Theory. In practical terms, it effectively means that the operator inside the colons (in this case the momentum operator) is the same operator but with all annihilation operators placed at the end:
$$
:\hat { P } ^ { \mu }: = \frac { 1 } { 2 } \int \frac { d ^ { 3 } k ^ { \prime } } { \left( 2 \pi ^ { 3 } \right) 2 E \left( \mathbf { k } ^ { \prime } \right) } k ^ { \prime \mu } \left( \hat { a } ^ { \dagger } \left( \mathbf { k } ^ { \prime } \right) \hat { a } \left( \mathbf { k } ^ { \prime } \right) + \hat { a } ^ { \dagger } \left( \mathbf { k } ^ { \prime } \right) \hat { a } \left( \mathbf { k } ^ { \prime } \right) \right) = \int \frac { d ^ { 3 } k ^ { \prime } } { \left( 2 \pi ^ { 3 } \right) 2 E \left( \mathbf { k } ^ { \prime } \right) } k ^ { \prime \mu } \left( \hat { a } ^ { \dagger } \left( \mathbf { k } ^ { \prime } \right) \hat { a } \left( \mathbf { k } ^ { \prime } \right) \right)
$$
And you can then proceed as usual from here. 
A: ::  means normal ordering of the creation and destruction operators inside the colons. The normal order happens when all destruction operators are placed to the right of the creation operators. So $:{a}(\textbf{k}){a}^\dagger(\textbf{k}):={a}^\dagger(\textbf{k}){a}(\textbf{k})$
