According to the commutation relation of annihilation and creation operators, $$[a,a^{\dagger}]=1$$ I would like to calculate the vacuum expectation value of the normal order of this commutator. We claim that since this commutator is just a c-number, then there is no effect from the normal ordering. We therefore get

$$\langle 0|(:[a,a ^\dagger]:)|0 \rangle=1. \tag{1}$$

However, if we expand the commutator first and do normal order later, we will get something like

$$\langle0|(:[a,a^\dagger]:)|0\rangle=\langle0|(:aa^\dagger-a^\dagger a:)|0\rangle=\langle0|(a^\dagger a-a^\dagger a)|0\rangle=0.\tag{2}$$ Which is contradicting with itself.

According to the commutation relation of annihilation and creation operators,

$$[a,a^{\dagger}]=1. \tag{1}$$

I would like to calculate the vacuum expectation value of the normal order of this commutator. We claim that since this commutator is just a c-number, then there is no effect from the normal ordering. We therefore get

$$\langle 0|(:[a,a ^\dagger]:)|0 \rangle=1. \tag{2}$$

However, if we expand the commutator first and do normal order later, we will get something like

$$\langle0|(:[a,a^\dagger]:)|0\rangle=\langle0|(:aa^\dagger-a^\dagger a:)|0\rangle=\langle0|(a^\dagger a-a^\dagger a)|0\rangle=0.\tag{3}$$ Which is contradicting with itself.

According to the commutation relation of annihilation and creation operators, $$[a,a^{\dagger}]=1$$ I would like to calculate the vacuum expectation value of the normal order of this commutator. We claim that since this commutator is just a c-number, then there is no effect from the normal ordering. We therefore get $$\langle 0|(:[a,a ^\dagger]:)|0 \rangle=1$$

However, if we expand the commutator first and do normal order later, we will get something like $$\langle0|(:[a,a^\dagger]:)|0\rangle=\langle0|(:aa^\dagger-a^\dagger a:)|0\rangle=\langle0|(a^\dagger a-a^\dagger a)|0\rangle=0$$ Which is contradicting with itself.

According to the commutation relation of annihilation and creation operators, $$[a,a^{\dagger}]=1$$ I would like to calculate the vacuum expectation value of the normal order of this commutator. We claim that since this commutator is just a c-number, then there is no effect from the normal ordering. We therefore get

$$\langle 0|(:[a,a ^\dagger]:)|0 \rangle=1. \tag{1}$$

However, if we expand the commutator first and do normal order later, we will get something like

$$\langle0|(:[a,a^\dagger]:)|0\rangle=\langle0|(:aa^\dagger-a^\dagger a:)|0\rangle=\langle0|(a^\dagger a-a^\dagger a)|0\rangle=0.\tag{2}$$ Which is contradicting with itself.