There seem to be two ways of defning what a vacuum is in QFT:
1. It is state $|0\rangle$ such that $$a_k|0\rangle = 0$$ for all anihilation operators $$a_k$$, with creation operators $$a_k^{\dagger}$$. Thus, it is defined in Fock space.
2. It is state $$|0\rangle$$ such that derivative...