  In the standard construction of Quantum Field Theory, a vacuum state is
required. The vacuum is a vector in a separable, infinite-dimensional Hilbert
space often referred to as Fock space. By definition the vacuum wavestate
depends on nothing and must be translationally invariant. We show that any such
translationally-invariant vector must have a norm that is either divergent or
equal to zero. It is impossible for any state to be both everywhere
translationally invariant and also have a norm of one. The axioms of QFT cannot
be made internally consistent.
