Why does the state space contain states with negative norm and what would be an example?

Question

My lecture script of Quantum Field Theory states that " the state space contains states with negative norm ".

Why does it have to be like this and what would be an example fo such a state?

Answer 1

First of all, it is not always true. Most of the time, people work with positive definite Hilbert spaces.

Second, the physical subspace of the Hilbert space has to be positive definite because the (squared) norms appear as probabilities and those can't be negative.

Third, one often works with an extended Hilbert space that also contains unphysical states of negative (squared) norm. This is omnipresent in theories with gauge symmetries. The reason is that the creation operators $a^\dagger$ obtained from Fourier modes of $A_\mu$, for example, obey $$ \langle 0 | a_\mu a^\dagger_\nu | 0 \rangle = c g_{\mu\nu} $$ by the Lorentz symmetry. The metric tensor is the only thing that may appear on the right hand side. So the norm of $a^\dagger_\mu |0\rangle$ may be both positive or negative because the metric tensor is an indefinite matrix.

To produce a Hilbert space of multi-photon states that directly contains the positive-norm states only, one has to break the manifest Lorentz symmetry of the description. Only two polarizations (the transverse spatial ones) out of the four remain as physical states. It doesn't mean that the actual dynamics of the theory starts to be Lorentz-violating; it's just that the Lorentz symmetry can't be easily verified by the tensor structure of the equations.