quantum field theory - Violation of unitarity: meaning and consequences

What is meant by unitarity and violation of unitarity of a QFT? For example, Fermi theory of beta decay is said to violate unitarity. How does violation of unitarity make a theory sick?

Answer

Unitarity is a central feature of all quantum theories, fields or no fields. Unitarity is simply the demand that the time evolution operator $U(t_1,t_2)$ from any time $t_1$ to any time $t_2$ be unitary, i.e. preserve the inner product $(\cdot,\,\cdot)$ of the Hilbert space of states $\mathcal{H}$, i.e. for any states $|\psi\rangle,|\phi\rangle$,

$$(U(t_1,t_2)|\psi\rangle,U(t_1,t_2)|\phi\rangle) = (|\psi\rangle,|\phi\rangle) \quad\forall t_1,t_2$$ which is easily seen to be equivalent to $U^\dagger U = \mathrm{id}_\mathcal{H}$ by definition of the adjoint.¹ It is also evident that it is bad if time evolution is not unitary, because, for example, the norm $\lvert(|\psi\rangle,|\psi\rangle)\rvert^2$ is, by the Born rule, the probability to find the (normalized) state $|\psi\rangle$ in the state $|\psi\rangle$. It's evident that that should better stay $1$ throughout all of time evolution.

Also, since unitarity means preserving the norm, and the norm is the squared sum of the coefficients in a basis (which are the probabilities to find a state in a basis state), non-unitarity would mean the sum of all probabilities exceeds $1$. Again, it is hopefully evident that that would be bad.

A kind of converse also holds: One might see that if we find probabilities (e.g. for scattering from initial to final states) exceeding $1$ anywhere in the theory although all initial states were normalized, then time evolution/the theory evidently is not unitary, which is, I believe, the case in the Fermi theory of beta decay.

---

^{1I do not use Dirac notation here because it obscures that preserving the inner product is a slightly different definition than $U^\dagger U = \mathrm{id}_\mathcal{H}$ a priori.}