# 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?

• Do you understand what unitarity means in quantum mechanics? QFT is no different. Jun 24, 2015 at 7:59
• Jun 5, 2018 at 22:50

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.
In Fermi theory of $$\beta$$-decay, the cross-section exhibit a divergent behaviour at scales above about 300 GeV. Such a problem is due to a poor description at low energies (i.e., to the approximation of point interaction). A more accurate theory, involving vector bosons, avoid this inconvenience.