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?

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    $\begingroup$ Do you understand what unitarity means in quantum mechanics? QFT is no different. $\endgroup$ – Meng Cheng Jun 24 '15 at 7:59
  • $\begingroup$ Should be helpful. $\endgroup$ – Cosmas Zachos Jun 5 '18 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.


Unitarity stems for "conservation of probability". It should be quite clear that quantum-mechanical evolution, as prescribed by Schrödinger equation, is due to the strongly unitary group generated by the Hamiltonian (when time-independent) or, more generally, by the corresponding unitary propagator (when the Hamiltonian is time-dependent). In each case, scalar products and hence probability amplitudes are preserved under time-evolution. Schrödinger equation is assumed to provide the rule for the evolution, so a violation of unitarity is unacceptable in the quantum-framework and when it happens one looks for a simpler solution, e.g. some wrong assumptions in the setting of the model.

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.


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