# Unitary quantum field theory

What do physicists mean when they refer to a quantum field theory being unitary? Does this mean that all the symmetry groups of the theory act via unitary representations? I would appreciate if one could provide some references where the definition of a unitary QFT could be found. Especially in the case where there might not be a special direction singled out as "time".

• No, that's a different story (unitary reps of symmetry groups). It means that all states $|n\rangle$ in the theory have positive norm, i.e. $\langle n|n \rangle > 0$, but the full answer is more involved and should involve a discussion of reflection positivity. May 24, 2013 at 19:45
• ...(and also the S-matrix and the Froissart bound, of course). May 24, 2013 at 20:30
• In a quantum theory (quantum mechanics or quantum field theory), unitarity means conservation of probability (or conservation of information), that is, if a state $|\psi>$ evolves in a state $|\psi'>$, you will have $<\psi|\psi> = <\psi'|\psi'>$. This means that the operator which transforms $|\psi>$ into $|\psi'>$ must be unitary. Unitarity is mandatory for the probabilistic coherence of the quantum theory. May 27, 2013 at 11:01

To expand on @user26374's answer a little, the phrase "A QFT is unitary" comes from the requirement that the $$S$$-matrix is unitary, i.e. $$S S^\dagger = S^\dagger S = 1$$ which is equivalent to the statement that sum of probabilities is 1. Unitarity implies several serious constraints on how a QFT can be formulated. For example, unitarity implies the Froissart bound, $$\sigma \leq s (\log s)^2$$ where $$\sigma$$ is the total cross-section and $$s$$ is the center of mass energy. It also implies that the propagator for a field must go no faster than $$\frac{1}{p^2}$$ at large $$p^2$$.
• Do you mean $\sigma(s)=O\left(s\,\log s\right)$? The LHS and RHS of that inequality don't have the same units also. May 7, 2019 at 23:02
• @xiaoshengyang - because we measure the $S$-matrix in a lab (actually, we measure cross sections and decay rates). Consequently, in order to verify whether a theory is actually unitary, we should attempt to recast it in terms of these measurable objects. Sep 9, 2022 at 5:54