What it means the theory violates unitarity

[Mesmerized]

What it means "the theory violates unitarity"

Hello, I know what unitary transformation is, but what does it mean that the theory does or does not violate unitarity? For example in some textbooks on QFT one can read that the Fermi theory of beta decay, which is not renormalizable, also violates unitarity. What it means - the unitarity of the theory?

 

[DrDu]

I suppose that the propagators aren't unitary. Bad slang of QFT people.

 

[Bill_K]

Unitarity means the S-matrix is unitary: SS^† = I. http://isites.harvard.edu/fs/docs/icb.topic473482.files/23-unitarity.pdf.

 

[Meir Achuz]

In Fermi's theory of weak interactions, the cross section for neutrino scattering would increase without limit as the energy increases, but cross sections are limited by unitarity.

 

[Mesmerized]

thanks.

thanks Bil_K, it seems there is an explanation of that in your link, I will once watch into it more carefully

 

[geoduck]

Unitarity means the S-matrix is unitary: SS^† = I. http://isites.harvard.edu/fs/docs/icb.topic473482.files/23-unitarity.pdf.

Do you know how the author of those notes derived the conclusion that cross-sections cannot be arbitrarily large from equation 70? You can have each of your Legendre coefficients less than 1, but that doesn't guarantee that the series converges.

 

[Demystifier]

http://isites.harvard.edu/fs/docs/icb.topic473482.files/23-unitarity.pdf.

The last paragraph is crucial:
"Keep in mind, this is not a statement that unitarity is violated in these theories. It says that unitary would be violated, if we could trust perturbation theory, which we can’t."

 

[vanhees71]

It's easy to understand that perturbation theory violates the unitarity of the S matrix. The S-matrix is (formally) derived from the time-evolution operator of states in the interaction picture, which reads
\hat{S}=\mathcal{T}_c \exp \left [-\mathrm{i} \int_{-\infty}^{\infty} \mathrm{d} t' \exp(-0^+ |t'|)\hat{H}_I(t') \right],
where I've put in the usual Gell-Mann-Low adiabatic switching of the interaction. Perturbation theory now uses the power expansion of the exponential, and this immediately makes clear that the approximate S matrix is not unitary at any finite order of perturbation theory. This explains why cross section, evaluated outside of the applicability of perturbation theory, violate unitarity constraints.

 

[DrDu]

Perturbation theory now uses the power expansion of the exponential, and this immediately makes clear that the approximate S matrix is not unitary at any finite order of perturbation theory.

Of course, but the question rather seems to be whether you can approximately resum partial series so that the approximate result is unitary or not.

 

[andrien]

unitary requirement is required so that when you calculate|S_fi|² for each possible final states,then the sum of all these |S_fi|² should be equal to 1.when one use perturbation theory then ,of course one is not taking into account many possible final states hence it just can not be unitary.

 

[Bill_K]

unitary requirement is required so that when you calculate|S_fi|² for each possible final states,then the sum of all these |S_fi|² should be equal to 1.when one use perturbation theory then ,of course one is not taking into account many possible final states hence it just can not be unitary.

Sure, but Ʃ|Sⁿ|² should still be less than 1. We say the theory violates unitarity when some |S_n|² > 1.