# Why S-matrix must be unitary?

Please teach me this:
Why scattering matrix in relativistic collision still must be unitary?Because in relativistic regime,the probability is not conservable.
Thank you very much in advanced.

## Answers and Replies

A. Neumaier
Please teach me this:
Why scattering matrix in relativistic collision still must be unitary?Because in relativistic regime,the probability is not conservable.
Thank you very much in advanced.

Probability must be conserved in each observer frame. This forces the requirement of unitarity.

If probability conservation were violated, there would be a positive probability that none of the outcomes happen. What could that mean? The in-particles would be lost. But this would just mean that the final state is the vacuum. But the vacuum is stable in time and because the dynamics is invertible, a final vacuum state implies a vacuum state at all earlier times. Thus in-particles cannot get lost - they must materialize as something.
Thus probabilities must sum to 1.

So,how is the probability conservable if we consider the process: 2 bodies--->n bodies scattering process.In this QTF theory process the probability is still conservable?

Unitarity is satisfied among quantum states.

In QFT, a quantum state is not associated with an individual particle, but with the whole system (vacuum + particles).

For example, unitarity ensures that
$$1 = \sum_{n=0}^{\infty} (\textmd{The probability of the initial two bodies getting scattered to be n bodies})$$.