# Why does $U(+\infty,-\infty)|0\rangle= e^{i \theta}|0\rangle$ hold ? i.e. time evolution of vacuum is still a vacuum

In quantum field theory, we always use $U(+\infty,-\infty)|0\rangle= e^{i \theta}|0\rangle$, where $$U(+\infty,-\infty)=\lim_{\epsilon\rightarrow +0} \mathcal{T}\exp\{i \int_{-\infty}^{+\infty} e^{-\epsilon|t'|} H_i(t') dt\}$$ That is we adiabatically switch on the interaction and then switch off. My question is why the infinite time evolution of free vacuum is still a free vacuum? Can anyone explain in the following case ?

In the case, the field is massless like Maxwell field. I know there is adiabatic theorem in quantum mechanics, but it requires there is always a gap from the orginal state. But in QED, above the vacuum these is a continuous spectrum since photon is massless. Why $U(+\infty,-\infty)|0\rangle= e^{i \theta}|0\rangle$ still holds?

The same problem also exists in the condensed matter field where the spectra of phonon is gapless. Why does every qft textbook or quantum many-body textbook just say it's an adiabatic theorem?

If this equation is not correct, then we cannot get the following formula:

$$\langle\Omega|\mathcal{T}A(t_1)B(t_2)\cdots|\Omega\rangle=\frac{\langle 0|\mathcal{T} U(+\infty,-\infty)A(t_1)B(t_2)\cdots| 0\rangle}{\langle 0| U(+\infty,-\infty) | 0\rangle}$$

• In massless theories that equation does not hold. But massless theories are ill-defined anyway, so you have to introduce an IR regulator that generates a mass-gap. So: in practice, massless theories are not really massless; they are to be regulated. – AccidentalFourierTransform May 31 '17 at 19:51
• @AccidentalFourierTransform Can you recommend any textbook talk about this? How to rigorously tackle this problem? – user153663 May 31 '17 at 20:09
• see for example the references in physics.stackexchange.com/a/330220/84967 – AccidentalFourierTransform May 31 '17 at 20:52
• – SRS May 31 '17 at 21:44