waves - Laser emission frequencies: does that physically represent the frequencies if it was turned on for infinite time?

Consider you have a laser that has a given emission spectrum.

Does this emission spectrum physically represents the electromagnetic field it would emit if it was turned on for an infinite amount of time ?

Indeed, if you turn on and off the laser for a time $T1$ or $T2$ the frequencies in the signal you would see would depend on those times because of Fourier transform properties.

Answer

If a peak in an emission spectrum was infinitely narrow then it would mean then a laser made from that transition would be truly monochromatic if the laser was on for infinite time. However no transition in the world is actually infinitely narrow. Every transition in fact has some linewidth so every laser has a finite linewidth. The narrowest lasers have linewidths on the order of $mHz$ so the laser must be on for a half an hour or so to ensure the measured linewidth is not Fourier limited. (A Fourier limited signal means that the width of your signal in Fourier space is “artificially” broadened due to the fact that you haven’t spent enough time measuring it to see the true spectrum).

If a laser had an infinitely narrow linewidth then any measurement of the linewidth of that laser would be Fourier limited because it is impossible to perform a measurement that takes infinite time.