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- How does one model an ensemble of particles in quantum field theory?

A. Neumaier said:But a fundamental description would have to come from relativistic quantum field theory, where there are no ensembles. One cannot repeatedly prepare a quantum field extending over all of spacetime.

I don't know of a single paper explaining how the transition in conceptual language from a single quantum field to an ensemble of particles can be justified from the QFT formalism.Maybe you can help me here with a reference?

vanhees71 said:Within QFT you can as well prepare single Ag atoms as you can in QM. QFT is also used since it's conception to describe scattering cross sections, and that's also due to the standard interpretation of the quantum state in terms of Born's rule. It's explained in any QFT textbook, e.g., Weinberg, QT of Fields vol. 1.

A. Neumaier said:One can model a single silver atom in this way. But how do you model an ensemble of 100 silver atoms moving at well separated times along a beam by quantum field theory? You cannot prepare multiple instances of a field extending over all of spacetime. The only use! of Born's rule in Weinberg's Vol. 1 (namely where he interprets the scattering amplitutes) doesn't address this issue - scattering has nothing to do with this question!

Then please explain how the transition in conceptual language from a single quantum field (extending all over spacetime, or at least over the lab during a day) to an ensemble of particles can be justified from the QFT formalism.vanhees71 said:Relativistic QFT has the same probabilistic interpretation as any QT (in fact there is only one overall conceptual framework and a non-relativistic (in both a "1st-quantization formulation" and a field-theoretical one as well as a special relativistic realization in terms of local QFTs).

Of course one can prepare ensembles within all kinds of QTs and, more importantly, in the lab. Relativistic QFT is among the best tested physical theories ever discovered. This would be impossible to achieve if it were not possible to prepare ensembles of the physical systems described by it, and these are particles and nuclei in particle accelerators, with which you can do scattering experiments with high precision. Another application is atomic physics. A specific quantum-field theoretical result is the explanation of the Lamb shift to several significant digits of accuracy, etc.

I don't understand, how one can claim that one cannot build ensembles within relativistic QFT, given all these successes. After all the first goal in all introductions to QFT is to establish the calculations of S-matrix elements, which precisely describe scattering processes, and obviously these can be realized with high accuracy in the lab.