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In classical mechanics, we assumed a particle to have a definite momentum and a definite position. Afterwards, with Quantum mechanics, we gave up the concept of a time-dependend position and momentum, and instead have propability distribution stuff and a Hilbert space containing all the information about the state of the particle. Still, we can "retrieve" the concept of a point particle, by stating that the quantum mechanical state gives a propability distribution for the (point-particle)-properties position and momentum, and moreover, the mean-values for position and momentum follow the classical rules.

What I am seeking for now is basically the same, but not for QM, but for QFT. For an abitrary Quantum-Field-Theory, is there a way to "construct" a particle-concept that gives position and momentum for one particle? I know that in QFT, particles are just excitations of the field, but still, is there a way to assign position and momentum to certain types of excitations?

For example, in Theories of free fields (see my other Question here), one can identify the hilbert-space with a fock-space, and by that one can "construct" some wave-package state, that then (in the context of many-body QM) has a localized position and momentum. Is something like that also possible in an interacting theory (despite the fact that the hilbert spaces of interacting theories are not in correspondence to a fock space)?

My thought is that this should be possible in principle, since the QFT is somewhat a generalization of the QM, that intendes to describe Nature better than QM. There should be something like "backward-compability".

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Quantumwhisp
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2 Answers2

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In QFT, a single particle does not scatter, hence its (renomalized) wave function in an interacting theory is the same as the corresponding asymptotic wave function in the asymptotic Fock space.

However, the multiparticle picture breaks down as the interacting Hilbert space cannot be identified with the asymptotic Fock space, by Haag's theorem. Thus multiparticle states make sense only asymptotically.

Arnold Neumaier
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If I'm understanding correctly, you're asking whether an arbitrary QFT admits an asymptotic Fock basis. If a QFT does admit a Fock basis, you can talk about particles and do scattering theory in momentum space and construct position operators for single particles.

But it's not guaranteed that a given QFT admits asympotic Fock bases. Quantum field theory in general deals with the quantum behavior of field systems. Sometimes the fields and interactions are such that you get particles, but this is a property that depends on the details of the QFT, not a property of QFTs in general. Conformal QFTs, for example, do not have asymptotic particle states or scattering matrices. Nor do theories which lack a mass gap.

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user1504
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