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A. Neumaier submitted a new PF Insights post
Causal Perturbation Theory
Continue reading the Original PF Insights Post.
Causal Perturbation Theory
Continue reading the Original PF Insights Post.
A. Neumaier said:But the perturbative expansion is not the whole story since there must be modifications in the IR, due to the fact that the physical electron is an infraparticle only.
A. Neumaier said:The physical electron in QED exists, but it is (like an electron in experiments, but unlike a - nonexistent - bare electron) inseparable from the electromagnetic field carried by its charge. This implies that the asymptotic states (which in perturbation theory are treated incorrectly as free Dirac states without a field) are in fact more complicated objects, called infraparticles. The latter exist in a far more real sense than the Dirac electrons. But their behavior is mathematically not fully understood. In a nonperturbative construction of QED, the infraparticle structure must be explicitly represented. This means that one would have to do perturbation theory starting in place of the Hilbert space of a free field with a Hilbert space featuring infraparticles instead. People have been trying to build such a Hilbert space but nobody so far has married it with a perturbative construction in the spirit of causal perturbation theory - except in case of an electron in an external electromagnetic field, which is already quite technical.
(Described in terms of bare stuff - which is frequently done though it is meaningless imagery - the infraparticle consists of a virtual bare electron plus a cloud of infinitely many virtual soft photons; this is manifested in traditional S-matrix calculations by summing over corresponding outgoing states to remove the IR divergences.)
A. Neumaier said:The physical electron in QED exists, but it is (like an electron in experiments, but unlike a - nonexistent - bare electron) inseparable from the electromagnetic field carried by its charge. This implies that the asymptotic states (which in perturbation theory are treated incorrectly as free Dirac states without a field) are in fact more complicated objects, called infraparticles. The latter exist in a far more real sense than the Dirac electrons. But their behavior is mathematically not fully understood.
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(Described in terms of bare stuff - which is frequently done though it is meaningless imagery - the infraparticle consists of a virtual bare electron plus a cloud of infinitely many virtual soft photons; this is manifested in traditional S-matrix calculations by summing over corresponding outgoing states to remove the IR divergences.)
You don't understand correctly. Both the electromagnetic field and the electron currents are real (measurable), photons are elementary excitations of the electromagnetic field, and electrons are elementary excitation of the electron current field, hence are as real. But they are not bare - the bare photons and electrons are meaningless auxiliary constructs that do not survive the renormalization limit.Feeble Wonk said:So, if I understand correctly, you are saying that the (bare) electron itself is a meaningless concept. According to the theory, the EM field is what is "real", and the electron is just a localized description of the field. Is that accurate?
Real in theoretical physics is what is gauge invariant, has a well-defined dynamics in time (since reality happens in time), approximates a real world situation (it is always to some extent an idealization).vanhees71 said:I don't know, what you mean by "real". It's a very confusing word
The whole construction happens in the asymptotic space, which even for an interacting theory is a Fock space, since asymptotic particles are free by definition. In Fock space, normal ordering is all that is required to render a polynomial, local operator meaningful. Thus you can start the induction with an arbitrary local polynomial in c/a operators.philosophus said:I do not understand the start of induction, unless some renormalization is hidden here to define the normal ordered power of the free fields
Is there a typo or omission in the last sentence of this paragraph:A. Neumaier said:In the light of the recent discussion starting here, I updated this Insight article, adding in particular detail to the section ''Axioms for causal quantum field theory''.
?Unfortunately, models proving that QED (or other interacting local quantum field theories) exists have not yet been constructed. On the other hand, there are also no arguments proving rigorously that such models exist. For a fully rigorous solution – a problem which for interacting 4-dimensional relativistic quantum field theories is open.
Thanks for pointing it out. Indeed, the paragraph was garbled. I corrected it and added some more information. Now the paragraph readsstrangerep said:Is there a typo or omission in the last sentence of this paragraph:
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Unfortunately, models proving that QED (or another interacting local quantum field theory in 4 spacetime dimensions) exists have not yet been constructed. On the other hand, constructions are available in 2 and 3 spacetime dimensions, and no arguments are known proving rigorously that such models cannot exist in 4 dimensions. Finding a fully rigorous construction for an interacting 4-dimensional local quantum field theory or proving that it cannot exist is therefore a widely open problem. My bet is that a rigorous construction of QED will be found one day.
A. Neumaier said:However, the whole procedure makes perturbative sense also without these requirements.
In particular, for quantum field theory in curved space-time one sacrifices condition 1, with success; see work by Stefan Hollands.
Yes, it is different. Hollands is not doing causal perturbation theory since, as I said, in his work condition 1 (i.e., covariance) is sacrificed, by using a cutoff.atyy said:Which papers of Stefan Hollands? I took a quick look at https://arxiv.org/abs/1105.3375 which introduces an ultraviolet cutoff, then takes it to infinity, so it seems a bit different from causal perturbation theory which "nowhere introduces nonphysical entities (such as cutoffs, bare coupling constants, bare particles or virtual particles)".