Natural Order and Science

[Alex K]

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I would like to come back to this statement, in which you seem to trace back the reason why particles and virtual particles have a different "ontological status" (one really exists, the other is an artefact of calculational method) to the fact that one appears as scattering matrix-elements, whereas the others don't. Since you are eager to discuss S-matrix theory, you must be aware that this distinction is problematic because

1. S-Matrix elements are infra-red divergent due to soft emission, and only lead to physically meaningful results if they are combined with virtual corrections in scattering cross-sections. From this it appears to me that real emission of particles alone isn't even defined, and that the distinction between the two classes of particles based on arguments from S-Matrix theory is problematic. What's your perspective?

2. The S-Matrix is a somewhat artificial construct obtained by assuming that ingoing and outgoing particles stop interacting, and then taking the time to +/- infinity and using the resulting asymptotic free states as "external particles". This works great when describing scattering processes in which isolated, massive and stable particles meet and part again, but if you depart from this idealized scenario, things like 1.) happen. Also, if you just look at the unitary time evolution operator without  taking it to asymptotic times, the distinction between what's an S-Matrix index and what isn't surely isn't as clear-cut any more.

Do you disagree? Or do you subscribe to an axiomatic S-Matrix approach to particle physics in which there is no local Lagrangian and no time evolution? Because that worked out so well?

The main idea of my previous response was to show that all Gauge Bosons are in fact forces that fermions carry on them. The interaction of fermions activated those bosons which then cause disturbances in the field of free flowing particles. These Bosons (termed as particles) are not detectable as particles in any sense because they are force fields that can be recorded as disturbances but not as particles.

As for s-matrix, a process like e+e− → μ+μ− is not well-defined in perturbation theory because of the infrared divergence.

Infra-red divergences are due to using UNPHYSICAL variables to describe in and out states and introduce gauge invariant, physical “dressed” states. In scattering calculations, it is a common tendency to make an assumption that the coupling constant can be set to zero at infinitely remote times. In QED this might appear reasonable while in QCD such calculations are often based on the premise of some parton-hadron duality. The results of scattering calculations are, though, plagued by infra-red (IR) divergences.

It has long been known that the soft divergences occurring in QED can be cancelled out in transition rates or cross sections computed for detectors with finite energy resolution: the soft divergences which occur in a scattering process due to the emission of an undetected soft real photons with total energy ≤ El exactly cancel out the soft divergences due to virtual photon corrections order by order in perturbation theory. This cancellation was first shown by Bloch and Nordsieck in QED and is referred to as Bloch-Nordsieck theorem. In ordinary (commutative) Yang-Mills theories just as in QED, there exist IR (soft) divergences due to MASSLESS GLUONS. This has been guaranteed by the theorems of Kinoshita and of Lee and Nauenberg known as KLN theorem which states that the transition rates are free of the collinear and soft divergences if we sum over initial and final states. This theorem is a fundamental quantum mechanical theorem on the basis of unitarity of S-matrix.

Field theories with space-time non-commutativity, for example, do not have a unitary S-matrix. The extra branch cuts in these theories, are developed in the loop diagrams which are responsible for the failure of the cutting rules and lack of unitary S-matrix.

In a scattering process computed up to one-loop order, if we consider the non-commutative soft photon emission, the non-commutative logarithmic IR divergence in the vertex correction will be cancelled in the cross section. However, there are additional non-Abelian type diagrams in which their non-commutative logarithmic and quadratic divergences cannot be cancelled out using the cross section method. This non-cancellation is attributed to an important difference between soft and non-commutative IR divergences. The soft divergences are associated with the classical limit but non-commutative IR divergences are completely a quantum mechanical effect and the soft divergences that only appear in the non-planar vertex correction to be cancelled out in the physical cross section to all orders.

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Ok, I'll for now act as if you were actually making an argument that you understand rather than just copying other people's material without attribution (which as an author, is very offensive to me).

Your new wall of text seems to be reinforcing my point. You must be aware that the KLN theorem, which "you" outline above, precisely requires the presence of "virtual" particles in loops in order to make sense of the sums over degenerate initial and final states.