Undergrad On-shell virtual particles and 'physical' Hilbert spaces....

[asimov42]

TL;DR

What is the relationship between on-shell virtual particles and physical Hilbert spaces?

Hi all,

Just a clarification question as I'm learning. It's possible to have Feynman diagrams where the internal lines (virtual particles) are in fact on shell. 'On shell' would imply 'observable,' (maybe?) but as noted in @A. Neumaier's great FAQ, only sets of Feynman diagrams have predictive power - I'm wondering what the relationship is between 'on shell' internal lines and actual, physical Hilbert spaces. E.g., shouldn't on-shell particles cause problems when summing diagrams? (i.e., an internal line still cannot represent a 'real' particle, correct?)

 

[A. Neumaier]

Summary:: What is the relationship between on-shell virtual particles and physical Hilbert spaces?

Virtual particles are off-shell with probability 1; it makes no sense to talk about on-shell virtual particles.

It's possible to have Feynman diagrams where the internal lines (virtual particles) are in fact on shell.

No. There is only one Feynman diagram per integral, not an infinite family, one for each collection of momenta.
In a Feynman diagram, the internal momenta are not numerical vectors BUT integration parameters ranging over Minkowski space. This is reflected by saying that the virtual particles are off-shell. The on-shell part of the integration region has Lebesgue measure zero.

shouldn't on-shell particles cause problems when summing diagrams?

The on-shell part of the integration region has Lebesgue measure zero, hence does not contribute to integral, except seemingly at singularities. The infinities appearing there are treated by renormalization.

 

[asimov42]

Ah thanks @A. Neumaier - that makes perfect sense.

Other question: can't tree-level diagrams have internal lines that are on shell in certain cases? (or rather still only at one point - but the integral does not explode?)

In the renormalization process, does one then effectively exclude the singular points? (just not being familiar with standard methods for renormalization - but knowing that the infinities must be removed somehow).

Given my extraordinarily limited knowledge, is this just the process regularization: introducing the cutoff, cancelling with cutoff-dependent counter terms, and then moving the cutoff to infinity?

 

[vanhees71]

If the momenta of internal lines get on shell, it's of course trouble, because an internal lines stands for a (free) propagator which is $\propto 1/(p^2-m^2+\mathrm{i} 0^+)$ if the momentum gets "on shell", i.e., $p^2=m^2$ you are obviously in trouble. This usually happens, when massless particles are involved. The most simple case is bremsstrahlung in QED (in leading order already at tree level). The phenomenon is known as IR and collinear singularities and have to be treated with appropriate soft-photon resummations (in QED a good keyword is Bloch and Nordsieck).

The formal reason is that in such cases the naive power counting of perturbation theory in terms of formal power series wrt. the coupling constant (or $\alpha$) breaks down, i.e., an infinite number of diagrams contributes at a given order since the powers from the verices in the numerator are canceled by the pole from the said IR and collinear singularities. You find a very good discussion of these issues in

Weinberg, Quantum Theory of Fields, Vol. 1.

 

[A. Neumaier]

Given my extraordinarily limited knowledge, is this just the process regularization: introducing the cutoff, cancelling with cutoff-dependent counter terms, and then moving the cutoff to infinity?

This is one of many ways of doing the renormalization; for asymptotically free QFTs (in general only if not resummed via the renormalization group or some other partial summation method) they all produce equivalent results. (In case of using a cutoff, regularization is just the process of introducing the cutoff; the remainder is called renormalization.)

 

[asimov42]

Ah yes thanks @vanhees71 - I'd read previously briefly about the problem of IR and collinear singularities (just didn't have sufficient context).

So, indeed, to confirm, (as @A. Neumaier said) that contributions in any perturbative calculation can only come from off shell 'virtual particles'; an on shell contribution always causes a divergence that needs to be canceled or removed in some other way? Just want to make sure I'm very clear on process.

 

[asimov42]

Related to the above is the following (quoting Luboš Motl): "The standard model is consistent in perturbative expansions."

If I understand correctly, this means that with renormalization (which is well defined and valid and not hocus pocus) all infinities discussed above cancel perturbatively to arbitrarily high orders (energies) - but that it breaks in the non-perturbative case at the Planck energy?

 

[vanhees71]

Ah yes thanks @vanhees71 - I'd read previously briefly about the problem of IR and collinear singularities (just didn't have sufficient context).

So, indeed, to confirm, (as @A. Neumaier said) that contributions in any perturbative calculation can only come from off shell 'virtual particles'; an on shell contribution always causes a divergence that needs to be canceled or removed in some other way? Just want to make sure I'm very clear on process.

The usual renormalization procedure is about UV divergences, i.e., the divergence of momentum integrals (from a loop in Feynman diagrams) at large energies and momenta. The occurance of these divergences is just an artifact of the very sloppy treatment of distributions by physicists, and the regularization procedures are just treating the distributions a bit more rigorous by first "smoothing them". Most physically is an approach, where you "smear out" the field operators. This is formalized in terms of the socalled "causal perturbation theory" approach, or Epstein-Glaser approach. For QED it's nicely elaborated in the textbook Scharf, Finite Quantumelectrodynamics.

The IR trouble is due to the fact that you have to resum the divergences occurring when the integral hits an on-shell pole in the propagators, which is artificial, because it can only occur, because free charges lead to an infinite range of the fields (the Coulomb potential only goes with $1/r$), and the asymptotic states of charged particles are not the naive ones of the free particles. The physical entities are rather "dressed particles", i.e., the bare particles with their own electromagnetic field. In the standard treatment this is handled by the said soft-photon resummations, which are "coherent sums" which can be identified with "coherent states" describing the charges' own electromagnetic field. This trouble you have already in the semiclassical theory (i.e., the particles are quantized and the em. field is kept classical) of non-relativistic QM, where it occurs in the divergence of the Coulomb-scattering cross section. A very pedagogical paper starting from this more simple example and then using the idea in QED is

P. Kulish, L. Faddeev, Asymptotic conditions and infrared divergences in quantum electrodynamics, Theor. Math. Phys. 4 (1970) 745.
https://dx.doi.org/10.1007/BF01066485