Renormalization and scale dependence

[TrickyDicky]

Since Wilson work in the 70s, the renormalization technique in QFT is physically justified with the concept of scale dependence(scale anomaly) of the parameters.
This apparently is akin to a universal version of the characteristic length usually applied to specific physical systems to define their scale.

Can anybody explain how is this scale dependence introduced(independently of the specific procedure:perturbative cutoff, dimensional, lattice...)? Where does it come from?

Does the Haag's theorem imply that this scale dependence technique is not even related to the QFT lagrangian?

 

[atyy]

A circle seen from far away may look like a point, so a point is an effective theory of a circle. But if one looks under the microscope, one will see that it is a circle. So the theory one uses - point or circle - depends on how closely or finely one looks. Scale dependence in quantum field theory is an analgous idea. If we probe the system using low energies and long wavelengths, then we will be looking at the system more coarsely. If we probe the system using high energies and short wavelengths, then we will be looking at the system more finely. Just like the microscope, as we change how finely we look, the theory changes in a way which is manifest in the scale dependence.

Of course, one can get much more dramatic changes than scale dependence, like uncovering new degrees of freedom.

The basic idea is described by Kadanoff's block spin picture https://www.amazon.com/dp/0521804426/?tag=pfamazon01-20

 

[bhobba]

This is actually a very deep question.

Murray Gell-Mann gave a talk on it - its actually tied up with beauty in physics and math:


Its this scale dependence that causes the same things to pop up over and over.

Thanks
Bill

 

[TrickyDicky]

A circle seen from far away may look like a point, so a point is an effective theory of a circle. But if one looks under the microscope, one will see that it is a circle. So the theory one uses - point or circle - depends on how closely or finely one looks. Scale dependence in quantum field theory is an analgous idea. If we probe the system using low energies and long wavelengths, then we will be looking at the system more coarsely. If we probe the system using high energies and short wavelengths, then we will be looking at the system more finely. Just like the microscope, as we change how finely we look, the theory changes in a way which is manifest in the scale dependence.

Yes, this is the usual metaphor. It basically amounts to trivially admitting we are stuck with the coarse view of the point and the circle scapes us so far. IOW that we don't have the right picture that Gell-Mann refers to in the talk linked by Bill.

The problem I see with this metaphor is that people take it too literally in the sense that theyseem to infer from it that the only way to advance towards the true (not just effective) field theory is by smashing matter with ever higher energies.

This is actually a very deep question.

Murray Gell-Mann gave a talk on it - its actually tied up with beauty in physics and math:


Its this scale dependence that causes the same things to pop up over and over.

Thanks
Bill

I remember that talk, it just touches upon the scale dependence issue, it is more concerned with the highly related concept of universality(when he talks about the similarity of the onion layers) that can be found in the renormalization group both in high energy physics and in condensed matter physics, i.e.: sameness of critical exponents in Kadanoff's second order phase transitions terms.

Probably both the scaling and the approximate self-similarity are sides of the same coin.

 

[atyy]

Yes, this is the usual metaphor. It basically amounts to trivially admitting we are stuck with the coarse view of the point and the circle scapes us so far. IOW that we don't have the right picture that Gell-Mann refers to in the talk linked by Bill.

The problem I see with this metaphor is that people take it too literally in the sense that theyseem to infer from it that the only way to advance towards the true (not just effective) field theory is by smashing matter with ever higher energies.

I think there are two other ideas out there. The first is to look for deviations from the present theory at low energy (eg. the discussion between Gross and Strassler reported by Motl http://motls.blogspot.com/2014/03/gross-vs-strassler-gross-is-right.html). The second is still to smash things at higher energy, but to realize that it may be too expensive for us, but maybe we can use cosmological observations (eg. BICEP2, if it pans out).

Anyway, yes, I believe we agree on scaling. Scaling is just one a manifestation of different effective theories at different scales. Universality has to do with fixed points occurring as one performs scaling or renormalization flow.

 

[TrickyDicky]

I think there are two other ideas out there. The first is to look for deviations from the present theory at low energy (eg. the discussion between Gross and Strassler reported by Motl http://motls.blogspot.com/2014/03/gross-vs-strassler-gross-is-right.html).

I would side with Strassler there.

Anyway, yes, I believe we agree on scaling. Scaling is just one a manifestation of different effective theories at different scales. Universality has to do with fixed points occurring as one performs scaling or renormalization flow.

I think it is interesting to contrast this scale dependence associated with quantum effects against the scale invariance of the classical theories. More commonly discussed in terms of(spontaneous) breaking of symmetries.
There is room to think that rather than symmetry beaking we might be facing the failure of approximate but not exact symmetries to account for the quantum effects.