what is renormalization and what does it do?

meopemuk

I am probably too conservative, but I think that we have too little (if any) reliable experimental data about black holes and their thermal spectrum. So, it is too early to use them as an argument.

I don't have any objections against using effective field theories in condensed matter and solid state physics. But I am not convinced that this is a viable approach for fundamental interactions.

According to quantum mechanics, the time evolution of a closed system is described by the time evolution operator

[tex] U(t) = \exp(\frac{i}{\hbar}Ht) [/tex]

In QED (and QCD) we know the Hamiltonian H pretty well. We know that it contains infinite renormalization counterterms and that it allows us to calculate the S-matrix very accurately. Now, let us take this Hamiltonian and try to solve some simple time-dependent problems. Let's not talk about "time varying symmetry breaking", which, as far as I know, is not well-understood yet. Let us consider the simplest possible task of calculating the time evolution of a 1-electron state [itex] | \Psi(0) \rangle = a^{\dag} |0 \rangle[/itex]. My point is that the naive quantum-mechanical expression

[tex] | \Psi(t) \rangle = \exp(\frac{i}{\hbar}Ht) | \Psi(0) \rangle [/tex]

just doesn't make sense in this case. For example, it would predict unphysical processes, like "electron -> electron + photon". Is there another (less naive) approach to this simple problem?

Eugene.

Slaviks

It would be great (although very surprising) to learn we've reach the bottom, but the problem is that we cann't tell!

No, it is not. Unless one makes such a belief system a requirement to be called "theoretical physicist".

BTW, there are even more deep-rooted reasons to believe that "the dream is dead" -- see the discussions of Leonard Susskind's work, e.g. http://rabett.blogspot.com/2006_01_01_archive.html


What are these clues?
What exactly do you mean by "it is known that two electrons are *exactly* identical"?

I don't see how an argument of particle identity can help you in extrapolating to arbitrary small scales which are never probed. E.g., two hydrogen atoms in their respective ground states are exactly identical, and there no way to prove their compositness once the energy of allowed experiments is well below the hyperfine splitting (the distance to the first excited state) = 21 cm wavelength. The very fact the we can have BEC means that the atoms being condensed are indistinguishable, once we cool the things cold enough.

For me a clear indication that QED alone has told everything it can is the fact that in the most recent tests of measured vs. calculated electron magnetic moment, the greatest uncertainty is in a tiny correction which come due to polarization of the hadronic vacua -- at high enough orders (energies), you cann't use QED alone.

If a more elegant formulation is possible, then it is always welcome! But as far as verifyable predictions are identical, the choice of philosophy it remains the matter of taste. For some, virtual particles are ugly and horrible, for other they may be quite inspirational. Everyone has his own intuition, experimental validity is the judge (that's what I like about physics).

nrqed

I may be wrong but it sounds as if you imply that an effective field theory approach implies the assumption of granularity of spacetime (I may have misinterpreted your words, if so I apologize). Saying that a theory is an eft does not imply that. It just implies that at some scale "new physics" arises. The nature of this new physics is quite arbitrary, it could be granularity of spacetime but it could be a new force, inner structure to the particles (including string-like structure) etc etc etc. So in that sense it is quite general.

A good example is of course the Fermi model of the weak interaction, which can be used as an eft as long as energies are much below the weak scale (the W mass, say), including in loop diagrams. The non-renormalizability of the theory indicated the need for new physics which had nothing to do with granularity of spacetime.

What are the conditions under which the dressed particle appraoach may be applied? Could it have been applied to cure the infinities of the Fermi model? In that case it would have missed the fact that there *was* a new underlying theory: the gauge weak interaction.

Finally, let me emphasize that the eft approach is extremely useful not only as a way to think of new physics but also to describe known theories at low wnergies. For example chiral perturbation theory, heavy quark eft and NRQCD for the strong force at low energies.. (there is also an equivalent to NRQCD called NRQED....you may recognize my handle). So the concept of eft has proven extremely successful as a tool that works very well to describe known theories. It suggests to me that it is a useful tool for describing known theories relative to "new physics".


But I don't know anything about the dressed particle approach and it would be certainly interesting to learn. I guess my first question qwould be: what are the conditions in which it is applicable? Could it has "solved" the Fermi model? Could it be used to decsribe QCD at low energies?


Regards

Patrick

meopemuk

You would probably agree that this "race to the bottom" kept theoretical physics vibrant for so many centuries. Of course, you are right, that this is a dream of *some* theoretical physicists, not *all* of them. Personally, I am not impressed by Susskind's logic, because of hugh amout of unfounded speciulations associated with it.

Agreed. That's why I used the word "clue" instead of "proof".


Alternative formalisms have their own merits, even if they lead to exactly the same predictions. A good example is provided by three formulations of quantum mechanics - Schroedinger's wave equation, Heisenberg's matrix mechanics, and Feynman's parth integral.

However, I believe that the "dressed particle" approach is not merely a different mathematical formalism. For me the biggest surprise was to learn that this approach predicts instantaneous (not retarded) Coulomb and magnetic interactions between charged particles. It appears that this conclusion does not contradict the usual field-based S-matrix approach, because, as I tried to point out earlier, the latter approach can't tell much about the time evolution of interacting systems and, therefore, about the speed of propagation of interactions. Moreover, at closer inspection, it appears that the possibility of faster-than-light interactions does not contradict any experimental evidence either. There are quite a few recent experiments (e.g., photon tunneling) which can be interpreted from the viewpoint of instantaneous interactions. So, in my opinion, this debate (which, supposedly, was closed 100 years ago) is now wide open.

Eugene.

meopemuk

Thank you for the correction. I used space-time "granularity" or "discreteness" as an example resembling the situation in condensed matter physics, where the "new physics" is associated with the crystal lattice. You are right that in theories of fundamental particles there could be other sources of "new physics": new heavy particles, strings, etc. The beauty (again, this is my personal view, and others may not see it as beautiful at all) of the "dressed particle" idea is that it allows us to formulate QFT self-consistently without relying on yet unknown "new physics".

If a theory is non-renormalizable (i.e., the number of different counterterm types is infinite), then the "dressed particle" formalism is powerless to change that. It can make the Hamiltonian of such a theory finite, but the number of independent parameters will remain infinite.


I agree that EFT is a very valuable tool for deriving some approximations (e.g., low-energy) of fundamental theories. The "dressed particle" approach is not an approximation. Its major idea is to apply a unitary (dressing) transformation to the field-theoretical Hamiltonian of renormalized QFT so that certain "bad" terms are eliminated. Examples of such "bad" terms are trilinear "electron -> electron + photon" and "vacuum -> electron +positron + photon" interaction operators responsible for "self-energies" and "vacuum polarization". It is important that the "dressing" transformation is carefully chosen so that the original (accurate) S-matrix is not changed. It can be also proven that this transformation can be chosen so that it cancels out all infinite counterterms in the Hamiltonian.

As a result, we obtain a finite Hamiltonian in which particles interact via instantaneous potentials. This Hamiltonian produces the same S-matrix as the original field-theoretical Hamiltonian. However, the advantage is that you'll not need regularization and renormalization. All loop integrals will be finite. Moreover, you can easily form the time evolution operator with this Hamiltonian, and you can diagonalize this Hamiltonian to get energies and wave functions of bound states, as is normally done in non-relativistic quantum mechanics. (These procedures were quite troublesome with the original field-theoretic Hamiltonian). The only significant difference with respect to ordinary quantum mechanics (where the number of particles was assumed fixed) is that interactions changing the number of particles are allowed as well, e.g., "2 electrons -> 2 electrons + photon"

Eugene.