# Etymology of “Renormalisation”

Just out of curiosity, does anyone know why "renormalisation" is so named? Who first came up with the term, and why was it used? I did a mathematics undergraduate so to me "normalisation" means rescaling a Hilbert space vector to have unit norm.

Heuristically the term renormalisation makes some sense, as it is a systematic process for "normalising" divergent quantities to give well-defined probabilities. But it seems like a bad name to me, because it's patently impossible to truly "normalise" a divergent quantity. As far as I know, the correct perspective is that renormalisation replaces bare parameters by physical ones, thus removing the infinities.

Is there something I have misunderstood, or is that another case of historical contingency clouding the terminology. Surely "reparameterization" would be a better name?

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You haven't missed something. The authors of QED took the term 'renormalization' (which makes sense when you are rescaling fields -- you fix their normalization to keep the kinetic term's overall scale fixed) and used it to name the whole suite of techniques (which doesn't make much sense, since coupling constants aren't normalized to begin with, and since you're changing the way you parametrize a model rather than altering a fixed set of parameters). – user1504 Jul 24 '13 at 18:48

Apparently, the term renormalization has been coined by several people in the 40's, as you can read on a CERN Courier from August, 2001 for instance. Below a quote of the relevant part:

The puzzle [how to tackle the infinities appearing on loop diagrams ?] was resolved in the late 1940s, mainly by Bethe, Feynman, Schwinger and Dyson. These famous theoreticians were able to show that all infinite contributions can be grouped into a few mathematical combinations, $Z_{i}$ (in QED, $i = 1,2$), that correspond to a change of normalization of quantum fields, ultimately resulting in a redefinition ("renormalization") of masses and coupling constants. Physically, this effect is a close analogue of a classical "dressing process" for a particle interacting with a surrounding medium.

but there is no reference to the original works. I guess they are the historical references introducing the full covariant perturbation theory, as you would have a good overview from the Nobel Prize website of Tomonaga, Schwinger and Feynman, or from the important paper by F.J. Dyson, The Radiation Theories of Tomonaga, Schwinger, and Feynman. Physical Review 75, 486–502 (1949).

As far as a conceive it, the name normalization has nothing really to do with norm, better with normal. You tend to make you're quantity normal (i.e. not wired or something, like a diverging physical quantity), you do not try to norm it.

More interesting is your second question, about a better name. This one already exists actually, and it is sometimes used: it's scaling invariance, scaling laws, or scaling something... This is cristal clear in the concept of renormalization group, from Wikipedia for instance, including all the historical relevant references, as for the CERN Courier cited above. For some reason the inventors of the renormalization group approach kept the name renormalization, but (re)scaling is definitely a better choice, as you rescale your theory actually (time-space length, mass, interaction constant, energy, ...)

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I'd be happy to accept your answer if you can find those references! Cheers! – Edward Hughes Jul 24 '13 at 16:18
@EdwardHughes Even if you don't accept this, it might be polite to upvote it. – user1504 Jul 24 '13 at 23:15
@EdwardHughes The first renormalization approach (i.e. not the renormalization-systematic-group approach) are described in the Nobel lectures by Feynman, Tomonaga and Schwinger. Dyson sketches the process in the paper I linked. If you find something interesting by yourself, you're wecolme to share :-). – FraSchelle Jul 25 '13 at 9:17

Given any quantum field theory one can construct the Feynman rules for calculating the Green's functions and S-matrix elements in perturbation theory. But in relativistic field theory one often encounters infinities in the calculation of diagrams containing loops. This is because the momentum variable in the loop integration ranges all the way from zero to infinity. In others words there is no intrinsic cut-off in momenta. These divergences will render the calculation meaningless. The theory of renormalization is a prescription which allows us to consistently isolate and remove all these infinities from the psychically measurable quantities. It should be emphasized that the need of renormalization is rather general and is not unique. Consider an electron moving inside a solid(lattice). Due to the interaction of the electron with tha lattice the effective mass of the electron m* is different from the mass of the electron m measured outside the solid. The electron mass (renormalized) from m to m* by the interaction. For relativistic field theory the situation is the same except for two distinctions.First, change(renormalization) due to interaction is infinite.Second, there is no way to switch off the interaction in order to measure both m and m*. This theory originally formulated for quantum electrodynamics by Feynman(1948), Schwinger, Tomonaga, Dyson and has been quite successful between theory and experiment.

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Sorry Snodrion, but I don't see how this is answering the question. – Ali Jul 21 '13 at 15:33
reparameterization is not a better name. renormalization has to do with interaction you don't replace bare parameters by physical ones. – Snodrion Jul 21 '13 at 15:37
@Snodrion - I think you are incorrect. See for example this. The idea of imposing a cutoff is called regularization and renormalization only comes into play when a regularized theory still has a divergence. Your example explicitly mentions a reparameterization - namely the rewriting of quantities in terms of m* rather than m! – Edward Hughes Jul 21 '13 at 18:24