-------------------------
S8a. Why renormalization?
-------------------------

Quantum field theory is what particle physicists define it is, and 
this includes many working interacting QFTs. But it is not a theory
in the mathematical sense. This is due to the freedom they take
when discussing the renormalization needed to remove formal
infinities from their theories.


Finite renormalization just refers to the fact that the coefficients 
in a Hamiltonian are not directly measurable but only computable as
function of some key observables. It is simply a consequence of the 
historical accident that these coefficients were given names (masses, 
charges) that sound like real properties, while they are in fact 
indirectly related to them.

Thus in solid state physics one gets bare masses of quasiparticles
from the coefficients of a Hamiltonian, but they are just parameters
and related to the measurable masses by some transformation, which is
dubbed the finite renormalization.

Infinite renormalization is needed in ordinary QM when the potential 
gets too singular, for example with delta-function potentials that 
model contact interactions. Hardly ever discussed in textbooks but 
important for understanding. See, e.g., hep-th/9710061, or Chapter I.3
in
   R. Jackiw,
   Diverse topics in theoretical and mathematical physics,
   World Scientific, Singapore 1995.
A paper by Dimock (Comm. Math. Phys. 57 (1977), 51-66) shows rigorously 
that, at least in 2 dimensions, delta-function potentials define 
the correct nonrelativistic limit of local scalar field theories.

In mathematical terms, infinite renormalization means that the 
interaction is a limit of regularized interactions related to fixed 
measurable quantities by finite transformations which, however, 
diverge when the regularization is removed. The limiting interaction 
remains, however, well-defined as a densely defined operator in 
Hilbert space.

For exactly the same reason it is needed in relativistic QFT, since
local fields imply singular interactions. But in 4 dimensions, the
limiting process is not well understood mathematically.

In 1+1 dimensions, everything is then well-defined mathematically
in terms of rigorous renormalization theory, for arbitrary polynomial 
interactions. (See the book by Glimm and Jaffe).

The 1+2-dimensional case is significantly more difficult and needs 
a restriction on the polynomial degree. There is a nontrivial 
renormalization theory for Phi^4 theory, which is mathematically 
well-understood. 

Only the 1+3 dimensional case is at present completely open.



What is loosely called 'infinite' in traditional discussions of 
renormalization means, strictly speaking, only that for the bare 
quantities, the limit where a cutoff goes to infinity does not exist. 
At any finite value of the cutoff, both the Hamiltonian and the 
counterterms are finite. If it were not so, one couldn't do 
renormalization and get something finite. 

The problem solved by Tomonaga, Schwinger and Feynman, for which they 
got the Nobel prize, was that they discovered how to
produce a well-defined limiting theory for cutoff to infinity
that allows to extract finite values for quantities comparable with 
experiment.

All renormalization until today follows the same pattern. 
One does certain formal computations at finite cutoff and moves, 
at some point where it no longer harms, the cutoff to infinity, 
being left with approximate formulas (at some fixed or variable 
loop order) that no longer contain a cutoff and have finite values.