In physics, renormalization is the process by which the "bare" parameters of a theory (those that appear in the equations) are related to their measurable quantities. In Quantum Field Theory, this is also used to get rid of infinities arising in loop calculations, and get reasonable results from the theory. But initially, renormalization has nothing to do with quantum theory or infinities, as the following example demonstrates.

Example - A Damped Oscillator

Consider the case of a damped harmonic oscillator, say, a mass on a spring. If there was no damping, the motion of the mass would be described by the equation (amplitude and phase are omitted because they are not relevant to this discussion):

x(t) = sin(ωt)
The parameter ω is the "bare" (unrenormalized) frequency of the oscillator. It is, however, not equal to the observed frequency of the oscillation because of the effects of damping.

A damped oscillator is described by a slightly different equation:

x(t) = et sin(Ωt)
Where γ is the damping coefficient and
Ω = (ω2 - γ2)1/2
is the actual measured frequency of oscillation, which is different from the bare frequency ω.

Now, ω is the frequency with which the mass would oscillate in the absence of damping. However, when we are dealing with a Real World experiment, there is no way to simply turn the damping off. No matter what kind of spring we use, it will always be there. Therefore, the bare frequency ω is not a directly observable quantity. The only parameter we can measure is the "physical" parameter Ω. Neither of them is infinite, and they are related to each other by the simple equation given above. So, once we have measured Ω (and the damping coefficient γ), we know how to recover the bare parameter ω in case we need it for some further calculations.

This is the basic idea of what renormalization is about. Now, on to Quantum Field Theory.

The Mass of the Electron

In Quantum Electrodynamics, the motion of an electron between two points in spacetime is described by a mathematical function called a propagator. The free propagator of the electron (i.e. in the absence of interactions) looks like this:

S0(p) = (p - m)-1
where p is the momentum of the electron in question1 and m is the bare (unrenormalized) electron mass.

If electrons didn't interact at all we wouldn't know of their existence, of course. So a realistic theory must include interactions. However, a single electron not only interacts with other electrons (or other charged particles in general), but also with itself. In classical physics, this is the result of the electron's electric charge interacting with the electric field generated by it. In Quantum Field Theory, the self-energy arises from the contributions of loop diagrams. For example, an electron can emit a photon, and reabsorb the same photon later. The diagram for this process has a closed loop.

The effect of these self-energy contributions is to change the observable mass of the electron. The full propagator (or, more correctly, the two-point Green's function) for the electron reads

S(p) = (p - m - Σ(p))-1
where the loop contributions are contained in Σ(p), which is called the proper self-energy. The physical (or renormalized) mass M of the electron is now
M = -S-1(0) = m + Σ(0).
Σ(0) can be calculated to arbitrary precision, so it seems we are done here. Unfortunately, the value obtained for Σ(0) is infinite (it involves divergent integrals), which means that M is also infinite for any finite value of the bare mass m. The measured value of M is, of course, finite, so the theory appears to be bogus.

The solution to this problem is the following: Since we can never observe a non-interacting electron, its bare mass m has no actual physical significance. It can be assigned an arbitrary value. The only requirement is that the chosen value for m yields the correct physical electron mass M from the formula above.

It is obvious that for M to be finite, m would have to be infinite, like Σ, but in such a way that their infinities cancel each other out. Of course, infinity is not a number, and infinity minus infinity is undefined, so a mathematically more rigorous way to get a grip on the infinities is needed here. This is called regularization.

The idea of regularization is to isolate the divergences of a function by introducing a parameter Δ so the function is finite unless Δ reaches some limit value (usually Δ→0), and to expand the function into a Laurent series in Δ around this value. The divergent part of the function then appears as a pole in the expansion.

Once Σ(0) is expressed as a Laurent series, with the infinite part separated as a pole, one can construct2 an m with the same pole but opposite sign, so the infinities cancel each other out in the sum, resulting in a finite value for M, as required. The electron mass is then correctly renormalized.

The renormalization procedure needs to be applied to all parameters of a quantum field theory (masses, coupling constants, mixing angles, etc.), and also to the fields themselves. If it can be proven that all occuring infinities can be absorbed in such a redefinition of these quantities from their bare values to their physical ones, the theory is called renormalizable.

1The p is really a "p-slash", or γμpμ, where the γμ form a Clifford algebra.
2This gets a little involved because m itself is used in the calculation of Σ, so the renormalization has to be done order-by-order in perturbation theory. Also, the wave function renormalization of the electron field has to be taken into account.