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In quantum field theory we have the concepts of regularization and renormalization. I'm a little confused about these two. In my understanding regularization is a way to make divergent integrals convergent and in renormalization you add terms to the Lagrangian which in turn cancel the divergent integrals.

Is there a connection between the two of these, or are these two separate ways of dealing with the infinities, i.e. do you use them both together to make the divergences disappear?

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Regularization and renormalization are conceptually distinct.

As you essentially indicate, regularization is the process by which one renders divergent quantities finite by introducing a parameter $\Lambda$ such that the "original divergent theory" corresponds to a certain value of that parameter. I put "original divergent theory" in quotations because strictly speaking, the theory is ill-defined before regularization.

Once you regularize your theory, you can calculate any quantity you want in terms of the "bare" quantities appearing in the original lagrangian (such as masses $m$, couplings $\lambda$, etc.) along with the newly introduced regularization parameter $\Lambda$. The bare quantities are not what is measured in experiments. What is measured in experiments are corresponding physical quantities (the physical masses $m_P$, couplings $\lambda_P$, etc.).

Renormalization is the process by which you take the regularized theory, a theory written in terms of bare quantities and the regularization parameter $(\Lambda, m, \lambda, \dots)$, and you apply certain conditions (renormalization conditions) which cause physical quantities you want to compute, such as scattering amplitudes, to depend only on physical quantities $(m_P, \lambda_P, \dots)$, and in performing this procedure on a renormalizable quantum field theory, the dependence on the cutoff disappears. So, in a sense, renormalization can be thought of as more of a procedure for writing your theory in terms of physical quantities than as a procedure for "removing infinities." The removing infinities part is already accomplished through regularization.

Beware that what I have described here is not the whole conceptual story of regularization and renormalization. I'd highly recommend that you try to read about the following topics which give a more complete picture of how renormalization is thought about nowadays:

  1. effective field theory
  2. wilsonian renormalization
  3. renormalization group
  4. renormalization group flows

You may also find the following physics.SE posts interesting/illuminating:

What exactly is regularization in QFT?

Regulator-scheme-independence in QFT

Why is renormalization necessary in finite theories?

Why do we expect our theories to be independent of cutoffs?

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    $\begingroup$ +1 for "renormalization can be thought of as more of a procedure for writing your theory in terms of physical quantities than as a procedure for "removing infinities."" And an additional physics.SE reference : physics.stackexchange.com/questions/73403/… $\endgroup$ – Adam Jun 18 '14 at 19:40
  • $\begingroup$ @Adam Thanks; I added your reference in the body of the answer as well. $\endgroup$ – joshphysics Jun 18 '14 at 19:44

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