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I am new to QFT, and so far I have studied dimensional regularization and Pauli-Villars regularization. These seem to be the only two regularization techniques discussed in most introductory textbooks. The Wiki page on regularization lists a few regularization techniques, one of which is lattice regularization. I would like to know more about this. However, the Wiki redirects to another page that does not discuss lattice regularization at all. I have also tried searching the web but without much luck.

What exactly is lattice regularization and how is it carried out? Are there any books that introduce this method to a beginner, especially one who only knows scalar $\phi^4$ theory and some QED (I don't know much at all about QCD)?

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The idea is to consider a theory defined on a discrete lattice instead of a continuous spacetime. By doing so, you introduce a minimum length scale—the lattice spacing—and consequently, you end up with a maximum momentum scale. Hence, your theory is now regulated in the ultraviolet limit because you ended up implementing a momentum cutoff.

I'm not acquainted with the details of lattice field theory, but I think the term "lattice regularization" might be unusual. You might find more information by looking up lattice field theories in more generality (maybe start with the question linked in the comments to your post).

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  • $\begingroup$ So by putting the theory and spacetime on a lattice everything is automatically regularized? $\endgroup$
    – CBBAM
    Commented Mar 13, 2023 at 17:26
  • $\begingroup$ @CBBAM At least in the ultraviolet, yes. Notice that putting the theory in a lattice does mean you are ignoring any degrees of freedom with length scales smaller than the lattice spacing, and hence you are already sort of introducing a cutoff, but in a different way. $\endgroup$
    – Níckolas Alves
    Commented Mar 13, 2023 at 17:43
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    $\begingroup$ Thank you, this really helped. I thought that lattice regularization was its own regularization procedure carried out after the theory had already been put on a lattice. It makes sense now. $\endgroup$
    – CBBAM
    Commented Mar 13, 2023 at 17:53

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