Editing Renormalization (section)

== Regularization ==
Since the quantity {{math|∞ − ∞}} is ill-defined, in order to make this notion of canceling divergences precise, the divergences first have to be tamed mathematically using the [[limit (mathematics)|theory of limits]], in a process known as [[regularization (physics)|regularization]] (Weinberg, 1995).

An essentially arbitrary modification to the loop integrands, or ''regulator'', can make them drop off faster at high energies and momenta, in such a manner that the integrals converge.  A regulator has a characteristic energy scale known as the [[Cutoff (physics)|cutoff]]; taking this cutoff to infinity (or, equivalently, the corresponding length/time scale to zero) recovers the original integrals.

With the regulator in place, and a finite value for the cutoff, divergent terms in the integrals then turn into finite but cutoff-dependent terms.  After canceling out these terms with the contributions from cutoff-dependent counterterms, the cutoff is taken to infinity and finite physical results are recovered.  If physics on scales we can measure is independent of what happens at the very shortest distance and time scales, then it should be possible to get cutoff-independent results for calculations.

Many different types of regulator are used in quantum field theory calculations, each with its advantages and disadvantages.  One of the most popular in modern use is ''[[dimensional regularization]]'', invented by [[Gerardus 't Hooft]] and [[Martinus J. G. Veltman]],<ref>{{Cite journal | last1 = 't Hooft | first1 = G. | last2 = Veltman | first2 = M. | doi = 10.1016/0550-3213(72)90279-9 | title = Regularization and renormalization of gauge fields | journal = Nuclear Physics B | volume = 44 | issue = 1 | pages = 189–213 | year = 1972 |bibcode = 1972NuPhB..44..189T | hdl = 1874/4845 | url = https://repositorio.unal.edu.co/handle/unal/81144 | hdl-access = free }}</ref> which tames the integrals by carrying them into a space with a fictitious fractional number of dimensions.  Another is ''[[Pauli–Villars regularization]]'', which adds fictitious particles to the theory with very large masses, such that loop integrands involving the massive particles cancel out the existing loops at large momenta.

Yet another regularization scheme is the ''[[Lattice field theory|lattice regularization]]'', introduced by [[Kenneth G. Wilson|Kenneth Wilson]], which pretends that hyper-cubical lattice constructs our spacetime with fixed grid size. This size is a natural cutoff for the maximal momentum that a particle could possess when propagating on the lattice. And after doing a calculation on several lattices with different grid size, the physical result is [[extrapolate]]d to grid size 0, or our natural universe. This presupposes the existence of a [[scaling limit]].

A rigorous mathematical approach to renormalization theory is the so-called [[causal perturbation theory]], where ultraviolet divergences are avoided from the start in calculations by performing well-defined mathematical operations only within the framework of [[Distribution (mathematics)|distribution]] theory. In this approach, divergences are replaced by ambiguity: corresponding to a divergent diagram is a term which now has a finite, but undetermined, coefficient. Other principles, such as gauge symmetry, must then be used to reduce or eliminate the ambiguity.