View source for Physics:Pauli–Villars regularization

{{Short description|Regularization technique in quantum field theory}}
__NOTOC__
{{Renormalization and regularization}}
In [[Physics:Theoretical physics|theoretical physics]], '''Pauli–Villars regularization''' ('''P–V''') is a procedure that isolates divergent terms from finite parts in loop calculations in field theory in order to [[Physics:Renormalization|renormalize]] the theory. [[Biography:Wolfgang Pauli|Wolfgang Pauli]] and [[Biography:Felix Villars|Felix Villars]] published the method in 1949, based on earlier work by [[Biography:Richard Feynman|Richard Feynman]], Ernst Stueckelberg and Dominique Rivier.<ref name="Schweber">{{cite book |last=Schweber |first=S. S. |title=QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga |url=https://archive.org/details/qedmenwhomadeitd0000schw |url-access=registration |publisher=Princeton University Press |location=Princeton, N.J. |year=1994 |isbn=9780691033273 }}</ref>

In this treatment, a [[Divergence|divergence]] arising from a [[Physics:Loop integral|loop integral]] (such as [[Physics:Vacuum polarization|vacuum polarization]] or electron self-energy) is modulated by a spectrum of auxiliary particles added to the [[Physics:Lagrangian (field theory)|Lagrangian]] or [[Physics:Propagator|propagator]]. When the [[Physics:Mass|mass]]es of the fictitious particles are taken as an infinite limit (i.e., once the regulator is removed) one expects to recover the original theory.

This [[Physics:Regularization|regulator]] is [[Physics:Gauge invariance|gauge invariant]] in an abelian theory due to the auxiliary particles being minimally coupled to the photon field through the [[Gauge covariant derivative|gauge covariant derivative]].  It is not gauge covariant in a non-abelian theory, though, so Pauli–Villars regularization cannot be used in QCD calculations.  P–V serves as an alternative to the more favorable [[Physics:Dimensional regularization|dimensional regularization]] in specific circumstances, such as in chiral phenomena, where a change of dimension alters the properties of the [[Gamma matrices|Dirac gamma matrices]].

[[Biography:Gerard 't Hooft|Gerard 't Hooft]] and [[Biography:Martinus J. G. Veltman|Martinus J. G. Veltman]] invented, in addition to [[Physics:Dimensional regularization|dimensional regularization]], the method of unitary regulators,<ref>G. 't Hooft, M. Veltman, Diagrammar, CERN report 73-9 (1973), see Secs. 2 and 5-8; reprinted in {{cite book |first=G. |last='t Hooft |title=Under the Spell of Gauge Principle |publisher=World Scientific |location=Singapore |year=1994 }}</ref> which is a Lagrangian-based Pauli–Villars method with a discrete spectrum of auxiliary masses, using the path-integral formalism.

==Examples==
Pauli–Villars regularization consists of introducing a fictitious mass term. For example, we would replace a photon propagator  <math> \frac{1}{k^2 + i \epsilon} </math>, by <math>  \frac{1}{k^2 + i \epsilon}  -  \frac{1}{k^2 - \Lambda^2+ i \epsilon} </math>, where <math>\Lambda</math> can be thought of as the mass of a fictitious heavy photon, whose contribution is subtracted from that of an ordinary photon.<ref>{{cite book |last1=Peskin |last2=Shroeder |title=An Introduction to Quantum Field Theory |publisher=Westview Press |edition=Reprint |date=1995 |isbn=0-201-50397-2 }}</ref>

== See also ==
* [[Physics:Dimensional regularization|Dimensional regularization]]
* Ghosts (physics)
* [[Physics:Regularization|Regularization]]

==Notes==
{{Reflist}}

==References==
*{{cite book |last1=Bjorken |first1=J. D. |last2=Drell |first2=S. D. |title=Relativistic Quantum Mechanics |publisher=McGraw-Hill |location=New York |year=1964 |oclc=534560 }}
*{{cite book |last=Collins |first=John |title=Renormalization |publisher=Cambridge University Press |location=Cambridge |year=1984 |isbn=0-521-24261-4 }}
*{{cite book |last=Hatfield |first=Brian |title=Quantum Field Theory of Point Particles and Strings |publisher=Addison-Wesley |location=Redwood, California |year=1992 |isbn=0-201-36079-9 }}
*{{cite book |last1=Itzykson |first1=C. |last2=Zuber |first2=J-B. |title=Quantum Field Theory |publisher=McGraw-Hill |location=New York |year=1980 |isbn=0-07-032071-3 }}
*{{cite journal |last1=Pauli |first1=W. |last2=Villars |first2=F. |title=On the Invariant Regularization in Relativistic Quantum Theory |journal=Reviews of Modern Physics |volume=21 |issue=3 |pages=434–444 |year=1949 |doi=10.1103/RevModPhys.21.434 |bibcode=1949RvMP...21..434P |doi-access=free }}

{{DEFAULTSORT:Pauli-Villars regularization}}
[[Category:Quantum field theory]]

{{Sourceattribution|Pauli–Villars regularization}}