Dangerously irrelevant operator: Difference between revisions

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{{refimprove|date=November 2010}}

In [[statistical mechanics]] and [[quantum field theory]], a '''dangerously irrelevant operator''' (or '''dangerous irrelevant operator''') is an [[operator (mathematics)|operator]] which is irrelevant, yet affects the [[infrared]] physics significantly because the [[vacuum expectation value]] (VEV) of some field depends sensitively upon the dangerously irrelevant operator.

==Example==

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:<math>V\left(\phi\right)=-a \phi^\alpha + b\phi^\beta</math>

Let us also suppose that ''a'' is positive and nonzero and <math>\beta</math> > <math>\alpha</math>. If ''b'' is zero, there is no stable equilibrium. If the scaling dimension of <math>\phi</math> is ''c'', then the scaling dimension of ''b'' is <math>d-\beta c</math> where ''d'' is the number of dimensions. It is clear that if the scaling dimension of ''b'' is negative, ''b'' is an irrelevant parameter. However, the crucial point is, that the [[Vacuum expectation value|<math>\mathrm{VEV}</math>]]

:<math>\langle\phi\rangle=\left(\frac{a\alpha}{b\beta}\right)^{\frac{1}{\beta-\alpha}}=\left(\frac{a\alpha}{\beta}\right)^{\frac{1}{\beta-\alpha}}b^{-\frac{1}{\beta-\alpha}}</math>.

depends very sensitively upon ''b'', at least for small values of ''b''. Because the nature of the IR physics also depends upon the <math>\mathrm{VEV}</math>, the IR physics looks very different even for a tiny change in ''b'' not because the physics in the vicinity of <math>\phi=0</math> changes much — it hardly changes at all — but because the <math>\mathrm{VEV}</math> we are expanding about has changed enormously.

In [[supersymmetry|supersymmetric]] models with a [[moduli space|modulus]], we can often have dangerously irrelevant parameters.