Dangerously irrelevant operator: Difference between revisions

Revision as of 19:39, 17 April 2007

Revision as of 10:41, 14 October 2007

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{{Orphan|date=September 2006}}

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

== Example ==

Let us suppose there is a field φφ with a [[potential]] depending upon two parameters, a and b.

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

Let us also suppose that a is positive and nonzero and &beta;β > &alpha;α. It is clear that if b is zero, there is no stable equilibrium. If the [[scaling dimension]] of &phi;φ 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 the VEV <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 VEV, 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 VEV we are expanding about has changed enormously.

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[[Category:Renormalization group]]

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