Dangerously irrelevant
Encyclopedia
In statistical mechanics
and quantum field theory
, a dangerously irrelevant operator (or dangerous irrelevant operator) is an operator which is irrelevant, yet affects the infrared
physics significantly because the vacuum expectation value
of some field depends sensitively upon the dangerously irrelevant operator.
with a potential
depending upon two parameters, a and b.
Let us also suppose that a is positive and nonzero and
> 
. If b is zero, there is no stable equilibrium. If the scaling dimension of 
is c, then the scaling dimension of b is 
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 
.
depends very sensitively upon b, at least for small values of b. Because the nature of the IR physics also depends upon the
, the IR physics looks very different even for a tiny change in b not because the physics in the vicinity of 
changes much — it hardly changes at all — but because the 
we are expanding about has changed enormously.
In supersymmetric
models with a modulus
, we can often have dangerously irrelevant parameters.
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
and quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, a dangerously irrelevant operator (or dangerous irrelevant operator) is an operator which is irrelevant, yet affects the infrared
Infrared
Infrared light is electromagnetic radiation with a wavelength longer than that of visible light, measured from the nominal edge of visible red light at 0.74 micrometres , and extending conventionally to 300 µm...
physics significantly because the vacuum expectation value
Vacuum expectation value
In quantum field theory the vacuum expectation value of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle...
of some field depends sensitively upon the dangerously irrelevant operator.
Example
Let us suppose there is a field with a potentialPotential
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...
depending upon two parameters, a and b.
Let us also suppose that a is positive and nonzero and
> . If b is zero, there is no stable equilibrium. If the scaling dimension of is c, then the scaling dimension of b is 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
In quantum field theory the vacuum expectation value of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle...
.depends very sensitively upon b, at least for small values of b. Because the nature of the IR physics also depends upon the
, the IR physics looks very different even for a tiny change in b not because the physics in the vicinity of changes much — it hardly changes at all — but because the we are expanding about has changed enormously.In supersymmetric
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
models with a modulus
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
, we can often have dangerously irrelevant parameters.