In
quantum field theory and
statistical mechanics, the
1/N expansion is a particular
perturbative analysis of quantum field theories with an SO(N) or
SU(N) internal symmetry.
This technique is used in
QCD (even though N is only 3 there) with the gauge group SU(3).
Another application is to the study of
AdS/CFT dualities.
Example
Let's start with a simple example. Let's look at the
O(N) φ4. The scalar field φ takes on values in the
real vector representation of O(N). Let's use the
index notation for the N "
flavors" with the Einstein summation convention. Because O(N) is orthogonal, no distinction will be made between covariant and contravariant indices. The Lagrangian density is given by
:
Note that N has been absorbed into the coupling strength λ. This is crucial here.
Let's introduce an auxiliary field F here.
:
Now, it's obvious in the
Feynman diagrams, the graph breaks up into disjoint cycles, each made up of φ edges of the same flavor and the cycles are connected by F edges.
Let's look at some examples:
Image:1_over_N1.png Image:1_over_N2.png
You get the idea.
Each 4-point vertex contributes λ/N and hence, 1/N. Each flavor cycle contributes N because there are N such flavors to sum over. Note that not all momentum flow cycles are flavor cycles!
It turns out, at least perturbatively, the dominant contribution to the 2k-point connected correlation function is of the order (1/N)
k-1 and the other terms are higher powers of 1/N. This means we can do a 1/N expansion, which gets more and more accurate in the large N limit. The vacuum energy density is proportional to N, but since we're not doing general relativity, that can be ignored.
Because of this structure, we can use a different graphical notation to denote the Feynman diagrams. Represent each flavor cycle by a vertex. There are also flavor paths connecting two external vertices. These too are represented by a single vertex. The two external vertices along the same flavor path are naturally paired and we can replace them by a single vertex and draw an edge (not an F edge) connecting it to the flavor path. Now, the F edges are edges connecting two
flavor cycles/paths to each other (or a flavor cycle/path to itself). The interactions along a flavor cycle/path have a definite cyclic order and so, this is a special kind of graph where the order of the edges incident to a vertex matters, but only up to a cyclic permutation, and since this is a theory of real scalars, also an order reversal (but if we have SU(N) instead of SU(2), order reversals aren't valid!). Each F edge is assigned a momentum (the momentum transfer) and there is an internal momentum integral associated with each flavor cycle.
QCD is an SU(3)
gauge theory involving
gluons and
quarks. The left-handed quarks belong to a triplet representation, the right-handed to an antitriplet representation (after charge-conjugating them) and the gluons to a
real adjoint representation. A quark edge is assigned a color (and an orientation!) and a gluon edge is assigned a color pair. There are also
ghosts with color pairs. Once again, we follow the colors.
In the large N limit, we only consider the dominant term. See
AdS/CFT.
See also
AdS/CFT
Category:Quantum field theory
Category:Quantum chromodynamics
Category:String theory