Renormalization Group — Mathematics Knowledge System

Overview

The renormalization group (RG) is a mathematical framework for studying how physical systems change under changes of scale. Starting from Wilson's formulation (1971), the RG acts on the space of Hamiltonians (or Lagrangians): integrating out short-distance degrees of freedom and rescaling gives a flow on this space. Fixed points of the RG flow correspond to scale-invariant systems (critical points, CFTs). The RG explains universality (why different microscopic models share the same critical exponents), and provides the foundation for quantum field theory, condensed matter physics, and the theory of phase transitions.

Intuition

Imagine zooming out from a magnet at its Curie point. At each scale you see the same pattern of correlated spins -- the system is scale invariant. The RG asks: if you 'integrate out' the finest scale features (high-momentum modes) and rescale, what happens to the effective coupling constants? If the couplings flow to a fixed point, the system is exactly scale invariant at long distances (a critical point). The fixed-point Hamiltonian determines the universality class: all systems with the same fixed point have the same critical exponents, regardless of microscopic details.

Formal Definition

Definition

The Wilson RG: start with a lattice model with Hamiltonian H(sigma; {K_i}) where {K_i} are coupling constants. Block-spin transformation: group spins into blocks, integrate over intra-block degrees of freedom, rescale. The result is a new Hamiltonian H'(sigma'; {K_i'}) with rescaled couplings K_i' = R(K_i). This defines an RG map R on the space of couplings. Fixed points K_i* = R(K_i*) are scale-invariant. The linearization of R around a fixed point gives critical exponents via the eigenvalues lambda_i of DRL (relevant: lambda_i>1, irrelevant: lambda_i<1, marginal: lambda_i=1). Critical exponent nu: lambda_{max} = b^{1/nu} where b is the rescaling factor.

K_i' = R_i(\{K_j\}),\quad K_i^* = R_i(\{K_j^*\})

RG map and fixed point

\beta(g) = \mu\frac{dg}{d\mu} = b_0 g^2 + b_1 g^3 + \cdots

Beta function (infinitesimal RG)

\lambda_i = b^{y_i},\quad y_i > 0 \text{ relevant},\quad y_i < 0 \text{ irrelevant}

Scaling exponents from linearized RG

\nu = 1/y_T,\quad \eta = 2-y_H,\quad \alpha = 2 - d\nu \text{ (hyperscaling)}

Critical exponents from RG eigenvalues

Notation

Notation	Meaning
$\beta(g)$	Beta function: rate of coupling change with scale
$g^*$	Fixed point coupling
$\nu, \eta, \alpha$	Critical exponents
$\epsilon = 4-d$	Deviation from 4 dimensions (Wilson-Fisher expansion)

Theorems

Theorem 1: Universality from RG Fixed Points

Different microscopic models (Ising, Heisenberg, phi^4 field theory) that flow to the same RG fixed point under the RG transformation have identical long-distance behaviour, including the same critical exponents. The critical exponents are determined by the eigenvalues of the linearized RG map at the fixed point, and are independent of the specific microscopic Hamiltonian.

Theorem 2: Wilson-Fisher Fixed Point

The phi^4 field theory in d = 4-epsilon dimensions has a non-trivial RG fixed point (the Wilson-Fisher fixed point) at coupling g* = epsilon/(6 * b_0) + O(epsilon^2). At this fixed point, the correlation length exponent is nu = 1/2 + epsilon/12 + O(epsilon^2) and the anomalous dimension eta = epsilon^2/54 + O(epsilon^3). This explains the critical exponents of the 3D Ising model (set epsilon=1).

Theorem 3: Zamolodchikov's c-Theorem

In 2D quantum field theory, there exists a function c(g) of the coupling constants that is monotonically decreasing along RG flows (from UV to IR) and equals the central charge at fixed points. This c-theorem constrains which CFTs can be connected by RG flows: one can flow from a CFT with central charge c_UV to one with c_IR only if c_UV >= c_IR.

Worked Examples

1
Action: S = integral (1/2)(partial phi)^2 + m^2/2 * phi^2 + g/4! * phi^4. Coupling g is dimensionless in d=4.
2
One-loop correction: a single loop of phi^4 interaction. The 1PI 4-point diagram is a 'fish' diagram with value ~ g^2 * integral d^4k/(k^4) ~ g^2 * log(Lambda).
$\delta g^{(1)} \sim \frac{3g^2}{16\pi^2}\log\Lambda$
3
The beta function is beta(g) = mu * dg/d mu = b_0 * g^2 + ..., where b_0 = 3/(16 pi^2) > 0.
$\beta(g) = \frac{3g^2}{16\pi^2} + O(g^3)$
4
Since beta(g) > 0 for small g > 0, the coupling increases under RG flow to the UV (g -> inf as mu -> inf). This is 'trivial' in the IR (g -> 0 at long distances) -- phi^4 in d=4 is IR-free.
$\beta(g) > 0 \Rightarrow g \text{ grows in UV (Landau pole)}$

✓ Answer

The one-loop beta function is beta(g) = 3g^2/(16*pi^2) > 0, meaning phi^4 theory in 4D is IR-free (Gaussian fixed point at g=0 is IR-stable).

Practice Problems

Hardfree response

Explain universality: why do the critical exponents of the 3D Ising model equal those of the liquid-gas critical point?

Common Mistakes

Common Mistake

Thinking the renormalization group is an actual group (a group in the mathematical sense).

The 'renormalization group' is actually a semigroup, not a group: the RG map (integrating out short-distance modes) is generally not invertible. Information is lost when modes are integrated out. Wilson himself noted this. The name 'group' is historical. The mathematical structure is a monoid or semigroup acting on the space of theories.

Quiz

A coupling constant g is called 'relevant' at a fixed point if:

The beta function vanishes at g*The eigenvalue of the linearized RG map satisfies lambda > 1 (i.e., it grows under rescaling)The coupling is dimensionlessThe coupling appears in the action with a positive coefficient

Historical Background

Kadanoff's block spin picture (1966) gave the physical intuition: group nearby spins into blocks and rescale. Wilson formalized the RG as a flow on coupling constants (1971), earning the Nobel Prize in 1982. The epsilon expansion (Wilson-Fisher, 1972) gave controlled perturbative computations of critical exponents. Polchinski's exact RG equation (1984) gave a functional differential equation for the Wilsonian effective action. In mathematics, the RG is connected to dynamical systems theory and, rigorously, to constructive field theory.

1966
Kadanoff introduces the block spin picture for second-order phase transitions
Leo Kadanoff
1971
Wilson formulates the renormalization group as a flow on coupling constants
Kenneth Wilson
1972
Wilson-Fisher epsilon expansion computes critical exponents in 4-epsilon dimensions
Kenneth Wilson, Michael Fisher
1982
Wilson wins Nobel Prize in Physics for the RG and critical phenomena
Kenneth Wilson
1984
Polchinski derives the exact RG equation for the Wilsonian effective action
Joseph Polchinski

Summary

The RG is a flow on the space of Hamiltonians: integrating out short-distance modes gives a new effective Hamiltonian.
Fixed points are scale-invariant theories; critical exponents = eigenvalues of the linearized RG map.
Universality: different microscopic models sharing the same RG fixed point have identical critical exponents.
Wilson-Fisher fixed point in d=4-epsilon gives controlled computations of 3D critical exponents.

References

BookGoldenfeld, N. Lectures on Phase Transitions and the Renormalization Group. Addison-Wesley, 1992.
BookWilson, K.G. and Kogut, J. The Renormalization Group and the Epsilon Expansion. Physics Reports, 1974.
WebsiteWikipedia -- Renormalization group