Introduction

Physicists are almost always concerned with a single length scale at a time. This is to say that if physicists are studying the properties of an atom, they typically aren’t concerned with whether that atom exists in a basketball or a star cluster (unless, of course, the temperature of the system is relevant to the study). And this is reflected in their theories: You don’t need to know quantum mechanics to study projectile motion or the motion of a pendulum, and you don’t need to understand the motion of a pendulum to do quantum mechanics.

Figure 1: Quantum mechanics can be used to study the properties of electrons that make up the carbon atoms of objects, but studying the dynamic motion of such objects doesn't require quantum mechanics.

Physical scales for systems are separated, and to the extent that physicists choose to do so, they can ignore everything both above and below their length scale of interest.

This is why the renormalization group [1] is seen as such a powerful construct in physics. The basic question the renormalization group answers is how a physical theory on one length scale relates to the same physical theory at another length scale, provided we can move from one to the other using the methods of the theory itself.

Figure 2: Fundamental idea of renormalization group refers to understanding how a physical theory \(X\) at one length scale relates to another theory \(X'\) at another.

The idea is sufficiently complex that it is often only encountered in graduate texts, but there are simpler representations of the theory that capture the basic idea. In what follows, we discuss such a simpler representation, constructing a toy model of a functional integral or partition function that allows us to implement the essence of the renormalization group transformation without the theoretical overhead typically associated with discussions of quantum field theory or statistical mechanics.

Basic Ideas and Gaussians

Say that we begin the following “action” or “energy”

\[ \mathcal{A}\left(\{\lambda_i\}_N\right) = \alpha \sum_{i =1}^{N} \lambda_{i}^2 - \beta \sum^{N}_{j =1}\lambda_{j}^2\lambda_{j+1}^2,\qquad (1) % \label{eq:toy_action} \]

where the degrees of freedom are \(\{\lambda_i\}\), and we impose \(\lambda_{N+1} = \lambda_{1}\).

In field theory and statistical physics, the action and the energy, respectively, define how the degrees of freedom in the system couple to one another. Seeking a local minimum of the action represents a “classical solution” to the field theory problem, while such a minimum of the energy is associated with a zero-temperature system that does not have thermal flucations.

To incorporate fluctuations (either quantum or thermal), the action or the energy need to be put into a specific integral. In quantum field theory this integral is called a “functional integral,” and in statistical physics it is a “partition function.” For the model we introduced in (1), such an integral has the form

\[ Z = \int^{\infty}_{-\infty} \left[ \prod_{i=1}^{N} d\lambda_{i}\right] \exp\left[- \mathcal{A}\left(\{\lambda_i\}_N\right)\right]. \qquad (2) % \label{eq:func_intgrl} \]

Following the analogy from quantum field theory or statistical physics, the integral \(Z\) gives us a way to organize averages over the random variables \(\lambda_i\). Averages are found by including a term like \(\sum_{i} h_i \lambda_i\) within the action, computing the integral, and then differentiating the result with respect to \(h_i\). Each differentiation brings down a factor of \(\lambda_i\), thus representing a factor that contributes to the average.

This is why the calculation of \(Z\) is crucial: It allows us to compute the averages that represnt the observables in a system. This is also where the renormalization group enters the picture: The way the renormalization group is implemented is to integrate \(Z\) in a specific way that makes scale changes apparent. The result of said integration is the answer to a question like:

How do the parameters \(\alpha\) and \(\beta\) change, as we change the length scale of the system?

This is the general question the renormalization group answers: How do the parameters that defined our initial theory change as we change the length scale in the system?

In the case of the toy example we’re building, the phrase “changing the length scale” doesn’t quite mean what it seems to mean. We don’t simply increase the length between the degrees of freedom. Instead, changing the length scale means “integrating out every other degree of freedom,” so that the only degrees of freedom left now have twice the effective length between them. This is the specific integration procedure mentioned above. This process is schematically represented in the figure below

Figure 3: Steps of Renormalization Transformation with \(N = 24\): Label half the degrees of freedom in the system, integrate or average over the labeled degrees of freedom, and then obtain a system similar to the first except with new interactions.

This is the essence of the renormalization group procedure in any form you find it:

1. Begin with one system
2. Itegrate (or “average”) out the degrees of freedom associated with certain parts of that system
3. Obtain a new system in which new parameters redefine old interaction coefficients

Figure 4: Renormalization group consists of beginning with one physical model and then "integrating out" or "averaging over" certain degrees of freedom to obtain another physical model.

This process is true in both statistical physics and quantum field theory, though it takes on different forms.

But now, what does it mean to “integrate out” degrees of freedom? The meaning follows from the definition of \(Z\). Taking \(Z\) to represent a probability normalization, integrating out degrees of freedom means “averaging over them,” that is, computing the integrations within \(Z\) for only those variables.

What would \(Z\) look like after such selective evaluation? How would \(\mathcal{A}\) change? Answering these questions will reveal the renormalization group nature of the system and show how the parameters \(\alpha\) and \(\beta\) change through this integrating-out procedure.

We’ll start by integrating out only half the degrees of freedom in Eq.(2), focusing on the \(\lambda_j\) with an even index \(j\). The resulting integration can then be represented as

$$ Z = \int^{\infty}_{-\infty} \left[ \prod_{i=1}^{N} d\lambda_{i}\right] \exp\big[- \mathcal{A}\left(\{\lambda_i\}_N\right) \big] = \int^{\infty}_{-\infty} \left[ \prod_{i=1}^{N'} d\lambda_{2i-1}\right] \exp\left[- \mathcal{A}'\left(\{\lambda_{2i-1}\}_{N'}\right)\right], \quad (3) $$

where \(N' = N/2\) and \(\mathcal{A}'\left(\{\lambda_{2i-1}\}_{N'}\right)\) is a new action which results from the integration. This new quantity represents what is known as the effective action [2].

The Effective Action

We need to find the effective action \(\mathcal{A}'\left(\{\lambda_{2i-1}\}_{N'}\right)\) through a series of integrations, but first we’ll write the original action in a more helpful form. Defining the resulting number of degrees of freedom as \(N'\) through \(2N' = N\), we can then write the original action as

$$\begin{aligned} \mathcal{A}\left(\{\lambda_i\}_N\right) & = \alpha \sum_{i =1}^{2N'} \lambda_{i}^2 - \beta \sum^{2N'}_{j =1}\lambda_{j}^2\lambda_{j+1}^2\\ & =\alpha \sum_{i =1}^{N'} \lambda_{2i-1}^2 + \alpha \sum_{i =1}^{N'} \lambda_{2i}^2 - \beta \sum^{N'}_{j =1}\lambda_{2j}^2\lambda_{j+1}^2- \beta \sum^{N'}_{j =1}\lambda_{2j-1}^2\lambda_{2j}^2\\ & = \alpha \sum_{i =1}^{N'} \lambda_{2i-1}^2 + \alpha \sum_{j = 1}^{N'} \lambda_{2j }^2M_{j} , \qquad \qquad \qquad \qquad \qquad \qquad (4) % \label{eq:act2} \end{aligned}$$

where we defined

$$ \qquad M_{j} \equiv \left(1 - \frac{\beta}{\alpha}(\lambda_{2j-1}^2+\lambda_{2j +1}^2)\right). \qquad (5) $$

In Eq.(4), the even and odd indexed \(\lambda_{k}\) have been separated, and we can perform the integral over the even degrees of freedom. Multidimensional Gaussian integrals evaluate cleanly, and we find

$$\begin{aligned} Z & = \int^{\infty}_{-\infty} \left[ \prod_{i=1}^{N'} d\lambda_{2i-1}\right] \exp\left[-\alpha \sum_{i =1}^{N'} \lambda_{2i-1}^2\right] \int^{\infty}_{-\infty} \left[ \prod_{i=1}^{N'} d\lambda_{2i}\right] \exp\left[- \alpha \sum_{j = 1}^{N'} \lambda_{2j }^2M_{j}\right]\\ & = \int^{\infty}_{-\infty} \left[ \prod_{i=1}^{N'} d\lambda_{2i-1}\right] \exp\left[-\alpha \sum_{i =1}^{N'} \lambda_{2i-1}^2\right] \prod_{i = 1}^{N'} \sqrt{\frac{\pi}{\alpha M_{i}}}\\ & = \int^{\infty}_{-\infty} \left[ \prod_{i=1}^{N'} d\lambda_{2i-1}\right] \exp\left[-\alpha \sum_{i =1}^{N'} \lambda_{2i-1}^2- \frac{1}{2} \sum_{i=1}^{N'} \ln \frac{\alpha M_{i}}{\pi}\right] \\ & \equiv \int^{\infty}_{-\infty} \left[ \prod_{i=1}^{N'} d\lambda_{2i-1}\right] \exp\left[- \mathcal{A}'\left(\{\lambda_{2i-1}\}_{N'}\right)\right] . \qquad (6) \end{aligned}$$

Thus, the effective action after our increase in scale is

$$ \mathcal{A}'\left(\{\lambda_{2i-1}\}_{N'}\right) = \alpha \sum_{i =1}^{N'} \lambda_{2i-1}^2+ \frac{1}{2} \sum_{i=1}^{N'} \ln \left(1 - \frac{\beta}{\alpha}(\lambda_{2i-1}^2+\lambda_{2i +1}^2)\right) + \frac{N'}{2} \ln \alpha/\pi. \quad (7) $$

Addendum: Effective Networks

In Eq.(7), we computed the effective action, but what does such a quantity represent conceptually?

Consider an analogous situation. Say you’re in a network of people of two types \(A\) and \(B\). Person \(A_j\) is connected to person \(B_k\), who is then connected to person \(A_{\ell}\) and so on, but \(A\) people are only directly connected to \(B\) people and vice versa. Let people pass messages to each other at various rates that we can calculate. Due to the connection properties of the system, there is a rate at which \(A_1\) passes messages to \(B_1\) but no base rate for \(A_1\) to pass messages to \(A_2\).

However, now consider all the \(B_j\)s to be random variables and average over all of them. From all of this averaging, we would get an effective network consisting entirely of \(A_j\)s, and we would have an effective rate for \(A_1\) to go from \(A_2\).

Figure 5: If you have a network of \(A\) and \(B\) nodes where the former can only move to the latter and vice versa, we can determine the effective rate at which \(A\) nodes transition to each other by integrating out (i.e., summing over) the intermediate \(B\) nodes in a two step interaction.

The effective action functions similarly to this effective rate. It shows how the remaining degrees of freedom couple to one another given the averaged out influence of the removed degrees of freedom. Thus from Eq.(7), we can infer how these new couplings contribute to new definitions of the quadratic and quartic couplings \(\alpha\) and \(\beta\).

The Perturbative Expansion and RG Equations

For subsequent simplicity, we define \(\lambda_{i}' \equiv \lambda_{2i-1}\). Then the integration Eq.(3) induces the transformation

$$\qquad \mathcal{A}\left(\{\lambda_i\}_N\right) \to \mathcal{A}'\left(\{\lambda_{i}'\}_{N'}\right).\qquad (8)$$

For the parameters, the renormalization group transformations induced by this integration are

$$ \qquad \alpha \to \alpha', \qquad \beta \to \beta' , \qquad (9) $$

where \(\alpha'\) and \(\beta'\) are the new interaction parameters for the new action

We can thus rephrase the renormalization group question from before. We are still concerned with changes in scale and parameters changes, but with the notation in Eq.(9) we can phrase this question specifically as,

For the transformation induced by Eq.(8), what are \(\Delta \alpha\) and \(\Delta \beta\) to lowest order?

From Eq.(7), the action on the right-hand side of Eq.(8) is

$$ \qquad \mathcal{A}'\left(\{\lambda_{i}'\}_{N'}\right) = \alpha \sum_{i =1}^{N'} \lambda_{i}'^{2}+ \frac{1}{2} \sum_{i=1}^{N'} \ln \left(1 - \frac{\beta}{\alpha}(\lambda_{i}'^{2}+\lambda_{i +1}'^{2})\right) + \frac{N'}{2} \ln \alpha/\pi. \qquad (10) % \label{eq:babyact} $$

Assuming ’low-coupling’ \(\beta/\alpha \ll 1\) and expanding the second term in Eq.(10) to second order in \(\beta/\alpha\), we find

$$\begin{aligned} \sum_{i} \ln & \left(1 - \frac{\beta}{\alpha}(\lambda_{j}'^{2}+\lambda_{j +1}'^{2})\right) \\ & = - \frac{\beta}{\alpha}\sum_{i = 1}^{N'} (\lambda_{i}'^2 + \lambda_{i+1}'^2) - \frac{1}{2}\left(\frac{\beta}{\alpha}\right)^2 \sum_{i=1}^{N'} (\lambda_{i}'^2 + \lambda_{i+1}'^2)^2 +\mathcal{O}\left(\beta^4/\alpha^4\right)\\ & = - \frac{2\beta}{\alpha}\sum_{i = 1}^{N'} \lambda_{i}'^2 - \left(\frac{\beta}{\alpha}\right)^2 \sum_{i=1}^{N'} \lambda_{i}'^2 \lambda_{i+1}'^2 - \left(\frac{\beta}{\alpha}\right)^2 \sum_{i=1}^{N'} \lambda_{i}'^4+\mathcal{O}\left(\beta^4/\alpha^4\right) \quad (11) \end{aligned}$$

The effective action can thus be written as

$$ \mathcal{A}'\left(\{\lambda_{i}'\}_{N'}\right) = \alpha' \sum_{i =1}^{N'} \lambda_{i}'^{2} - \beta' \sum_{i =1}^{N'} \lambda_{i}'^{2}\lambda_{i+1}'^2-\frac{\beta^2}{2\alpha^2}\sum_{i=1}^{N'} \lambda_{i}'^4 +\mathcal{O}\left(\beta^4/\alpha^4\right) \qquad (12) % \label{eq:babyact2} $$

where

$$ \alpha ' \equiv \alpha - \frac{\beta}{\alpha}, \qquad \beta' \equiv \frac{\beta^2}{2\alpha^2}. $$

Renormalization group equations are typically defined in terms of changes in parameters rather than raw transformations. So defining \(\Delta \alpha\) and \(\Delta \beta\) via

$$ \alpha' = \alpha + \Delta \alpha\, \quad \beta' = \beta + \Delta \beta $$

we then find

$$ \qquad \Delta \alpha = - \frac{\beta}{\alpha} \, , \quad \Delta \beta = \frac{\beta^2}{2\alpha^2} - \beta. \qquad (13) $$

These are the “renormalization group equations” for this system. They tell us how the parameters in \(\mathcal{A}\) change as we change the “scale” of the system. We can imagine iterating this procedure, continually redefining the degree of freedom labels and continually integrating out half of them. The parameters \(\alpha\) and \(\beta\) will continually change during this process in such a way that their values represent a “flow” in the space of coupling constants [3].

That is basically it. You average over certain degrees of freedom and you get new coupling constants. The renormalization group idea has different manifestations in different subjects, but the essence of the above procedure remains the same wherever it is applied.

Why of the Renormalization Group

Now that we have worked through this toy example and explored the “what” and the “how” of the renormalization group, we now consider the “why,” namely why is this question of how scale changes parameters important?

There are different perspectives on this answer depending on whether you look at quantum field theory or statistical physics:

• Quantum Field Theory (QFT): Knowing how a parameter changes as we change length scale tells us in what regimes perturbation theory is relevant. If a parameter decreases at small length scales (or high energy), then we call the associated theory “asymptotically free”, and perturbation theory applies at high energies. Conversely, if a parameter decreases at large length scales (or low energy) then we call the associated theory “infrared free," and perturbation theory applies at low energies. QCD is the prototypical asymptotically free theory and QED is the prototypical infrared free theory. There are also cases where the parameter remains the same in which case we say we are at a fixed point.
  
  For the integration leading to the effective action, it allows us to derive low energy effective theories from high energy ones, as one does when obtaining the Four-fermi theory of beta decay from the more fundamental electroweak theory of weak interactions. We could then even apply the renormalization group to this four-fermi interaction to understand how it changes with energy scale.

Figure 6: In quantum field theory, the renormalization group is implemented by integrating out high-energy degrees of freedom.The diagram depicts the tree-level (i.e., classical) interactions, but the full effective action also includes loop corrections from quantum fluctuations.

• Statistical Physics: Knowing how a parameter changes as we change length scale often tells us where phase transitions occur within systems. For the continuous version of the renormalization group transformations, we sometimes find that parameters reach finite and non-zero values which do not change as we make successive transformations. Such points are called “non-trivial fixed points” (“non-trivial” because the parameters are not zero or infinite and “fixed” due to their unchanging nature) and are associated with phase transitions in the system. At such points, the physics of the system exhibits “universality” in which all other systems with the same number of dimensions and the same symmetries exhibit the same scaling properties for observables, regardless of how different the two systems are.
  
  For example, we find that the Ising model’s non-trivial fixed point is associated with a diverging correlation length which results in all of the degrees of freedom being coupled together. This coupling has a parallel manifestation in liquid-gas phase transitions which exhibit the critical opalescence that turns a liquid mixture from transparent to opaque (the opacity reflecting the fact that light is scattered at all wavelengths in a system in which degrees of freedom are coupled at all length scales).

Figure 7: In the statistical mechanics of the Ising model on a square lattice, the renormalization group is implemented by summing over the states of every other degree of freedom.

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Footnotes

[1] The phrase "renormalization group" suffers from two false associations that belie the true meaning of what it represents. The notion of **renormalization** developed in the mid-1940s as a way to handle the divergent integrals in quantum field theory through high-energy cutoffs (and later by taking dimensions to certain limits). The fact that these cutoffs led to parameter changes was discovered later, but it's possible to have parameter changes without infinities, and one can seek to regularize infinities without having parameter changes in a theory.

In mathematics a group is a set with a binary operation that takes elements of said set and transforms them into another of that set. These transformations must be associative, have an identity operation, and be invertible. The equations and operations that define the "renormalization group" do not fit this definition since the "integrating out" operation we discuss in the text is not invertible; it's like saying we can go from a sample of elements to an average, but we cannot reverse construct the sample from the average. However, the parameter transformations are associative and there is an identity transformation, so we could call it a semi-group.

Candidly, a better name to replace "renormalization group" would be "parameter-scale-theory," but that admittedly sounds less fancy than the chosen name.

[2] To be precise, this quantity isn't quite the analog of the effective action by the typical definition in quantum field theory. Typically the effective action is computed by introrducing an external field term like \(\sum_{i=1} h_i \lambda_i\) into the action, computing the integral defining the partition function, taking the logarithm and then performing a legendre transform to move from a function depending on the external field to one depending on a classical field. After all that work, what we cited as the effective action represents just one term in a long series of terms for the true effective action.

[3] One thing to note about these changes is that although we can construct a flow that considers coupling constants at both high and low energy, the essence of the integration procedure proceeds in one direction: When implementing the renormalization group approach, we transition from a small length scale to a large length scale theory, or from a high energy to a low energy theory, but we cannot proceed in the reverse direction. We cannot begin with a theory and try to divine what more microscopic theory it is a consequence of. Many different possible microscopic theories can result in the same "effective theory." This is partly why finding more fundamental theories is so difficult.

Another reason is that our intuitions about theory construction are often limited by the theories we have encountered, theories that more fundamental ones have no requirement to resemble.