Functional Generators on the Lattice

We saw in section 3.5 that the quantum theory, in a way analogous to the classical theory, establishes a functional relation between the expectation value $v$ of the field and the external source $j$, which means that in order to calculate $v$ at a given point it is necessary to know $j$ at all points, not only at that particular point. It was mentioned there that this relation can be used to explore the properties of the quantum theory. In this section we will introduce the functional generators of the correlation functions of the quantum theory on the lattice, which are the instruments that can be used for this type of analysis. The final objective of this section is to arrive at the concept of the effective action. Observe that, differently from what we have been doing up to this point in the discussion of the quantum theory, the development presented in this section is of a very general character. Instead of using the theory of the free scalar field to develop in detail concepts whose qualitative nature may then be generalized, we will make the development directly in the general case, for any arbitrary model of scalar fields.

In this section we will make a temporary change in notation and denote the expectation value of the field by $\varphi_{(c)}$, not by $v$. We will also refer to $\varphi$ as the fundamental field, to distinguish it from $\varphi_{(c)}$, which will also be a type of field. In order to develop the formalism we will assume that we have some model of scalar fields defined by an action $S[\varphi]$, without external sources, which has the property that $S[\varphi]=S[-\varphi]$ and, therefore, the property that $\langle\varphi\rangle=0$. We will then add explicitly to this action a linear term with the external source, given as usual by the product of $j$ and $\varphi$,

$$S_{(j)}[\varphi]=S[\varphi]-\sum_{\vec{n}}j(\vec{n})\varphi(\vec{n}).$$

When we do this the source causes the generation of a non-vanishing expectation value for the field, which is a function of $\vec{n}$ and a functional of $j$,

  $$\varphi_{(c)}[j]=\langle\varphi(\vec{n})\rangle_{(j)}= \frac{\dis... ...\varphi(\vec{n})e^{-S_{(j)}}}{\displaystyle \int[{\bf d}\varphi]e^{-S_{(j)}}}.$$ (3.6.1)

The index $j$ on the expectation value means that it is calculated in the distribution of the theory with the external source, defined by $S_{(j)}$, instead of that defined only by $S$. The expectation value $\varphi_{(c)}$ is also referred to as the “classical field” of the quantum theory. This does not mean, however, that it can always be measured directly, because the expectation value of the field at a single site is an ultra-local object, not an extended object on the lattice. For example, for a point source in the theory of the free field $\varphi_{(c)}$ is the Green function and therefore has a divergence at the origin, in the continuum limit. On the other hand, since this “classical field” is an expectation value its value does not fluctuate like the fundamental field, that is, it does behave basically like a classical quantity. By and large we may think of this classical field $\varphi_{(c)}$ as an observable of the quantum theory, and that will be enough for the purposes of this chapter. A deeper discussion of this topic will be presented later on, when we introduce the concept of block variables.

An important point about the functional relation between $\varphi_{(c)}$ and $j$ is that it is a bijection, that is, given a $j(\vec{n})$ a certain function $\varphi_{(c)}(\vec{n})$ is uniquely determined, and vice-versa, given a certain $\varphi_{(c)}(\vec{n})$ there is a unique function $j(\vec{n})$ that corresponds to it. The first part of this statement is rather obvious, because a single cause $j$ cannot produce two different consequences $\varphi_{(c)}(\vec{n})$. Regarding the second part, in the case of the classical theory this is a simple consequence (problem 3.6.1) of the uniqueness of the solution of a differential equation. In the quantum theory we may show this in the following way: since $S$ is invariant by the transformation $\varphi\rightarrow-\varphi$, it follows that $S_{(j)}$ is invariant by the joint change of sign of $\varphi$ and $j$, which also has the effect of changing the sign of $\varphi_{(c)}$. It is clear then that, if a certain $j$ and a certain $\varphi_{(c)}$ are related by the functional relation established by the quantum theory, then $-j$ and $-\varphi_{(c)}$ are as well. In addition to this, it is clear that any non-vanishing external source affects the expectation value of the field in some way, so that only $j=0$ is related with $\varphi_{(c)}=0$. Given all this, it follows that there cannot be two different sources $j_{1}$ and $j_{2}$ that produce the same $\varphi_{(c)}$, because otherwise there would be a non-vanishing source $j=j_{1}-j_{2}$ that is related to $\varphi_{(c)}=0$ by the functional relation.

We will assume, for simplicity, that the models are defined on a finite lattice within a box, with periodical boundary conditions. The basic functional generator that we wish to define is a functional of the external source $j$, traditionally denoted by $Z[j]$,

  $$Z[j]=\left\langle e^{\sum_{\vec{n}}j(\vec{n})\varphi(\vec{n})}\ri... ...e \int[{\bf d}\varphi]e^{-S_{(j)}}}{\displaystyle \int[{\bf d}\varphi]e^{-S}}.$$ (3.6.2)

Note that we have here an expectation value in the measure (or distribution) of $S$, without the term with the external source. One may also say that $Z[j]$ is the ratio of two measures, one with $j$ present and the other without it. Given $j$, $Z$ is a real number, a simple functional of $j$. As we will show later on, in general $\langle S\rangle$ diverges in the continuum limit, so that $Z$ is a possibly singular ratio in that limit, except if $j=0$, in which case $Z=1$ both on finite lattices and in the continuum limit. However, the value of $Z$ itself is not actually very important, what really matters is how it varies when we vary $j$. In any case it is a finite quantity on finite lattices, where it can therefore be used for the operations to be described below, and anyway we should always take the limit only at the final step of any given calculation, by which time $Z$ will be gone from the picture.

We will consider, then, the functional derivatives of $Z$ with respect to $j(\vec{n})$, that is, the variations of $Z$ when we vary $j$ at a single point $\vec{n}$. On the lattice functional differentiation is no more than partial differentiation with respect to the variables related to the degrees of freedom of the system. There is one $j(\vec{n})$ for each degree of freedom $\varphi$ of the system, and each value of $j(\vec{n})$ at the various sites is a variable that may be changed independently, so that we have for our functional variations

$$\frac{\mbox{\boldmath$\mathfrak{d}$}j(\vec{n}_{1})}{\mbox{\b... ...frak{d}$}j(\vec{n}_{2})} =\delta^{d}(\vec{n}_{1},\vec{n}_{2}),$$

where we used the symbol $\mbox{\boldmath$\mathfrak{d}$}$ to indicate the functional derivatives. Observe also that only $\varphi_{(c)}$ depends on $j$, the fundamental field $\varphi$ is independent of the external sources. Taking $n$ functional derivatives of $Z$ we obtain

$$\frac{\mbox{\boldmath$\mathfrak{d}$}^{n}Z[j]}{\mbox{\boldmat... ...i_{n} e^{-S_{(j)}}}{\displaystyle \int[{\bf d}\varphi]e^{-S}},$$

where we denoted the dependencies with the positions $\vec{n}_{i}$ by means of indices $i$, for simplicity of notation. Observe that the expectation value is taken in the measure of $S$, without external sources. For $j=0$ we recover from this formula the correlation functions of the model in the theory without external sources,

$$\frac{\mbox{\boldmath$\mathfrak{d}$}^{n}Z[j]}{\mbox{\boldmat... ...ngle\varphi_{1}\ldots\varphi_{n}\right\rangle =g_{1,\ldots,n}.$$

However, for $j\neq 0$ we do not obtain the true correlation functions in the presence of $j$, because the expectation values are taken in the measure of $S$. The true correlation functions in the presence of $j$ are given by the ratios

$$\frac{1}{Z[j]}\;\frac{\mbox{\boldmath$\mathfrak{d}$}^{n}Z[j]... ...playstyle \int[{\bf d}\varphi]e^{-S_{(j)}}}=g_{(j)1,\ldots,n},$$

where the index $(j)$ indicates that the expectation value is taken in the measure of $S_{(j)}$ and where we do not make $j=0$. In particular, the expectation value $\varphi_{(c)}$ of the field is given by

$$\varphi_{(c)1}[j]=\frac{1}{Z[j]}\;\frac{\mbox{\boldmath$\mat... ...\mathfrak{d}$}}{\mbox{\boldmath$\mathfrak{d}$}j_{1}}\ln(Z[j]).$$

This motivates the definition of another functional, related to $Z$ by exponentiation,

$$W[j]=\ln(Z[j]),\mbox{~~that is,~~}Z[j]=e^{W[j]}.$$

The “classical field” $\varphi_{(c)1}$ may now be written as

  $$\varphi_{(c)1}[j]=\langle\varphi_{1}\rangle_{(j)}= \frac{\mbox{\b... ...frac{\mbox{\boldmath$\mathfrak{d}$}Z[j]}{\mbox{\boldmath$\mathfrak{d}$}j_{1}}.$$ (3.6.3)

In a way analogous to $Z$, the functional $W$ also generates correlation functions of the theory. However, we are looking in this case at a different set of functions. While $Z$ generates the full correlation functions $g_{1,\ldots,n}$, $W$ generates functions that are called the connected correlation functions $g_{(c)1,\ldots,n}$. As we saw above, the first derivative of $W$ with respect to $j$ gives us the one-point function, the expectation value of the field. In order to examine in more detail the nature of these functions, let us take one more functional derivative of $W$. Starting from (3.6.3) we get,

  $$\frac{\mbox{\boldmath$\mathfrak{d}$}^{2}W[j]}{\mbox{\boldmath$\ma... ...h$\mathfrak{d}$}j_{2}} =g_{(j)1,2}-\varphi_{(c)1}\varphi_{(c)2} =g_{(c,j)1,2}.$$ (3.6.4)

Here $g_{(j)1,2}$ is the complete propagator in the presence of $j$ and $g_{(c,j)1,2}$ is the connected propagator in the same conditions. Note that, for $j=0$ in a theory which is symmetrical by reflection of the fields, as we assume here, we have that $\varphi_{(c)}=0$ and then the two propagators coincide. However, for $j\neq 0$ or in cases where $j=0$ does not imply that $\varphi_{(c)}=0$, it is the connected propagator $g_{(c,j)1,2}$ given by $W$, not the full propagator $g_{(j)1,2}$ given by $Z$, which is the true correlation function of the theory, as we discussed in section 3.2. In order to obtain the correlations between $\varphi_{1}$ and $\varphi_{2}$ in circumstances in which $\langle\varphi\rangle\neq 0$ it is necessary to subtract the product of the expectation values of the two fields. We see therefore that $W$ is a functional with a more direct significance than that of $Z$. In particular, the functional $W$ can be used to write the functions of three, four or more points of the theory (problem 3.6.2), which are related in a more direct way to the existence within it of true physical interactions.

Up to this point, the structure that we have is that the functionals $Z$ and $W$ depend on the external source $j$ and that functional derivatives with respect to it produce from these functionals all the correlation functions of the theory. Since the physics of a model in the quantum theory is encoded in the set of its correlation functions, these functionals may be understood as abbreviated condensations of all the properties of the model. To calculate completely these functional is equivalent to solve completely the theory, which usually is not an easy thing to do. We will proceed now with the development of the formalism of the functional generators, with the intent of obtaining a description of these properties in terms, not directly of $j$, but of the classical field $\varphi_{(c)}$ that appears as a consequence of the introduction of the external sources. Note that we may write the definitions of $Z$ and $W$ as

$$Z[j]=e^{W[j]}=\frac{\displaystyle \int[{\bf d}\varphi]e^{\su... ...i(\vec{n})} e^{-S}}{\displaystyle \int[{\bf d}\varphi]e^{-S}},$$

so that we may think of $W$ as something like a “renormalized version” of the term in the fundamental action involving the external sources, $\sum_{\vec{n}}j(\vec{n})\varphi(\vec{n})$. Since there is a definite relation between each $j$ and each $\varphi_{(c)}$, we are led to think that it should be possible to write $W$ as a functional of $\varphi_{(c)}$ instead of $j$. Let us recall that the first functional derivative of $W$ is given by

$$\varphi_{(c)}[j]=\frac{\mbox{\boldmath$\mathfrak{d}$}W[j]}{\mbox{\boldmath$\mathfrak{d}$}j},$$

so that the total variation of $W$ due to variations of $j$ at each point may be written in the form of a functional differential

$${\bf d}W=\sum_{\vec{n}}\frac{\mbox{\boldmath$\mathfrak{d}$}W... ...ec{n}) =\sum_{\vec{n}}\varphi_{(c)}(\vec{n}){\bf d}j(\vec{n}),$$

where ${\bf d}j$ are the arbitrary variations of $j$ at all points. We may now define a new functional $\Gamma$ by means of a Legendre transformation applied to $W$,

$$\Gamma=\sum_{\vec{n}}j(\vec{n})\varphi_{(c)}(\vec{n})-W[j].$$

Note that the first term is simply the expectation value of $\sum_{\vec{n}}j(\vec{n})\varphi(\vec{n})$ because, since $j$ does not depend on $\varphi$, we have

$$\left\langle\sum_{\vec{n}}j(\vec{n})\varphi(\vec{n})\right\r... ...\rangle_{(j)} =\sum_{\vec{n}}j(\vec{n})\varphi_{(c)}(\vec{n}).$$

We consider now the variation of $\Gamma$ due to an arbitrary variation of $j$, and therefore of $\varphi_{(c)}$, obtaining

$$\begin{eqnarray*} {\bf d}\Gamma & = & \sum_{\vec{n}}j(\vec{n}){\bf d}\varphi_{(c... ... \\ & = & \sum_{\vec{n}}j(\vec{n}){\bf d}\varphi_{(c)}(\vec{n}), \end{eqnarray*}$$

where we used the Leibniz rule and the form of the differential of $W$. The conclusion is that $\Gamma$ is a functional directly of $\varphi_{(c)}$, because it depends only indirectly on $j$, its functional derivative being given by

$$\frac{\mbox{\boldmath$\mathfrak{d}$}\Gamma[\varphi_{(c)}]}{\mbox{\boldmath$\mathfrak{d}$}\varphi_{(c)}}=j$$

and its functional differential by

$${\bf d}\Gamma=\sum_{\vec{n}}j(\vec{n}){\bf d}\varphi_{(c)}(\vec{n}),$$

showing that $\Gamma$ is a functional only of $\varphi_{(c)}$. We may now write for our functionals that

$$e^{W[j]}= e^{\sum_{\vec{n}}j(\vec{n})\varphi_{(c)}(\vec{n})}... ...i(\vec{n})} e^{-S}}{\displaystyle \int[{\bf d}\varphi]e^{-S}}.$$

At this point it starts to appear that $\Gamma[\varphi_{(c)}]$ has something to do with a kind of “classical action” for the “classical field” $\varphi_{(c)}$. The functional $\Gamma[\varphi_{(c)}]$ is called the effective action of the theory and we will see later on that this interpretation is correct and can be in fact very useful. We may write its complete definition in the form

$$e^{-\Gamma[\varphi_{(c)}]}=\left\langle e^{\sum_{\vec{n}}j(\... ...})\right]}\;e^{-S}}{\displaystyle \int[{\bf d}\varphi]e^{-S}}.$$

Note that, since $\Gamma$ is a functional directly of $\varphi_{(c)}$, not of $j$, the external sources that appear in this expression should be understood as functionals $j[\varphi_{(c)}]$. As we shall see in what follows, the effective action is related directly to the classical limit of the theory, as well as to its main properties relative to propagation phenomena and to the physical interactions that may exist in the theory.

Problems

1. Basing your arguments in the famous theorem relative to the uniqueness of the solution of a differential equation under certain conditions, show that the mapping between the classical solutions and the external sources in the classical theory of fields is a bijective or one-to-one map, that is, show that each external source $j$ corresponds to a unique classical solution $\varphi_{(c)}$.

2. Using the definition of the connected three-point correlation function,

   
   

   $$g_{(c,j)1,2,3}=\frac{\mbox{\boldmath$\mathfrak{d}$}^{3}W[j]}... ...math$\mathfrak{d}$}j_{2}\mbox{\boldmath$\mathfrak{d}$} j_{3}},$$
   
   

   in a theory with a non-vanishing external source $j$, show that it relates to the complete three-point and two-point functions by

   
   

   $$g_{(c,j)1,2,3}=g_{(j)1,2,3}-g_{(j)1,2}\;\varphi_{(c)3} -g_{(... ...arphi_{(c)2} +2\varphi_{(c)1}\,\varphi_{(c)2}\,\varphi_{(c)3}.$$
   
   

   Substituting the complete propagators $g_{(j)i,j}$ in terms of the connected propagators $g_{(c,j)i,j}$, with the use of the relation shown in equation (3.6.4), show that the connected three-point function relates to the complete three-point function by

   
   

   $$g_{(c,j)1,2,3}=g_{(j)1,2,3}-g_{(c,j)1,2}\;\varphi_{(c)3} -g_... ...varphi_{(c)2} -\varphi_{(c)1}\,\varphi_{(c)2}\,\varphi_{(c)3},$$
   
   

   which corresponds to the subtraction from the complete function of all the possible factorizations in terms of the connected functions of smaller number of points.

3. Show, starting from the definition of $\varphi_{(c)}[j]$ in terms of $j$ in equation (3.6.1), that the variations of $\varphi_{(c)}(\vec{n})$ and $j(\vec{n})$ at the same point $\vec{n}$ are related by

   
   

   $${\bf d}\varphi_{(c)}=\left[\langle\varphi^{2}\rangle_{(j)} -... ...rphi\rangle_{(j)}^{2}\right]{\bf d}j=\sigma_{(j)}^{2}{\bf d}j,$$
   
   

   where $\sigma_{(j)}$ is the square of the local width of the distribution of values of $\varphi$ at any given site $\vec{n}$,

   
   

   $$\sigma^{2}= \langle\varphi^{2}(\vec{n})\rangle-\langle\varphi(\vec{n})\rangle^{2}.$$
   
   

   This is always a positive and non-zero number, which shows that, given the sign we chose for the definition of $S_{(j)}$, $\varphi_{(c)}$ always increases with $j$ if both refer to the same point $\vec{n}$.

4. Repeat the procedure in problem 3.6.3 for the case in which the two quantities involved are at different points, $\varphi_{(c)}(\vec{n}_{1})$ and $j(\vec{n}_{2})$ with $\vec{n}_{1}\neq\vec{n}_{2}$. Show that $\varphi_{(c)}$ also increases with $j$ in this case, so long as the model at issue has the property that its propagator in position space is always positive, $g_{(c)}(\vec{n}_{1},\vec{n}_{2})\geq 0$, for any $\vec{n}_{1}$ and $\vec{n}_{2}$. If we recall that the propagator is related to the Green function of the classical theory, we see that this is intuitively a very reasonable property for all models that we may want to examine.