Physical Significance of the Effective Action

As we saw in sections 3.5 and 3.6, both the classical theory and the quantum theory establish well-defined functional relations between $\varphi_{(c)}$ and $j$. In the classical theory this relation is established by the location of the minimum of the action $S$ as a function of the external source $j$, while in the quantum theory it is established by the modification of the relative statistical weights $\exp(-S)$ of the field distribution in configuration space, caused by the introduction of $j$. Unlike what happens in the theory of the free scalar field, in general these two functional relations are different. We may then ask whether there is in the quantum theory a functional whose minimum establishes between $\varphi_{(c)}$ and $j$, in the fashion of what happens in the classical theory, the functional relation defined by this quantum theory. Clearly, this has to be some functional of $\varphi_{(c)}$, so that we may consider its variations when we vary $\varphi_{(c)}$ around the value defined by the quantum theory.

The objective of this section is to show that the effective action $\Gamma[\varphi_{(c)}]$ is such a functional, besides analyzing its properties and establishing its role as a kind of abstract of the properties of the quantum theory. Just as we did in section 3.6, we will establish these facts in a general way, for any models of scalar fields, not only for the free theory. We already saw in section 3.6 that the knowledge of the functionals $Z[j]$ and $W[j]$ allows us to obtain all the correlation functions of the quantum theory and, since from the physical standpoint the quantum theory may be understood as the set of its correlation functions, we have through these functionals a complete image of the quantum theory and its consequences. We will see here that the effective action also has this same role, but that it presents the structure of the quantum theory in a more synthetic and direct way. Obtaining $\Gamma$ in closed form corresponds to the complete solution of the theory and, therefore, is not usually an easy task. However, it is often possible to formulate testable hypothesis about the form of $\Gamma$ or of parts of it, based on symmetry arguments or other types of reasoning, that are very useful to guide us in our explorations of the structure of the models.

Our first task is to show that $\Gamma$ is indeed related to the solution of the quantum theory through a minimization process. In order to put in a clearer perspective our procedure in this first part, let us point out that the classical theory and the quantum theory act on very different spaces of functions with respect to the field $\varphi$. When we study the classical theory through the principle of minimum action, we assume that, in the continuum limit, the possible fields are continuous and differentiable functions at almost all points, that is, all except a collection of isolated singularities, usually associated to the presence of point sources, which are no more than mathematically convenient fictions. In contrast to that, in the quantum theory we start by assuming that the fields can assume values in a much larger space, the space of all possible functions, without any restrictions of differentiability or even of continuity. As we shall see in detail later on, this space of configurations is a space of functions which are typically discontinuous at all points, even in the continuum limit. The imposition on it of the statistical distribution of a given model attributes to each element of the space different statistical weights but does not change the character typically discontinuous of the configurations.

Figure 3.7.1: A diagram of the function spaces involved in the classical and quantum theories.

On the other hand, the space of the expectation values $\varphi_{(c)}$ of the field in the quantum theory is much more limited than the space of the fields $\varphi$, because the statistical averaging process over all the possible configurations has a strong effect of eliminating the discontinuities and non-differentiabilities of the configurations, usually resulting, in the continuum limit, in continuous functions for $\varphi_{(c)}$, which in general are also differentiable except for a set of isolated singularities associated to singularities in the external sources $j$ that are included in the theory. As we saw before, both the classical theory and the quantum theory establish functional relations between the sources $j$ and the continuous and mostly differentiable fields. The relations among these spaces are illustrated in figure 3.7.1. In this figure ${\rm J}$ is the set of possible external sources, ${\rm C}$ is the set of possible configurations of the fields in the classical theory and ${\rm Q}$ the set of possible configurations of the fields in the quantum theory. The symbols $R_{\rm C}$ and $R_{\rm Q}$ represent the relations established between ${\rm J}$ and ${\rm C}$ by the classical and quantum theories, respectively. The symbol $<\;>$ represents the averaging process of the quantum theory, which takes us from ${\rm Q}$ into ${\rm C}$.

In general, for $j=0$ we have $\varphi_{(c)}=0$ and, for each $j\neq 0$ some fairly well-behaved $\varphi_{(c)}$. Se see in this way that the possible configurations for $\varphi_{(c)}$ are determined by the possible configuration of $j$ that we may introduce in the theory. Since $j$ are classical external sources, we may think about the configurations of $j$ as those that we usually associate to distributions of sources or charges in the classical theory, that is, densities represented by continuous and mostly differentiable functions to which we may superpose arbitrary but finite sets of isolated point sources or charges. Once this is settled, from this set of possible sources $j$ the quantum theory determines for each model a certain space of possible classical fields $\varphi_{(c)}$, which is a subspace of ${\rm Q}$ and which we may take as the space that is relevant in the classical limit of the theory. The classical limit is the limit of large wavelengths in which the quantum fluctuations are ignored and the averages such as $\varphi_{(c)}$ represent what can be observed in the theory.

We see in this way that, starting from the most general possible set of configurations, the dynamical structure of the quantum theory itself automatically defines the subspace of configurations which is relevant for the corresponding classical theory, eliminating any need of imposing by hand the properties of this space in the classical limit of the theory. By and large, we may think of ${\rm J}$ and ${\rm C}$ as copies of the same space, the space where classical field-like objects exist. After all, an external source is no more than a representation of an expectation value within some other model or some other part of a more general model, representing a part of the physical world whose quantum behavior is not under direct scrutiny. Fundamentally, everything in nature has an underlying quantum behavior and an ultimate theory in its most fundamental form should describe the quantum interaction between all parts of nature without any explicit reference to external classical objects such as external sources.

It is in the context of this subspace of the possible configurations of the classical field $\varphi_{(c)}$, defined by the averaging process in the quantum theory, that we will study the behavior of $\Gamma$ in the immediacy of the configuration $\varphi_{(c)0}$ associated to a given $j_{0}$ by the functional relation established by the dynamics of the quantum theory. At this point it is convenient to recall the definition of $\Gamma$,

  $$\Gamma[\varphi_{(c)}]= -\ln\left\{ \frac{\displaystyle \int[{\bf ... ...vec{n})\right]}\;e^{-S}}{\displaystyle \int[{\bf d}\varphi]\;e^{-S}} \right\},$$ (3.7.1)

as well as the fact that $\Gamma$ is a functional of $\varphi_{(c)}$ alone, so that in this expression $j$ should be understood as just a functional of $\varphi_{(c)}$, through the functional relation established between them by the quantum theory. Hence, given the effective action $\Gamma[\varphi_{(c)}]$ defined in this way for a given but otherwise arbitrary $\varphi_{(c)}$, we will now define the effective action in the presence of an arbitrary external source $j_{0}$, not necessarily the one that is related with $\varphi_{(c)}$ by the functional relation established by the quantum theory, as

$$\Gamma_{(j)}=\Gamma[\varphi_{(c)}] -\sum_{\vec{n}}j_{0}(\vec{n})\varphi_{(c)}(\vec{n}).$$

Note that this new external source is a source for $\varphi_{(c)}$, not for $\varphi$, so we are asking ourselves how would the action $\Gamma$ behave as a classical action under the introduction of a source. Given a fixed $j_{0}$, this equation defines $\Gamma_{(j)}$ for an arbitrary $\varphi_{(c)}$, so that its functional variation is given by

$${\bf d}\Gamma_{(j)}={\bf d}\Gamma[\varphi_{(c)}] -\sum_{\vec{n}}j_{0}(\vec{n}){\bf d}\varphi_{(c)}(\vec{n}).$$

In order to establish that the solution of the theory in the presence of $j_{0}$ is given by a local minimum of $\Gamma_{(j)}$, we will now consider the variations of this functional around the point $\varphi_{(c)0}$ which is the value of $\varphi_{(c)}$ that is related with $j_{0}$ through the functional relation established by the quantum theory. In this case the differential of $\Gamma_{(j)}$ is the expression given above with

$${\bf d}\varphi_{(c)}(\vec{n}) =\varphi'_{(c)}(\vec{n})-\varphi_{(c)0}(\vec{n}).$$

It is necessary to make very clear in which way we should analyze the variations of $\Gamma_{(j)}$. We assume that one makes small but otherwise arbitrary variations ${\bf d}\varphi_{(c)}(\vec{n})$ of the classical field and we ask what is the corresponding variation of $\Gamma_{(j)}$. In the second term of the equation above the external source is kept fixed at a given function $j_{0}(\vec{n})$ while $\varphi_{(c)}$ varies, but in the first term we should calculate the variation of $\Gamma$ that comes from its definition. Hence, when we calculate the variation of $\Gamma$, the $j[\varphi_{(c)}]$ that appears in the definition, being a functional of $\varphi_{(c)}$, does not remain fixed but rather varies with $\varphi_{(c)}$ according to the functional relation established between them by the quantum theory.

Let us calculate then this variation of $\Gamma$, making a variation $\varphi_{(c)0}\rightarrow\varphi'_{(c)}$ of the classical field around the value $\varphi_{(c)0}$. The variation of $\Gamma$ is given by

$${\bf d}\Gamma=\Gamma[\varphi'_{(c)}]-\Gamma[\varphi_{(c)0}],$$

so that, using the definition of $\Gamma$, we obtain

$${\bf d}\Gamma= -\ln\left\{ \frac{\displaystyle \int[{\bf d}\... ...hi(\vec{n})-\varphi_{(c)0}(\vec{n})\right]}\;e^{-S}} \right\},$$

where the variation of $\varphi_{(c)}$ around $\varphi_{(c)0}$ corresponds to a variation of $j$ around $j_{0}$, $j_{0}\rightarrow j'[\varphi'_{(c)}]$. In general both these variations are dependent on position. We will now write $j'$ as $j'=j_{0}+{\bf d}j$ and expand to first order the exponential that appears in the denominator, getting

$$e^{\sum_{\vec{n}}j'(\vec{n}) \left[\varphi(\vec{n})-\varphi'... ...n})-\varphi'_{(c)}(\vec{n})\right] {\bf d}j(\vec{n}) \right\},$$

from which it follows that

$$\begin{eqnarray*} \hspace{-2em} & & {\bf d}\Gamma= \\ \hspace{-2em} & & -\ln\lef... ...\sum_{\vec{n}}j_{0}(\vec{n})\varphi(\vec{n})}\;e^{-S}} \right\}, \end{eqnarray*}$$

where we took off the functional integrals factors that do not depend on $\varphi$. Writing this expression in terms of $S_{(j)}=S-\sum_{\vec{n}}j_{0}(\vec{n})\varphi(\vec{n})$ and manipulating it a little we obtain

$$\begin{eqnarray*} {\bf d}\Gamma & = & \sum_{\vec{n}}j_{0}(\vec{n}) \left[\varphi... ..._{\vec{n}}{\bf d}\varphi_{(c)}(\vec{n}){\bf d}j(\vec{n})\right], \end{eqnarray*}$$

since we have that $\langle\varphi(\vec{n})\rangle_{(j)}=\varphi_{(c)0}$ and that ${\bf d}\varphi_{(c)}=\varphi'_{(c)}-\varphi_{(c)0}$. Expanding now the logarithm to first order we obtain

$${\bf d}\Gamma=\sum_{\vec{n}}j_{0}(\vec{n}){\bf d}\varphi_{(c... ...+\sum_{\vec{n}}{\bf d}\varphi_{(c)}(\vec{n}){\bf d}j(\vec{n}).$$

We have therefore for the variation of $\Gamma_{(j)}$, from its definition,

$$\begin{eqnarray*} {\bf d}\Gamma_{(j)} & = & \sum_{\vec{n}}j_{0}(\vec{n}){\bf d}\... ... & \sum_{\vec{n}}{\bf d}\varphi_{(c)}(\vec{n}){\bf d}j(\vec{n}). \end{eqnarray*}$$

If we recall now the result obtained in problem 3.6.3, according to which we may write for the variations of $j$ and $\varphi_{(c)}$ at the same arbitrary point $\vec{n}$ that

$${\bf d}j(\vec{n})=\frac{1}{\sigma_{(j)}^{2}}{\bf d}\varphi_{(c)}(\vec{n}),$$

we see that we can write our final result for the variation of the effective action in the presence of external sources,

$${\bf d}\Gamma_{(j)}=\sum_{\vec{n}} \frac{[{\bf d}\varphi_{(c)}(\vec{n})]^{2}}{\sigma_{(j)}^{2}},$$

which means that, given a certain external source $j_{0}(\vec{n})$ and a certain functional $\Gamma[\varphi_{(c)}]$, the corresponding functional $\Gamma_{(j)}$ always increases, for any variation ${\bf d}\varphi_{(c)}(\vec{n})$ around the function $\varphi_{(c)0}(\vec{n})$ determined by the quantum theory from $j_{0}(\vec{n})$. It follows that the functional $\Gamma_{(j)}[\varphi_{(c)}]$ is at a minimum when ${\bf d}\varphi_{(c)}(\vec{n})\equiv 0$, that is, when $\varphi_{(c)}(\vec{n})$ is the function determined by the quantum theory.

In this way, we conclude that $\Gamma$ describes how the quantum theory responds to the introduction of external sources, in the same way in which $S$ does the same thing in the classical theory. We see therefore that, in the limit of large wavelengths, that is, for distances which are much larger than those finite correlation lengths that appear in the theory, in situations where the quantum fluctuations can be ignored, $\Gamma$ is indeed the classical action that describes the classical limit of the model, thus describing its classical behavior, which exists as a consequence of the underlying quantum structure of the model.

In order to continue to elucidate the significance of $\Gamma$ we will now examine its functional derivatives with respect to $\varphi_{(c)}$. We saw in section 3.6 that the functional derivatives of $Z$ and $W$ with respect to $j$ give us directly all the correlation functions of the theory. Let us now see how the derivatives of $\Gamma$ can help us to probe the structure of the theory. We already know the first derivative, which is

$$\frac{\mbox{\boldmath$\mathfrak{d}$}\Gamma[\varphi_{(c)}]}{\... ...\boldmath$\mathfrak{d}$}\varphi_{(c)1}} =j_{1}[\varphi_{(c)}],$$

where we are using again the notation of the dependency on $\vec{n}_{1}$ by means of numerical indices. It is tempting to differentiate this a second time directly in terms of $\varphi_{(c)}$, but it is more convenient and clearer, causing less confusion, to proceed in another way. We should always remember that the relation between $j$ and $\varphi_{(c)}$ is not local and that the derivative of the right-hand side of this equation is not as simple as it may seem at first sight. We will differentiate this equation with respect to $j$ first, not with respect to $\varphi_{(c)}$, obtaining

$$\frac{\mbox{\boldmath$\mathfrak{d}$}}{\mbox{\boldmath$\mathf... ... j_{1}}{\mbox{\boldmath$\mathfrak{d}$}j_{2}}=\delta^{d}_{1,2}.$$

We now use the chain rule in order to rewrite the derivative in terms of $\varphi_{(c)}$,

$$\frac{\mbox{\boldmath$\mathfrak{d}$}}{\mbox{\boldmath$\mathf... ...box{\boldmath$\mathfrak{d}$}\varphi_{(c)1}} =\delta^{d}_{1,2}.$$

Now, from equations (3.6.3) and (3.6.4) we have that

$$\frac{\mbox{\boldmath$\mathfrak{d}$}\varphi_{(c)3}}{\mbox{\b... ...rak{d}$}W}{\mbox{\boldmath$\mathfrak{d}$}j_{3}} =g_{(c,j)3,2},$$

from which it follows that

  $$\sum_{3}g_{(c,j)3,2} \frac{\mbox{\boldmath$\mathfrak{d}$}^{2}\Gam... ...\varphi_{(c)3}\mbox{\boldmath$\mathfrak{d}$}\varphi_{(c)1}} =\delta^{d}_{1,2}.$$ (3.7.2)

This result tells us that the second functional derivative of $\Gamma$ is the inverse of the propagator in position space, both the propagator and its inverse considered as operators in space-time, with matrix representations such as the ones we saw before, in section 2.3, for the finite-difference operators. We will give this new operator a name, which is suggestive in the case of the free theory,

  $$\frac{\mbox{\boldmath$\mathfrak{d}$}^{2}\Gamma[\varphi_{(c)}]} {\... ...=\raisebox{0.6ex} {\mbox{\fbox{\rule{0ex}{0.2ex}\rule{0.2ex}{0ex}}}}_{(c)1,2},$$ (3.7.3)

where the propagator is the inverse of the operator $\raisebox{0.6ex} {\mbox{\fbox{\rule{0ex}{0.2ex}\rule{0.2ex}{0ex}}}}_{(c)}$, that is, $g_{(c,j)1,2}=\raisebox{0.6ex} {\mbox{\fbox{\rule{0ex}{0.2ex}\rule{0.2ex}{0ex}}}}_{(c)1,2}^{-1}$. Let us exemplify this with the theory of the free scalar field. One can show (problem 3.7.1) that, in the theory of the free scalar field defined by the action $S_{0}$,

$$S_{0}[\varphi]=\frac{1}{2}\sum_{\ell}(\Delta_{\ell}\varphi)^{2} +\frac{\alpha_{0}}{2}\sum_{s}\varphi^{2}(s),$$

the effective action has exactly the same form as $S_{0}$, written in terms of the classical field,

  $$\Gamma[\varphi_{(c)}]=\sum_{\vec{n}}\left\{ \frac{1}{2}\sum_{\mu}... ...}(\vec{n})\right]^{2} +\frac{\alpha_{0}}{2}\varphi_{(c)}^{2}(\vec{n})\right\},$$ (3.7.4)

where we decomposed the sum over links as usual. This fact explains in a clear way why the propagator of the free quantum theory is equal to the Green function of the free classical theory. We may now take the functional derivatives of $\Gamma$, which we will do in a rather symbolic and formal way, leaving a more detailed approach to the reader, in problem 3.7.2. Taking the first functional derivative we obtain

$$\frac{\mbox{\boldmath$\mathfrak{d}$}\Gamma[\varphi_{(c)}]}{\... ...1,3}\right] +\alpha_{0}\varphi_{(c)3}\delta^{d}_{1,3}\right\},$$

where we once again are using the notation of numerical indices for the dependencies on position, and we used the fact that the variables $\varphi_{(c)}$ are independent at all the points, so that

$$\frac{\mbox{\boldmath$\mathfrak{d}$}\varphi_{(c)1}}{\mbox{\boldmath$\mathfrak{d}$}\varphi_{(c)2}}=\delta^{d}_{1,2}.$$

Integrating the first term by parts, which does not produce any surface terms due to the periodical boundary conditions, we then use the delta functions to eliminate the sums and obtain

$$\frac{\mbox{\boldmath$\mathfrak{d}$}\Gamma[\varphi_{(c)}]}{\... ...}(-\Delta^{2}_{1,3}+\alpha_{0}\delta^{d}_{1,3})\varphi_{(c)3}.$$

The second functional differentiation is now immediate and results in

$$\frac{\mbox{\boldmath$\mathfrak{d}$}^{2}\Gamma[\varphi_{(c)}... ...delta^{d}_{3,2} =-\Delta^{2}_{1,2}+\alpha_{0}\delta^{d}_{1,2}.$$

We see that in this case the operator $\raisebox{0.6ex} {\mbox{\fbox{\rule{0ex}{0.2ex}\rule{0.2ex}{0ex}}}}_{(c)}$ is directly related to the Euclidean Klein-Gordon operator. Note that, just as the Laplacian operator $\Delta^{2}$, the operator $\raisebox{0.6ex} {\mbox{\fbox{\rule{0ex}{0.2ex}\rule{0.2ex}{0ex}}}}_{(c)}$ is not diagonal. We may write the content of this result in the form of operators in configuration space,

$$\raisebox{0.6ex} {\mbox{\fbox{\rule{0ex}{0.2ex}\rule{0.2ex}{0ex}}}}_{(c)1,2}=(-\Delta^{2}+\alpha_{0}I)_{1,2},$$

where $I$ is the identity operator. The fact that this operator is the inverse of the propagator is translated in this language into the fact that the propagator is the Green function of the Euclidean Klein-Gordon operator, satisfying the finite-difference equation

$$\sum_{3}(-\Delta^{2}+\alpha_{0}I)_{1,3}g_{(c)3,2}=\delta^{d}_{1,2},$$

which may be written in a more familiar form, in the continuum limit, as a differential equation with a Dirac delta function in the non-homogeneous term, so that we have, in terms of the corresponding dimensionfull quantities,

$$\left(-\sum_{\mu}\partial_{\mu}^{2}+m_{0}^{2}\right)G_{(c)}(x_{1}-x_{2}) =\delta^{d}(x_{1}-x_{2}).$$

Observe that, since $\Gamma$ is quadratic on the fields, any higher-order functional derivative of $\Gamma$ vanishes, showing that the propagator and hence the phenomenon of propagation are the only physical content of this model. In general these higher-order derivatives are related to the so-called irreducible functions with more than two points, that is, with the physical interactions that exist in the models. Their absence in this case is the mathematical translation of the fact that this is a theory of free fields. In future volumes it will be seen that in non-linear models the higher-order derivatives of the effective action will give us directly the renormalized coupling constant, whose value describes the intensity of the physical interactions that exist in the quantum theory.

Problems

1. Starting from its definition, given in equation (3.7.1), calculate the effective action $\Gamma$ of the theory of the free scalar field defined by $S=S_{0}$, obtaining the result given in the text, in equation (3.7.4). In order to do the calculation transform the action to momentum space and use the explicit relation between $\widetilde\varphi _{(c)}$ and $\widetilde\jmath$ given in equation (3.5.3), which in our current notation may be written as

   
   

   $$\widetilde\varphi _{(c)}(\vec{k})=\frac{\widetilde\jmath (\vec{k})}{\rho^{2}(\vec{k})+\alpha_{0}}.$$
   
   

2. Using the explicit form of the effective action $\Gamma$ of the free theory, given in equation (3.7.4), for the one-dimensional case $d=1$, calculate its second functional derivative with respect to the classical field, in two positions $n_{1}$ and $n_{2}$,

   
   

   $$\frac{\mbox{\boldmath$\mathfrak{d}$}^{2}\Gamma[\varphi_{(c)}... ...)}(n_{1})\mbox{\boldmath$\mathfrak{d}$} \varphi_{(c)}(n_{2})},$$
   
   

   showing that the result is $2+\alpha_{0}$ if $n_{2}=n_{1}$, $-1$ if $n_{2}=n_{1}+1$ or if $n_{2}=n_{1}-1$ and $0$ in any other case. In this way, it becomes clear that this operation of functional differentiation does in fact recover the matrix elements of the matrix representation of the Euclidean Klein-Gordon operator in configuration space.