Perturbation Theory

In this section we will develop in detail a perturbative approximation technique for the $\lambda \varphi ^{4}$ model which we introduced in section 1.1. As we shall see later on, it will allow us to confirm the qualitative behavior of the model, which was described in a heuristic way in that section. Let us recall that the model is defined by the action

$$S[\varphi]=\frac{1}{2}\sum_{\ell}(\Delta_{\ell}\varphi)^{2} ... ...um_{s}\varphi^{2}(s) +\frac{\lambda}{4}\sum_{s}\varphi^{4}(s),$$

containing a quartic interaction term. Due to the presence of the quartic term we do not know how to solve the model analytically. However, without this term the model becomes Gaussian and then we are able to solve it completely. It becomes clear then that the results of the complete model should converge to the corresponding results of the free theory when we make $\lambda\rightarrow 0$ since, in the continuum limit, this implies that we must approach the Gaussian point in the parameter plane of the model.

The main idea of perturbation theory is to develop an expansion for the complete model around the soluble Gaussian model. Presumably, for small values of the coupling constant the results of the complete model are not very different from the results of the free theory and hence we may understand the interaction term as a small perturbation applied to the Gaussian model. In this way, maybe we will be able to use the expansion in order to obtain useful approximations for the complete model near the Gaussian point in the critical diagram. This is just the usual expectation that one has for an approximation scheme, but a word of warning is in order here. Although we will see that it is in fact possible to calculate some useful approximations, things are not as simple as one may think at first, and the approximation scheme does not work quite in the way that one would expect.

The first step in the development of the perturbative technique is the separation of the action in two parts, which we shall denominate $S_{0}$ and $S_{I}$,

$$S=S_{0}+S_{I},$$

where $S_{0}$ is a purely Gaussian action. For the time being we will not be very specific about the detailed form of each one of the two parts. We have, for an arbitrary observable ${\cal O}$ of the complete model,

$$\langle{\cal O}\rangle=\frac{\displaystyle \int[{\bf d}\varp... ...{I}}}{\displaystyle \int[{\bf d}\varphi]e^{-S_{0}}e^{-S_{I}}}.$$ (1.2.1)

We may now write this in terms of the measure of the free theory defined by $S_{0}$, dividing both numerator and denominator by $\int[{\bf d}\varphi]e^{-S_{0}}$ and thus obtaining

$$\langle{\cal O}\rangle=\frac{\left\langle{\cal O}\;e^{-S_{I}}\right\rangle_{0}} {\left\langle e^{-S_{I}}\right\rangle_{0}},$$

where the subscript $0$ denotes expectation values of the theory defined by $S_{0}$,

$$\langle{\cal O}\rangle_{0}=\frac{\displaystyle \int[{\bf d}\... ...phi]e^{-S_{0}}}{\displaystyle \int[{\bf d}\varphi]e^{-S_{0}}}.$$

The term $S_{I}$ of the action is the one that contains the parameter $\lambda$, that we presume to be small. However, in general $S_{I}$ may contain also other parameters, so that in order to enable us to do the development of the perturbation theory in a more organized and explicit fashion it is convenient to use, instead of $\lambda$, a new expansion parameter $\varepsilon$ that we introduce as follows,

$$f(\varepsilon)=\frac{\left\langle{\cal O}\;e^{-\varepsilon S... ...e_{0}} {\left\langle e^{-\varepsilon S_{I}}\right\rangle_{0}}.$$ (1.2.2)

We have therefore that $f(0)=\langle{\cal O}\rangle_{0}$ and $f(1)=\langle{\cal O}\rangle$. Perturbation theory consists of making a series expansion, which we denominate the perturbative expansion, of $f(\varepsilon)$ around $\varepsilon=0$, up to a certain order, followed by the use of the resulting expressions at the point $\varepsilon=1$. Of course this can only be a good approximation to the complete theory if $S_{I}$ is a small quantity. Classically we can make $S_{I}$ small by adjusting the values of $\lambda$ and any other parameters that it may contain but, as we shall see in what follows, this is not possible in the quantum theory. This is the basic fact that is at the root of all difficulties with the perturbative approach to quantum field theory.

In order to understand the origin of the difficulties it is necessary to recall some important properties of the theory of the free scalar field, since we are writing our quantities here in terms of the expectation values of that theory. As we saw in the section in [7], in the case of the dimensions $d\geq 3$ which are the ones of interest for quantum field theory, the quantity $\sigma^{2}=\langle\varphi^{2}\rangle$, which we denote here by $\sigma_{0}^{2}$ to record the fact that it is a quantity relating to the free theory, is a finite and non-zero quantity both on finite lattices and in the continuum limit. In addition to this, we showed in the section in [8] that the quantity $\langle[\Delta_{\mu}\varphi]^{2}\rangle$ is also finite and non-zero both on finite lattices and in the limit, in which case it has the value $1/d$. Still in the section in [8] these facts were used to show that both the expectation value of the kinetic part $S_{K}$ of the action,

$$S_{K}=\frac{1}{2}\sum_{\ell}(\Delta_{\ell}\varphi)^{2},$$

and the expectation value of the part $S_{M}$ of the action containing the mass term,

$$S_{M}=\frac{\alpha_{0}}{2}\sum_{s}\varphi^{2}(s),$$

diverge as powers of $N$ in the continuum limit, even if we keep the models within boxes with finite volumes. In the case of $S_{K}$ we have $\langle S_{K}\rangle\sim N^{d}/2$, while in the case of $S_{M}$ we have $\langle S_{M}\rangle=(m_{0}L)^{2}N^{d-2}\sigma_{0}^{2}/2$. In addition to this, it is possible to show that in the free theory the following relation holds,

$$\langle\varphi^{4}\rangle_{0}=3\langle\varphi^{2}\rangle_{0}^{2},$$

which is the result indicated in the problem in [9]. From these consideration it follows that, assuming that the general form of $S_{I}$ is given by $S_{V}$,

$$S_{V}=\sum_{s}\left( \frac{\alpha}{2}\varphi^{2}+\frac{\lambda}{4}\varphi^{4}\right),$$

where $S=S_{K}+S_{V}$, we have for its expectation value

$$\langle S_{V}\rangle_{0}=\frac{\sigma_{0}^{2}}{2} \left(\alpha+\frac{3}{2}\sigma_{0}^{2}\lambda\right)N^{d}.$$

This means that, so long as the factor within parenthesis is not zero in the limit, $\langle S_{V}\rangle_0$ diverges as $N^{d}$ in the continuum limit.

At first sight it may seem that the expression in parenthesis may indeed vanish in the limit, since we must remember that, as was discussed in section 1.1, $\alpha$ is necessarily negative in the limit, while the factors contained in the second term of the expression are all positive. In fact, this expression is similar to our heuristic estimate for the equation of the critical curve, which was $\alpha+C_{0}^{2}\sigma_{0}^{2}\lambda=0$. However, one can verify a-posteriori that the expression is not identical to the equation of the critical curve, either by numerical means or by the the approximations in which we will calculate the equation of the curve later on. For example, in the case of the perturbative approximation we will verify that the two expressions differ by the extra factor of $1/2$ that appears in the second term in the parenthesis in the expression of $\langle S_{V}\rangle_0$.

In any case, even if the expression in parenthesis did coincide with the equation of the critical line, it would not be equal to zero on finite lattices, but would only approach zero in the $N\rightarrow\infty$ limit, with some inverse power of $N$. Since the expression is multiplied by a factor of $N^{d}$, it would have to go to zero very fast in order to avoid the divergence. As we saw in section 1.1 and will confirm quantitatively later on, the equation of the critical curve is directly related to the value of $\alpha _{R}$, so that it must go to zero exactly as $N^{-2}$, which is not enough to eliminate the divergence in the dimensions of interest, $d\geq 3$. Furthermore, even if everything worked out and $\langle S_{V}\rangle_0$ did go to zero in the limit, if we consider that we also have that $\langle S_{K}\rangle_0$ diverges as $N^{d}$ in the limit, we see that the resulting theory could not possibly fail to become trivial in that case, since the interaction term would then become vanishingly small in the continuum limit, when compared to the remaining part of the action.

The conclusion to which we are forced is that $\langle S_{V}\rangle_0$ in fact diverges in the continuum limit as $N^{d}$. It is important to observe once more that this divergence is not due to an integration over an infinite volume, because we can do the complete development of the theory within a finite box without any change in this result. This divergence is a property of the continuum limit, an ultraviolet characteristic of the theory, in which the influence of the high-frequency and short-wavelength modes of momentum space predominate. It is not a property of the infinite-volume limit, that is, of the infrared regime of the theory, in which the low-frequency and long-wavelength modes predominate. It becomes clear, therefore, that it is not possible to keep $S_{V}$ small by mere changes in the parameters $\alpha$ and $\lambda$, except if we make them converge rapidly to zero in the continuum limit, which takes us back to the Gaussian point, where all the results are already known, constituting the theory of the free scalar field.

At this point it does not seem that this perturbative technique can end up having any practical use but, in any case, let us proceed with our analysis of the situation. If we consider for a moment the denominator of equation (1.2.1) it is clear that we will have, in the continuum limit,

$$\left\langle e^{-S_{I}}\right\rangle_0\rightarrow 0,$$

while the perturbative expansion of this quantity, obtained by the series expansion of the exponential function, will contain divergent terms if we keep $\varepsilon$ finite and non-zero when we take the limit,

$$\left\langle e^{-\varepsilon S_{I}}\right\rangle_{0}\approx ... ...,\mbox{   where   } \langle S_{I}\rangle_{0}\rightarrow\infty.$$

We see here that a simple and naive expansion within such a singular structure can make a vanishing quantity appear as a collection of infinities in the terms of the expansion. We can now see that the limit of equation (1.2.1) for $N\rightarrow\infty$ is a limit of the form $0/0$. However, it certainly exists, so long as the theory is well defined, which we expect to be true so long as we keep the parameters of the theory within the stable region of the critical diagram. The denominator can be understood as the ratio of the measures of the interacting model and of the free theory,

$$\left\langle e^{-S_{I}}\right\rangle_{0}=\frac{\displaystyle... ...(S_{0}+S_{I})}}{\displaystyle \int[{\bf d}\varphi]e^{-S_{0}}},$$

so that the conclusion we arrive at is that these two measures are related in a singular way in the continuum limit. On any finite lattice the expectation value $\langle S_{I}\rangle_{0}$ is finite and we can improve the approximation by decreasing somewhat the parameters $\alpha$ and $\lambda$. However, in the continuum limit the only form to avoid the divergence is to make both $\alpha$ and $\lambda$ approach zero very quickly, thus making the model return to the Gaussian point.

This behavior of $S_{I}$ is the basic cause that is behind all the divergences that appear in the perturbative expansion of the model. It is directly related to the strong fluctuations undergone by the fields in the continuum limit, as well as with the fact that the dominating field configuration are discontinuous in the limit, as we studied in the section in [8]. Despite all this, it is still very reasonable to think that the observables $\langle{\cal O}\rangle$ of the complete model are continuous functions of the parameters of the model, because the observables are defined by means of statistical averages that eliminate the fluctuations and discontinuities which are characteristic of the fundamental field. In other words, it is reasonable to think that $f(\varepsilon)$ is at least a continuous and differentiable function of $\varepsilon$, so that there should be at least a reasonable first-order approximation for $f$ near $\varepsilon=0$, and it could even be that $f$ is an analytical function of $\varepsilon$ (problem 1.2.1).

We are faced here by a rather strange situation: on the one hand, it is reasonable to think that there is an approximation up to some order for the observables of the complete model in the vicinity of the Gaussian point but, on the other hand, we see that this approximation may not be accessible by means of the perturbative expansion starting from the definition of the quantum theory, due to the divergences that appear. Observe that this apparent conflict is related to a exchange of order of two limits, involving the continuum limit and the limit of the summation of the perturbative series. We may argue that on finite lattices the perturbative series can be summed, since all the quantities involved are finite and well-behaved in this case. Hence, in principle we may sum the perturbative series on finite lattices and after that take the continuum limit. However, when we write the series only up to a certain term of finite order and then take the continuum limit, we are inverting the order of the two limits. Although it is reasonable to think that, once the continuum limit is taken, the resulting observables should have convergent expansions in terms of the parameters of the model, there is no guarantee that these expansions are those obtained by the exchange of the order of the limits. In fact, the divergences that appear show us that the two procedures must have very different results.

At this point it is important to observe that the equation (1.2.1) which defines the observables of the quantum theory is a ratio of two quantities involving $S_{I}$ and that, due to this, it is possible that some or even all the divergences due to this quantity end up by cancelling each other, between those coming from the numerator and those coming from the denominator, if we make a careful expansion of the ratio, that is, a careful expansion of $f(\varepsilon)$. We will verify later on that it is indeed possible to obtain in this way a useful approximation for some of the observables of the complete model, despite the divergences that are involved in the limit, but we should keep in mind that we are dealing with a singular expansion, so that it should come as no surprise it not everything works out perfectly as expected. It is in this context that the idea of renormalization appears for the first time with a recognizable meaning. Unfortunately, this term is used for several different things in the structure of the theory, but here it really has to do with renormalizing something in the usual sense. In fact, one can treat the problem at hand by making a change in the normalization of both the numerator and the denominator of equation (1.2.1), eventually obtaining the same results that we will obtain here in a more direct way (problem 1.2.2).

We will examine here the first-order and second-order terms in $\varepsilon$ for the expansion of $f(\varepsilon)$, for which we obtain

$$f(\varepsilon)=f(0)+\varepsilon f'(0)+\frac{1}{2}\varepsilon^2 f''(0)+\ldots,$$

where the first three terms contain (problem 1.2.3)

$$\begin{eqnarray*} f(0) & = & \langle{\cal O}\rangle_{0}, f'(0) & = & -\left[\l... ..._{0} -\langle{\cal O}\rangle_{0}\langle S_{I}\rangle_{0}\right]. \end{eqnarray*}$$

Making $\varepsilon=1$ we obtain

$$\langle{\cal O}\rangle \approx \langle{\cal O}\rangle_{0}-\left[\langle{{\cal O}} S_{I}\rangle_{0} -\langle{\cal O}\rangle_{0}\langle S_{I}\rangle_{0}\right]$$

$$+\frac{1}{2}\left\{\left[\langle{{\cal O}} S^{2}_{I}\rangle_{0} -... ...\rangle_{0} -\langle{\cal O}\rangle_{0}\langle S_{I}\rangle_{0}\right]\right\}.$$ (1.2.3)

This is the approximation for $\langle{\cal O}\rangle$ up to the order $\varepsilon^{2}$, that is, effectively up to the order $\lambda^{2}$. We will use it later on to calculate perturbative approximations for some of the observables of the model.

Observe that it is not to be expected that this expansion may produce a convergent series for the observables of the model. An alternative way to see this is to observe that there cannot be a non-vanishing convergence radius for the series of $f(\varepsilon)$ around $\varepsilon=0$ in the complex $\varepsilon$ plane, because a non-vanishing convergence disk around zero would include negative values of $\varepsilon$, which correspond to points in the unstable region of the parameter plane of the model, where we know that it does not exist. At most what we can hope to obtain are reasonable approximations up to a certain order, which hopefully will be good enough to allow us to form a correct qualitative idea about the behavior of the model. Note that the model would be clearly more useful if it did not cease to exist when we exchange the sign of the coupling constant. One is led to recall that this is the expected situation in electrodynamics, in which we can have charges of either sign.

In order to complete the development of our perturbative ideas, we must now return to the issue of the separation of the action $S$ in parts $S_{0}$ and $S_{I}$. This separation will depend on whether we want to perform calculations in one or the other of the two phases of the model, the symmetrical phase or the broken-symmetrical phase, whose existence and nature we discussed in section 1.1. In any case $S_{0}$ must satisfy the two essential conditions: it must be no more than quadratic on the fields and it must be stable, which means that it must correspond to a well-behaved theory of free fields, having therefore a lower bound.

The issue of stability must be examined carefully at this point. As we saw in section 1.1, in any continuum limit that does not approach the Gaussian point the parameter $\alpha$ will become strictly negative. Therefore we cannot include the $\alpha$ term in $S_{0}$, because this quadratic action would become unbounded from below and the corresponding measure well be ill-defined even on finite lattices. The alternative of including only the derivative term in $S_{0}$ and of simply including the $\alpha$ term in $S_{I}$ is also not adequate, since the free massless theory that results from this has a zero mode that could be absent from the complete model, leading to the possibility of the appearance of spurious infrared divergences.

In order to avoid possible infrared problems we will introduce into the model a new parameter $\alpha_{0}\geq 0$ associated to a quadratic term containing $\varphi^{2}$, in such a way that the model is not actually changed. Dealing first with the case in which we are in the symmetrical phase, we will choose for $S_{0}$ the action of the free theory as we have studied it since the section in [10],

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

The interaction part $S_{I}$ of the action will contain the remaining terms of the original action and a term containing $\alpha_{0}$ with the opposite sign, so that the sum of $S_{0}$ and $S_{I}$ continues equal to the original action. It follows that in this symmetrical phase we will have for $S_{I}$

$$S_{I}=\sum_{s}\left[\frac{\alpha-\alpha_0}{2}\varphi^{2}(s) +\frac{\lambda}{4}\varphi^{4}(s)\right].$$

The parameter $\alpha_{0}$ is clearly irrelevant in the exact model and the final results should be independent of it. We will see later on that this is indeed the case but, since $\alpha_{0}$ appears both in $S_{0}$ and in $S_{I}$, which will be treated in very different ways during the development of the approximation technique, we will also see that there are some subtleties relating to the role played by $\alpha_{0}$. Up to this point it seems that we are free to keep the parameter $\alpha_{0}$ finite and non-zero in the $N\rightarrow\infty$ limit, but it is not very reasonable to do this because this procedure would correspond to a diverging mass $m_{0}$ for the distribution defined by $S_{0}$ in the limit. Instead of that, we will choose $\alpha_{0}=m_{0}^{2}/N^{2}$ and work with an $m_{0}$ which is kept finite in the limit, rather than diverging. What we are hinting at here is that perhaps it is possible to improve the quality of the approximation by a suitable choice of the free parameter $\alpha_{0}$. If we knew beforehand the value $m_{R}$ of the renormalized (physical) mass of the complete model in the limit, we could even consider making $m_{0}=m_{R}$. Although it is not apparent at this moment that we should do this, or that we could do it, since we do not yet know $m_{R}$, we will see later on that this is, in fact, a natural and very convenient choice.

In the broken-symmetrical phase we expect that the expectation value of the field $\langle\varphi\rangle$ will be different from zero and, in order to enable us to develop the perturbative approximation is a simpler way, it is convenient to first rewrite the model in terms of a shifted field $\varphi'$ given by

$$\varphi'=\varphi-v_{R},\mbox{  }\varphi=\varphi'+v_{R},\mbox... ...\langle\varphi'\rangle=0,\mbox{  }\langle\varphi\rangle=v_{R},$$

where $v_{R}$ is the expectation value of the field which, in the absence of any external sources breaking the discrete translation invariance of the lattice, as is our case here, should be constant, having the same values at all the sites. Since $v_{R}$ is a constant it follows that the derivative term of the action remains unchanged. The polynomial terms which are quadratic and quartic on the fields, however, are transformed according to the relations

$$\begin{eqnarray*} \varphi^{2} & = & \varphi'^{2}+2v_{R}\varphi'+v_{R}^{2}, \v... ...arphi'^{3} +6v_{R}^{2}\varphi'^{2}+4v_{R}^{3}\varphi'+v_{R}^{4}. \end{eqnarray*}$$

We may neglect the constant terms, that do not depend on the field, since the exponentials of these terms are constant factors that appear both in the numerator and in the denominator of the ratio that defines the observables, thus cancelling off and not affecting in any way the statistical distribution of the model. Doing this we obtain for the complete action of the model

$$S=\sum_{s}\left[\frac{1}{2}\sum_{\mu}(\Delta_{\mu}\varphi')^... ...lambda v_{R}\varphi'^{3}+\frac{\lambda}{4}\varphi'^{4}\right].$$

Since we know that $\alpha$ will always be strictly negative, we introduce now the parameter $\alpha_{0}\geq 0$ and separate the action into a free part

$$S_{0}=\frac{1}{2}\sum_{\ell}(\Delta_{\ell}\varphi')^{2} +\frac{\alpha_{0}}{2}\sum_{s}\varphi'^{2}(s)$$ (1.2.4)

and an interaction part

$$S_{I} = \sum_{s}\left[v_{R}\left(\alpha+\lambda v_{R}^{2}\right)\varphi'(s) +\frac{\alpha-\alpha_{0}+3\lambda v_{R}^2}{2}\varphi'^{2}(s)\right.$$

$$\left.\;\;\;\;\;\;\;\;+\lambda v_{R}\varphi'^{3}(s)+\frac{\lambda}{4}\varphi'^{4}(s)\right].$$ (1.2.5)

This is the form of the interaction term to be used in the broken-symmetrical phase of the model. We have therefore a completely well-defined scheme for trying to obtain approximations for the observables of the complete model in the vicinity of the Gaussian point, both in the symmetrical phase and in the broken-symmetrical phase. We must now perform in detail the calculation for some particular observables of the model, always keeping in mind that this is a very singular approximation scheme and that it may turn out that not everything will work as we might hope, in order to verify what we may learn about the structure of the model by means of the use of this technique.

Problems

1. ($\star$) Determine whether the function $f(\varepsilon)$ defined in equation (1.2.2) is or is not analytical as a function of $\varepsilon$. In order to do this, first extend $\varepsilon$ to the complex plane, $\varepsilon=x+\imath y$ with real $x$ and $y$, writing the function $f$, now complex, as $f=u(x,y)+\imath v(x,y)$. Verify then whether $u(x,y)$ and $v(x,y)$ satisfy the two Cauchy-Riemann conditions: $\partial_{x}u(x,y)=\partial_{y}v(x,y)$ and $\partial_{y}u(x,y)=-\partial_{x}v(x,y)$. Perform the verification both on finite lattices and in the continuum limit.

2. ($\star\star$) It is argued in the text that the problems with the perturbative expansion originate from the fact that $\langle S_{I}\rangle_0$ diverges as $N^{d}$ in the continuum limit. This causes, for example, the denominator of equation (1.2.1), which defines the observables, to behave in the limit as

   
   

   $$\left\langle e^{-S_{I}}\right\rangle_0\rightarrow 0.$$
   
   

   One could imagine that one way to try to get around this problem is to add to the action a field-independent term $\zeta(\alpha,\lambda,N)$, which corresponds to multiplying both the numerator and the denominator of equation (1.2.1) by a number $Z(\alpha,\lambda,N)=\exp[\zeta(\alpha,\lambda,N)]$. This corresponds to a renormalization of the statistical averages that define the expectation values of the complete model in terms of the expectation values of the free theory, leading to

   
   

   $$\langle{\cal O}\rangle=\frac{\displaystyle \int[{\bf d}\varp... ...{\displaystyle \left\langle e^{\zeta-S_{I}}\right\rangle_{0}}.$$
   
   

   Naturally, this does not change the observables. However, we are now free to choose $\zeta$ in any way we choose, and we may consider choosing it so that the quantity $\zeta-S_{I}$ acquires a small or even a vanishing average value, rather than diverging as $N^{d}$ in the limit. It is clear that in this case $\zeta$ will have to be chosen so as to diverge in the limit and hence cancel the divergence of the average value of $S_{I}$. Observe however that in this way we can control only the average value of the difference $\zeta-S_{I}$, we cannot control the fluctuations of this quantity, because $\zeta$ cannot depend on the fields.

   If we recall that, as was seen in the text, the large-$N$ limit of equation (1.2.1) is of the type $0/0$, it is reasonable to think that a general criterion or renormalization condition for the choice of $\zeta$ would be

   
   

   $$\left\langle e^{\zeta-S_{I}}\right\rangle_{0}=1,$$
   
   

   which causes the limit to cease to be of the type $0/0$, but which is a very complicated condition to implement. To first order, we may think that the condition $\langle\zeta-S_{I}\rangle_{0}=0$ should be sufficient, and it is a condition which is much simpler to deal with. About this type of renormalization procedure we have the following tasks to propose:

   1. Show that this scheme would be sufficient to make the perturbative series well-behaved, with finite terms in the continuum limit, so long as, besides keeping at zero the average value of the quantity $\zeta-S_{I}$, we could also keep the fluctuations of this quantity at some finite average sizes around zero. Please note that we are not talking about the series being convergent, but only about its individual terms not diverging in the limit.

   2. Show that it is not possible to satisfy this condition in this model. In order to do this consider the calculation of $\langle(\zeta-S_{I})^{2}\rangle_{0}$ under the condition that $\zeta=\langle S_{I}\rangle_{0}$, that is, calculate the quantity

      
      

      $$\langle S_{I}^{2}\rangle_{0}-\langle S_{I}\rangle_{0}^{2}$$
      
      

      and show that it does not have a finite limit when $N\rightarrow\infty$.

   3. Repeat the first-order calculations presented in the text, using these ideas and the condition $\langle\zeta-S_{I}\rangle_{0}=0$ in order to determine $\zeta$, thus showing that exactly the same results presented in the text are obtained in this context.

3. Perform explicitly the expansion of $f(\varepsilon)$ up to the order $\varepsilon^{2}$ and derive the form of the three terms that appear in equation (1.2.3).