Definition of the Model

In the previous volume of this series [2] we studied in detail the theory of the free scalar field. That model was sufficiently simple to allow us to calculate analytically all the predictions of the theory. As we saw, both in the case of the classical theory and in the case of the quantum theory this simplicity follows from the linearity of the model. We also saw that this same linearity is responsible for the fact that the model does not contain the concept of interactions between particles, and hence that the only physics that it does contain is the propagation of free particles. This was shown by the factorization of all the correlation functions in terms of the propagator, and also by the fact that the energies of the particles are simply additive, that is, the energy of a state containing two particles is the sum of the energies of the two corresponding single-particle states, implying the absence of any kind of interaction energy.

We will make here a first trial at including interactions in the theory, for which it will be necessary to break the linearity of the model, including in the action terms with more that two powers of the field. We will therefore examine the model defined by

$$S[\varphi]=\frac{1}{2}\sum_{\ell}(\Delta_{\ell}\varphi)^{2} ... ...um_{s}\varphi^{2}(s) +\frac{\lambda}{4}\sum_{s}\varphi^{4}(s),$$ (1.1.1)

which we denominate the $\lambda \varphi ^{4}$ polynomial model. We therefore choose to break the linearity of the model by the introduction of a new ultra-local term into the action, leaving untouched the term containing the derivatives. This is the simplest example of a model that, in the classical theory, contains interacting fields. Our task here is to determine the nature of the corresponding quantum theory. This new action still has the same symmetry of the action of the free theory, namely, it is invariant by the sign inversion of the fields. In addition to this, it has a lower bound so long as the parameter $\lambda$, which we call the coupling constant, is positive and not zero. On the other hand, unlike what happened in the case of the free theory, the parameter $\alpha$ may be negative in this model, so long as $\lambda$ satisfies these conditions.

Note that the addition to the action of the free theory of a single cubic term is out of the question for two reasons: it would break the symmetry and, more importantly, would cause the action not to have a lower bound. This second problem is much more serious than the failure of the action to be invariant by the symmetry transformations, since it would imply the non-existence of the corresponding quantum theory. We could, on the other hand, include a cubic term together with the quartic term, thus obtaining a non-symmetrical but stable theory. If we want to have a stable theory and keep the symmetry, we should restrict the discussion to terms with even powers of the field. We will do this here, for simplicity and ease of presentation, and motivated by the fact that, in general, symmetries have an important role to play in physics. We will discuss explicitly the case $\varphi^{4}$, but almost everything that we will do can also be done for the cases $\varphi^{2p}$, $p=2,3,4,\ldots$, with analogous results.

We say, in the classical theory defined by the action given above, that the field $\varphi$ is self-interacting. As we may see in future volumes of this series, it is also possible to define models with fields having several components that interact with each other, and that involve invariance by groups of symmetry transformations which are larger and more complex than the simple sign reflections that we have in the model with a single component. It is also possible to define manageable models with different types of field that interact with one another, which are, of course, the most important models for real physics. However, for our objectives here we may limit ourselves to the model with a single field component, postponing to a future opportunity the discussion of the more complex models.

Unlike what happened in the case of the free field, in the non-linear models there is no known way to calculate the predictions of the quantum theory in exact analytical form. In this section we will limit ourselves to the qualitative description of the behavior of the model by means of heuristic arguments based on extensive experience with its numerical treatment. Later on we will develop a technique of approximate perturbative calculations that will allow us to determine in a quantitative and fairly reliable way some of the main characteristics of the model. In general, in the case of the non-linear models it will always be necessary to make use of some approximation technique or of computer simulations in order to determine the behavior of the models.

In our approach to the subject, the computer simulations will often be the main tool for the exploration of the models. Once one becomes well acquainted with the technique of stochastic simulation, it can become a language for the understanding of the models, sometimes leading one to the solution of problems, sometimes suggesting new ideas, new observables and even new models. The ideas and techniques involved in the methods of stochastic simulation constitute a rather extended topic with a very technical character, and will be developed in detail in a separate volume of this small series of books.

The character of the classical theory determined by our action is clear, and it is not necessary to examine it in detail. The definition of the classical theory is the same as before, the classical solution is the configuration $\varphi$ that minimizes the action. The fact that it exists is guaranteed by the conditions we impose on the parameters of the model: $\lambda>0$ with any $\alpha$ or $\lambda=0$ with $\alpha\geq 0$. We may derive, in a way analogous to the one used before for the free theory, the corresponding classical equation of motion, which will be, of course, a non-linear differential equation for $\varphi$ (problem 1.1.1). In order to begin the examination of the behavior of the quantum theory, we recall that it is defined by the probability distribution over all possible configurations of $\varphi$, given by

$$\frac{\displaystyle [{\bf d}\varphi]\:e^{-\sum_{\ell}\frac{1... ...{\alpha}{2}\varphi^{2}+ \frac{\lambda}{4}\varphi^{4}\right)}},$$

Figure 1.1.1: A typical ultra-local distribution of the fields, given by the potential, that is, by the polynomial terms of the action, in the case $(\alpha<0,\lambda>0)$.

where we grouped separately the term containing the derivatives and the ultra-local part, containing the polynomial terms, both the quadratic one and the quartic one, which we call the interaction term. If we recall that the measure $[{\bf d}\varphi]$ is a product of differentials over all the sites $s$, representing the fact that in this measure the stochastic variables $\varphi$ have uniform probability distributions, we see that we can include the ultra-local terms in the measure, writing the distribution as

$$\frac{\displaystyle \left[{\bf d}\varphi\;e^{-\left(\frac{\a... ...ght] \:e^{-\sum_{\ell}\frac{1}{2}(\Delta_{\ell}\varphi)^{2}}}.$$

In this new measure the variables $\varphi$ no longer have uniform probability distributions, but have instead the probability distribution given by the exponential of the potential. A typical example of such a distribution can be seen in figure 1.1.1. We see in this way that a possible way to understand our model is to think of it as constituted of the dynamics implemented by the derivative term, but applied indirectly to new random variables $\chi$ located at the sites, with uniform probability distributions, which are given in terms of the variables $\varphi$ by the differential relation

$${\rm d}\chi={\rm d}\varphi\: e^{-(\frac{\alpha}{2}\varphi^{2}+\frac{\lambda}{4}\varphi^{4})}.$$

This means that we may write the stochastic variable $\chi$, which has an uniform probability distribution within a closed interval, in terms of the stochastic variable $\varphi$, which has a non-uniform probability distribution over the whole real line, as

$$\chi\left(\varphi\right)=\int_{0}^{\varphi}{\rm d}\varphi'\:... ...(\frac{\alpha}{2}\varphi'^{2}+\frac{\lambda}{4}\varphi'^{4})}.$$ (1.1.2)

In this way the theory is reduced to the study of the effect of the derivative term on the local distributions of either $\chi$ or $\varphi$ at each site. The inverse of relation (1.1.2), which usually can only be obtained numerically, gives us $\varphi(\chi)$ and enables us to obtain $\varphi$ with the correct non-uniform distribution, starting from the variable $\chi$ with an uniform probability distribution within a closed interval, which is not difficult to generate numerically. In this way a part of the structure of the model, the part of the distribution given by the ultra-local terms of the action, is implemented in an exact way. This is, in fact, one of the ways in which one can simulate this model in practice, by producing values of $\chi$ at each site with the correct distribution, getting from them the corresponding values of $\varphi$, and simulating the dynamics of the derivative term by the use of stochastic techniques. The use in stochastic simulations is the main application of this decomposition, which usually is not very useful as an analytical approach.

In a very general way, the complete local distribution that rules the fluctuating values assumed by the fields at an arbitrarily given site is given by the combination of the effects of the potential and of the derivative term. In order to discuss the behavior of the model at an intuitive level, it is very useful to invert the decomposition described above, representing the effect of the derivative term by the Gaussian local distribution that it implies, which was studied in detail in the first volume of this series of books, and considering directly the effects that the potential term may have on it. This allows us to use in the analysis of the model our classical intuition concerning the behavior of an object within the potential, by extending the object from a simple point body to a fluctuating statistical distribution. In this way we are able to build an almost-classical intuition in order to understand the behavior of the model.

We can imagine that we draw a copy of the potential at each site and that we put inside it the value of the field at that site, while the terms $(\Delta_{\ell}\varphi)^{2}$ interconnect each pair of neighboring sites. The stochastic dynamics implemented by these derivative terms will cause the values of the fields to fluctuate at each site, so that we will actually have a distribution of values within each potential. In the theory of the free field these local distributions are simple Gaussians, whose width $\sigma_{0}$ is a number of the order of $1$ for any lattice size, a number that determines the value of the physical mass (also called the renormalized mass), which in the case of the free theory is simply $m_{R}=m_{0}$. In our current model the complete local distributions also have a width $\sigma\approx\sigma_{0}$ of the order of $1$, but their format may no longer be exactly Gaussian. The renormalized mass is defined in this model, just as in the free theory, as the inverse of the correlation length of the model, measured through the two-point correlation function, and presumably it is also related to $\sigma$, as is the case for the free theory. What we propose to do in this section is to understand the complete dynamics of the model by considering directly the influence of the potential over the Gaussian local distribution implemented by the derivative term.

Let us recall that the relation between the two-point function and the renormalized mass was studied in detail, in the case of the free field, in a section of the first volume [4]. The exponential decay of the two-point function for large distances $r$, given by $\exp(-m_{0}r)$ as was studied in that section, is a general property of all massive theories, either free or otherwise. The behavior of the two-point function of the interacting theories for large distances is in general of this same type, except for the exchange of the parameter $m_{0}$ for another parameter $m_{R}$ which usually differs from $m_{0}$. In the numerical approach we usually measure this new parameter by means of a curve-fitting process applied to the numerical propagator of the theory in momentum space rather than position space, that is, to the Fourier transform of the two-point function, which is technically easier to do, and also more efficient. We do this in the expectation, to be confirmed a posteriori, that the form of this function in the interacting models is not very different from its format in the free theory, and it usually works very well.

In the model that we are introducing here, given values of $N$, $\alpha$ and $\lambda$, we will have not only a resulting value for the parameter $\alpha _{R}$ related to the renormalized mass, but also some resulting value for the renormalized coupling constant $\lambda_{R}$, which is the physical coupling constant whose nature and precise definition we will examine in more detail later on. Unlike what happened in the free theory, in general $\alpha _{R}$ will not be equal to $\alpha$, and in addition to this neither will $\lambda_{R}$ be equal do $\lambda$. In fact, let us recall the fact that the parameter $\alpha$ can be negative in this model, while the parameter $\alpha_{R}=(m_{R}a)^{2}$ is necessarily non-negative, and should tend to zero in the continuum limit. The rule of the game now is that neither $\alpha$ nor $\lambda$ have any direct physical meaning and that we are free to do with them whatever is necessary, within the constraints of the stability of the theory, in order to have $\alpha _{R}$ and $\lambda_{R}$ assume physically acceptable and significant values in the limit $N\rightarrow\infty$.

Since we have two free parameters to adjust in the model, that is, two functions $\alpha(N)$ and $\lambda(N)$ of the increasing size $N$ of the lattice that we may define, it may seem at first sight that we may always choose these functions so as to obtain any physically acceptable values of $\alpha_{R}(\alpha,\lambda,N)$ and $\lambda_{R}(\alpha,\lambda,N)$ in the limit $N\rightarrow\infty$. However, this is not necessarily so because, besides the stability constraints that we must impose on the basic parameters of the theory, it may be that the dynamics of the theory itself imposes over the renormalized parameters $\alpha _{R}$ and $\lambda_{R}$ other constraints, with the consequence that not all the possibilities are actually realized in practice. In an extreme case, it is possible that there are no choices of the functions $\alpha(N)$ and $\lambda(N)$ for which the values of $\alpha _{R}$ and $\lambda_{R}$ are physically acceptable in the limit, in which case we say that the quantum theory of the model does not exist. In a more general way, it may be that not all pairs of physically acceptable values for $\alpha _{R}$ and $\lambda_{R}$ are reachable by means of some path $[\alpha(N),\lambda(N)]$ with increasing $N$, in the space of parameters of the theory. For example, it may be that a constraint between $\alpha _{R}$ and $\lambda_{R}$ is established in the limit, preventing us from choosing both of them freely, in which case we may say that the parameters $\alpha$ and $\lambda$ become degenerate in the limit.

In what follows we will describe a qualitative way of understanding the behavior of the model which, despite the fact that it is purely intuitive and heuristic, based on the phenomenology of computer simulations, will give us qualitatively correct results, as we will verify later on by means of approximate calculations^1.1. In order to do this we will, as was mentioned above, represent the effect of the derivative term of the action by the fluctuations that it implies for the values of the field $\varphi$ at a given site, resulting on a Gaussian local distribution of values with a width of the order of $1$. Let us recall that in the case of the free field the width $\sigma_{0}$ did not depend on $m_{0}$ in the continuum limit. On finite lattices the width did depend on $\alpha_{0}$, but not very strongly, so long as $\alpha_{0}$ was not zero on finite lattices. In an analogous way, in our case here we expect the width $\sigma$ not to depend on $m_{R}$ in the limit, while on finite lattices it should not depend too strongly on either $\alpha$ or $\lambda$.

Figure 1.1.2: The potential and the local distribution due to the derivative term, in the case of the free theory.

In this way, in first approximation we may imagine that the width of the local distribution behaves like a semi-rigid body with finite dimensions which are almost constant along the continuum limit. If the width is ``squeezed'' to any value below its normal size, this gives rise to a non-zero value for the renormalized mass $m_{R}$. A zero squeezing force corresponds to zero $m_{R}$, and the larger the squeeze, the larger the renormalized mass. As the lattice size $N$ increases the width becomes more ``rigid'', in the sense that the same squeezing force corresponds to a larger value of $m_{R}$, until it becomes infinitely rigid in the continuum limit, in which any non-zero squeezing force gives rise to an infinite $m_{R}$. Let us imagine now that we insert this local distribution inside the potential defined by the ultra-local terms of the action at each site, as shown in figure 1.1.2 for the case of the free theory, with a Gaussian distribution and a quadratic potential. If we were examining the classical theory, we would put inside the potential a point body representing the value of the classical solution for the field, and it would come to rest at the minimum of the potential. The examination of the behavior of the quantum theory corresponds to the introduction into the potential well of an extended object which can be represented heuristically by our semi-rigid body, which becomes rigid in the continuum limit in the sense explained above.

Figure 1.1.3: The potential and the local distribution due to the derivative term, in the case of the non-linear theory, in the symmetrical phase.

The width of the local distribution in the absence of the potential is determined only by the derivative term and corresponds to the value $\alpha_{R}=0$, that is, to a zero renormalized mass. When we put the distribution under the action of the potential on a finite lattice what happens is that it tends to concentrate the values of the field around the minimum, and hence squeezes the distribution, decreasing its width, because it is statistically unfavorable for the field to exist in the positions where the potential is larger. This squeeze of the width of the distribution gives rise to a finite and non-zero value for $\alpha_{0}$ and hence for the renormalized mass. The decrease in the width of the distribution is never very large, and it is still a quantity of the order of $1$. The difference in width due to the squeeze goes to zero in the continuum limit because, as we saw in the first volume of this series, in this limit it is necessary that $\alpha_{0}$ go to zero in the free theory, making the potential well become infinitely wide and flat at the position of the minimum. The fact is that such a vanishing effect over the width is sufficient to give to the renormalized mass, in the limit, any positive value we wish.

Figure 1.1.4: The potential and the local distribution due to the derivative term, in the case of the non-linear theory, in the broken-symmetrical phase.

In our model with a quartic term, in the case where we have both $\alpha$ and $\lambda$ positive or zero, we should expect a qualitatively similar behavior, since we have the same derivative term in the action and a potential well with a similar form, although the detailed format of the curve is not exactly the same. In this case, in order for the potential well to become infinitely wide in the continuum limit, thus allowing $\alpha _{R}$ to go to zero and $m_{R}$ to approach a finite value, it is necessary that both $\alpha$ and $\lambda$ tend to zero in the limit. In this way we have just made, without too much effort, a prediction with very serious consequences regarding the behavior of the model: if we limit ourselves to the case in which $\alpha\geq 0$ and $\lambda\geq 0$ for all $N$, it will be necessary to make both $\alpha\rightarrow 0$ and $\lambda\rightarrow 0$ in the limit, which takes the model back to the critical point of the Gaussian model and therefore eliminates any possibility that $\lambda_{R}$ be different from zero in limits of this type.

Except for the case $d=3$, this implies that there are in fact no interactions between particles in the quantum theory of this model, in any limits that stay within the quadrant given by $\alpha\geq 0$ and $\lambda\geq 0$. We say that in this case the model has only the trivial limit, leading to the theory of the free field, or that the theory is trivial in this sector of the space of parameters of the model. The case $d=3$ is a little different because, since in this case the physical coupling constant has dimensions of mass, it is possible that there are interactions even if the model approaches the Gaussian point, e phenomenon that we will discuss later on.

We conclude that, if we are to have any chance of finding an interesting limit in this model, it will be necessary for at least one of the two parameters to be negative. Since we cannot make $\lambda$ negative due to the stability constraints, it follows that the parameter $\alpha$ fill be necessarily negative, in any continuum limit of this model that has any chance of not being trivial. In fact, in this case something very interesting happens, because the potential of the model acquires a double well, as shown in figure 1.1.3, which alters completely the behavior of the model, since now a new relevant parameter related to the potential arises, given by the distance between the two minima, which can be easily calculated from the potential. We see that we now have two different widths at play in the problem, the width of the local distribution and the distance between the two minima. We also have two widths related to the potential, the width of each one of the two wells and the total width of the two wells, which are related by a factor of approximately two. The positions of the two minima of the potential are given by $\varphi=\pm\sqrt{-\alpha/\lambda}$, while the value of the potential at the minima is given by $-\alpha^{2}/(4\lambda)$.

We see at once that now the statistical disadvantage of the rise of the potential at each side of the double well, which tend to squeeze the distribution, can be compensated by the statistical advantage due to the two local minima of the potential, which may tend to widen the distribution. Another way to put it is to say that the central bump of the potential tends to ``un-squeeze'' the distribution, working against the squeezing tendency of the potential rises at the two sides of the double well. If we tune our parameters in an appropriate way, it may be possible to end up with a vanishing squeeze in the limit, without the need for an infinitely wide well. In this way the possibility arises that we may have in this case $\alpha_{R}=0$ without it being necessary that $\alpha$ or $\lambda$ approach zero in the continuum limit. In other words, the possibility arises that there are certain non-zero pairs of values $(\alpha_{c},\lambda_{c})$ that we can approach in the continuum limit so that $\alpha_{R}\rightarrow 0$ in the limit, a behavior which, as we discussed before in the section in [5], is typical of second-order phase transitions. Refining a little our analysis we verify that indeed such a phase transition happens in this model, related to a process of spontaneous symmetry breaking.

As we saw, both the distance between the local minima and the width of each one of the two wells around them are proportional to $\sqrt{-\alpha/\lambda}$, which may be made as large as we wish by choosing $\alpha$ negative and with absolute value much larger than $\lambda$. In this way, by adjusting the parameters we can make the two potential wells much wider than the width of the local distribution, which is always of the order of $1$, thus making it no longer statistically favorable for the distribution to stay centered around $\varphi=0$. The depth of the two wells is given by $\alpha^{2}/(4\lambda)$ and also increases when we make the absolute value of $\alpha$ larger than $\lambda$, contributing to make it statistically favorable for the distribution to shift to one of the two sides, thus falling into one of the two wells. Since the two wells are identical, this happens in a random way, spontaneously to one of the two sides, which therefore spontaneously breaks the symmetry which so far implied that the expectation value of the field had to be zero, $\langle\varphi\rangle=0$. Note that the local distribution must fall to the same side at all sites, otherwise the derivative term would make a huge unfavorable contribution to the statistical weights. The situation of broken symmetry is illustrated qualitatively in figure 1.1.4.

Figure 1.1.5: Critical diagram of the non-linear $\lambda \varphi ^{4}$ model.

We discover in this way that a process of spontaneous symmetry breaking occurs in this model, giving origin to two phases in the space of parameters of the model, in each one of which the behavior of the model is of a certain type, different from the other one. The expectation value $v_{R}=\langle\varphi\rangle$ is the order parameter of the transition, being equal to zero in one of the phases, the symmetrical one, and different from zero in the other phase, the broken-symmetrical one. Since the parameter space is part of a two-dimensional plane, we expect that the two phases be separated by a one-dimensional curve. In fact, we can easily estimate the locus of this phase-transition curve. The argument that leads us to verify that the two phases exist depends crucially on the width of the potential, which is proportional to $\sqrt{-\alpha/\lambda}$. This quantity does not change so long as $\alpha$ and $\lambda$ are proportional to each other, $\alpha=-\beta\lambda$, so that the location of the points where the transition occurs, separating the two phases, should depend only on the proportionality constant $\beta$.

We can estimate this quantity assuming that at the transition the distance between the two wells is of the order of the width $\sigma_{0}$ of the local distribution, whose value is determined predominantly by the derivative term of the action. It is clear that, if the distance between the wells is significantly smaller than $\sigma_{0}$, the local distribution will tend to remain centered around $\varphi=0$, with its width somewhat reduced, while if the distance is significantly larger than $\sigma_{0}$, the local distribution will tend to shift sideways and fall into one of the wells. In order to better understand this argument it is useful to think about the extreme cases, the one in which the total width of the potential is much smaller than $\sigma_{0}$ and the local distribution is highly squeezed within it, and the one in which the total width of the potential is much larger than $\sigma_{0}$ and the local distribution is completely free to move within the potential and therefore to fall into one of the two wells. This estimate gives us the relation $\sigma_{0}\approx\sqrt{-\alpha_{c}/\lambda_{c}}$ for the critical values of the parameters or, more precisely, $C_{0}\sigma_{0}=\sqrt{-\alpha_{c}/\lambda_{c}}$, where $C_{0}$ is some positive constant of the order of $1$ that can only be obtained by a more complete calculation, and which means that $\beta_{c}=C_{0}^{2}\sigma_{0}^{2}$. We have therefore an equation determining the pairs of values $(\alpha_{c},\lambda_{c})$,

$$C_{0}^{2}\sigma_{0}^{2}\lambda_{c}+\alpha_{c}=0,$$

indicating that the locus of the phase transition is a critical line with a negative slope, which extends from the Gaussian point $(\alpha=0,\lambda=0)$ all the way to infinity, within the quadrant $(\alpha<0,\lambda>0)$ in the parameter plane of the model. Of course it is unlikely that the critical curve is exactly a straight line, because we did not take into consideration, in this qualitative argument, the changes in the depth of the wells due to the variation of the parameters, but we will see later that a straight line is in fact quite a reasonable approximation. Observe that the symmetrical phase occupies all the quadrant $(\alpha\geq 0,\lambda\geq 0)$ and part of the quadrant $(\alpha\leq 0,\lambda\geq 0)$, differing therefore from the classical expectation that making $\alpha <0$ would always break the symmetry. This is, of course, a direct consequence of the exchange of the classical point body by an extended quantum object within the potential well.

We may now draw a critical diagram for the model, illustrating in this way the two phases and the critical curve, like the one that can be seen in figure 1.1.5. The half-axis $(\lambda=0,\alpha<0)$ and the lower half-plane $\lambda<0$ are not included in the diagram, of course, since the model is unstable in these regions. The choice of two functions $\alpha(N)$ and $\lambda(N)$ that determines a particular continuum limit of the model corresponds to a path drawn in this diagram, which may start at any point within the stable region but which must necessarily end at some point of the critical curve, which is the locus where we have $\alpha_{R}(\alpha,\lambda)=0$ in the limit. These paths are called flows, or renormalization flows of the model. The Gaussian point is the critical point of the theory of the free field and the continuum limits of that model are represented by flows that go along the semi-axis $(\lambda=0,\alpha>0)$ in the direction of $\alpha=0$. We can see here, once again, that any limits staying within the quadrant $(\alpha\geq 0,\lambda\geq 0)$ must approach the Gaussian point. Flows can approach the same point of the critical line from either the symmetrical phase or the broken-symmetrical phase, possibly producing different results.

The slope of the critical curve at the Gaussian point is finite and non-zero and can be calculated by a perturbative approximation, as we shall see later in this chapter. The slope of the curve at the asymptotic region is also finite and non-zero, and can be related to critical points of other models of scalar fields, the so-called non-linear sigma models, as we shall also see later on. In addition to this, the qualitative properties of this curve can also be confirmed by means of another process of approximation that we will examine in detail later on, namely the so-called mean-field techniques. By and large the nature of the critical curve is rather well established and understood in any dimension $d\geq 3$, and the analysis can be extended without any important qualitative changes to the models $\lambda\varphi^{2p}$ for $p\geq 3$, as well as to multi-component models which are invariant under larger symmetry groups. In this last case the presence of more field components does introduce some new elements into the structure, of course. Usually more precise calculations of the position of the critical lines will involve some rather intensive and possibly difficult computer work.

Observe that there are many other flows that approach the Gaussian point, besides those defined directly by the free-field model. For example, we have a class of flows that go along the vertical half-axis $(\lambda>0,\alpha=0)$ in the direction of $\lambda=0$. This class of flows can produce trivial limits in which the resulting renormalized mass is determined by the dimensionless coupling parameter $\lambda$, instead of by $\alpha$. This phenomenon, which we can see here in a very simple way, is also known by the name of ``dimensional transmutation''. It is also possible to approach the Gaussian point from the broken-symmetrical phase, from under the critical curve. In this way we may define trivial limits in which the field, although free, has non-zero expectation values $V_{R}=\langle\phi\rangle$. Except in the case $d=3$ any limits that are candidates to not being trivial must approach some other point of the critical curve, and not the Gaussian point. If all possible flows of the model turn out to be trivial then we say that the model is trivial. Then, except for the introduction of the concept of spontaneous symmetry breaking, such models are just another way to produce the theory of the free scalar field in the continuum limit. They may still be useful test models on finite lattices, though.

Observe that what we have obtained here is a type of critical behavior just like the one described in the section in [5], where the quantity $v_{R}=\langle\varphi\rangle$ plays the role played by the magnetization in the case of statistical mechanics, as shown in the first figure of that section. However, we can do a little better here, and continue our heuristic argument in order to get an estimate for $v_{R}$ as a function of $\alpha$ and $\lambda$. First of all let us point out that we should be able to get only the absolute value of $v_{R}$ and not its sign, since the symmetry can break to either side. Let us therefore estimate $v_{R}^{2}$ and not $v_{R}$. It must be zero at the critical line, that is, when the squared width of the potential, $-\alpha/\lambda$, is equal to the squared width of the local distribution, $C_{0}^{2}\sigma_{0}^{2}$. It must also be equal to zero whenever the potential is less wide than that, that is,

$$v_{R}^{2}=0\mbox{    for    }-\frac{\alpha}{\lambda}\leq C_{0}^{2}\sigma_{0}^{2},$$

characterizing the symmetrical phase. To this we may add that, if the potential is wider than the width of the local distribution, then the distribution should be able to shift to one side by something like the difference between the two, giving as the corresponding estimate for $v_{R}^{2}$ the value

$$v_{R}^{2}=-\frac{\alpha}{\lambda}-C_{0}^{2}\sigma_{0}^{2} \m... ...  for    }-\frac{\alpha}{\lambda}\geq C_{0}^{2}\sigma_{0}^{2},$$

characterizing the broken-symmetrical phase. Note that this formula gives the correct value $v_{R}^{2}=0$ at the critical line. It also gives the correct value in the case of an extremely wide potential, in which case $C_{0}^{2}\sigma_{0}^{2}$, which is always of the order of $1$, can be neglected by comparison with $-\alpha/\lambda$, and we should have the average $v_{R}$ of the local distribution sitting at the minimum of the potential, that is, $v_{R}^{2}=-\alpha/\lambda$. In short, we have for $v_{R}^{2}$ the result

$$\lambda v_{R}^{2}=-(\alpha+C_{0}^{2}\lambda\sigma_{0}^{2}),$$

whenever the quantity in parenthesis is negative, and zero otherwise. Note that the quantity in parenthesis coincides with the equation of the critical line, thus showing that indeed $v_{R}^{2}$ is zero over that line. Note also that the result for $v_{R}^{2}$ contains $1/\lambda$, possibly indicating that a calculation which is purely perturbative in $\lambda$ may not be sufficient to obtain this result.

We may also obtain estimates for the dimensionless squared mass $\alpha _{R}$, using arguments similar to the ones above. Starting with the symmetrical phase, which contains the possibility that $\lambda=0$, with $\alpha>0$, we know two things about $\alpha _{R}$: first, it must be zero over the critical line, and second it must be equal to $\alpha$ when $\lambda=0$. Since the equation of the critical line contains a term linear in $\alpha$, it is clear that in order to satisfy both these criteria we must make $\alpha _{R}$ equal to that equation, thus obtaining

$$\alpha_{R}=\alpha+C_{0}^{2}\lambda\sigma_{0}^{2} \mbox{    for    }\alpha+C_{0}^{2}\lambda\sigma_{0}^{2}\geq 0,$$

where the condition is the same as before, characterizing the symmetrical phase. In the broken-symmetrical phase we must work a little more to get the result. First of all, let us discuss why there should be a non-zero physical mass in this case. This is so because, after the symmetry breaks and the local distribution falls within one of the two wells, it will become squeezed by it, leading to an increase in the renormalized mass, is a way similar to what happens in the free theory. So it follows that $\alpha _{R}$ should have its minimum of zero at the critical line and increase when one goes away from it on either side.

In order to estimate the value that $\alpha _{R}$ should have in the broken-symmetrical phase, let us go deeply into it, making $-\alpha\gg\lambda$, so that the potential is very wide and the local distribution is sitting around one of the two local minima. Under these conditions we may approximate the potential in the relatively small region where the local distribution is significantly different from zero by a parabola. If we calculate the second derivative of the potential at the minimum, which gives the curvature of this parabola, we get for it the value $-2\alpha$, which is positive because $\alpha$ is negative. By comparison with the situation in free theory we see now that deep in the broken-symmetrical phase we should have $\alpha_{R}=-2\alpha$. Adding to this that in this phase $\alpha _{R}$ still must be zero over the critical line, we see that we must make $\alpha _{R}$ proportional to the equation of the line, with a constant of proportionality that will bring about the correct value in the deeply broken regime. With all this it is not difficult to see that we must have

$$\alpha_{R}=-2(\alpha+C_{0}^{2}\lambda\sigma_{0}^{2}) \mbox{    for    }\alpha+C_{0}^{2}\lambda\sigma_{0}^{2}\leq 0,$$

where the condition is once again the same as before, characterizing the broken-symmetrical phase. Note that the value $-2\alpha$ for the curvature of the parabola near the minimum only goes to zero is we make $\alpha=0$ and hence go back to the Gaussian point. Therefore the potential never becomes infinitely flat around its local minima in the broken-symmetrical phase, except in limits that approach the Gaussian point. There are, therefore, no other possible continuum limits with finite masses in this phase, except those that tend to the Gaussian point or to some other point of the critical line.

As a last refining touch of our argument, we may point out its relation to the question of the triviality of the model. Let us look back at figure 1.1.3 and imagine that it represents the critical situation, in which the potential is just wide enough not to squeeze the local distribution. This means that the local distribution is almost free to move, but there is no space for it to actually do so. In this situation the squeezing action of the outer walls of the potential and the spreading action of the central bump are exactly balanced, so that the net action over the width of the local distribution vanishes. However, even without changing the width, the potential can tend to change the shape of the distribution. In fact, if we consider in which parts of the local distribution the central bump and the outer walls act, we realize that the bump tends to flatten and spread the top of the Gaussian, while the two outer walls tend to increase the slopes on the two sides of the distribution. This change of shape is exactly what one would expect if the local distribution tended to be more like the function $\exp(-\lambda_{R}\varphi^{4}/4)$ than like a Gaussian.

In fact, one would expect $\lambda_{R}$ to manifest itself by affecting the form of the complete local distribution. Since $\alpha _{R}$ is related to the second moment of the distribution, its width, it is reasonable to expect that $\lambda_{R}$ is correspondingly related to the fourth moment of the distribution, and hence to the shape of the curve that describes it. However, our experience with $\alpha _{R}$ shows that, due to the derivative term of the action, the local distribution has a very rigid character in the continuum limit, as a consequence of the requirements of propagation, which requires $\alpha _{R}$ to go to zero in the limit. There is therefore legitimate doubt that we can have this distribution significantly changed in shape in the continuum limit, to allow for a non-zero value of $\lambda_{R}$, without disturbing the dynamics of propagation and thus ending up with infinitely massive particles and no propagation. Note that if it is true that we must have $\lambda_{R}\rightarrow 0$ in the limit, then it is imperative that $\lambda_{R}$ be zero exactly over the critical line, where we already know that $\alpha _{R}$ is zero, otherwise there would be no possible continuum limits except those going to the Gaussian point.

Of course we cannot resolve this difficult matter with only heuristic arguments. In fact, we will see that perturbation theory is also not enough to handle this issue, and we will have to use computer simulations in order to explore it. However, we can say that, if indeed it turns out that we must have $\lambda_{R}\rightarrow 0$ in the limit, then only theories that can be non-trivial with $\lambda_{R}$ going to zero still have a chance of being truly interacting quantum theories. This involves the scaling relations between $\lambda_{R}$ and its dimensionfull version $\Lambda_{R}$. As is discussed in problem 1.1.8, in the classical case, this leaves, of all polynomial models $\varphi^{2p}$, $p=2,3,4,\ldots$, in all dimensions $d=3,4,5,\ldots$, a single possibility: the $\lambda \varphi ^{4}$ model in $d=3$.

We end this section with a short discussion of the continuum limit of the classical theory, which requires rewriting the action of the model in terms of dimensionfull quantities. If we recall our discussion about the physical significance of the block variables in the section in [6], we will see that it is the dimensionfull variables and parameters that have a more direct physical relevance in the quantum theory. While the dimensionless local variables and parameters, usually numbers of the order of $1$ that do not scale significantly in the continuum limit, are convenient both for establishing mathematical facts about the internal structure of the models and for dealing with them in a practical way in computer simulations, the dimensionfull variables include scale factors that cause them to scale in the continuum limit in the correct way in order to represent the superpositions of the dimensionless observables over the large and increasing numbers of sites contained within the blocks. As we discussed in that section, such superpositions constitute the only type of quantity within the theory that can in fact be directly observed.

The definition of the dimensionfull field in terms of the dimensionless field in the $\lambda \varphi ^{4}$ model is the same as in the theory of the free field, $\phi=a^{(2-d)/2}\varphi$, since it is determined only by the derivative term. We do not have in this case a mass term properly speaking, since $\alpha$ can be both positive and negative, but in a way similar to that of the theory of the free field we may introduce a parameter $m$ with dimensions of mass by means of the relation $m^{2}=\vert\alpha\vert/a^{2}$. A simple analysis of the quartic term gives us, finally, the definition of the dimensionfull coupling constant $\Lambda=a^{d-4}\lambda$. The treatment of the sums over links and sites and of the finite differences in the continuum limit of the classical theory, in terms of integrals and derivatives, is identical to the one discussed in the case of the free theory, so that we obtain for the action $S[\phi]$ in the continuum limit, in terms of the dimensionfull quantities,

$$S[\phi]=\int_{V}{\rm d}^{d}x\left\{ \frac{1}{2}\sum_{\mu}\le... ...\phi^{2}(\vec{x})+\frac{\Lambda}{4}\phi^{4}(\vec{x}) \right\},$$

where the sign of the quadratic term depends on the sign of $\alpha$. Observe that the relation existing between the parameters $\lambda$ and $\Lambda$ of the classical theory implies that, since $\Lambda$ must remain finite in the limit, $\lambda$ must behave in different and definite ways in each space-time dimension. For $d\geq 5$ it is necessary that $\lambda$ diverge to infinity in the limit in order that $\Lambda$ be different from zero, which shows that these classical theories have a rather singular behavior in this case. For $d\leq 3$, on the other hand, it is necessary that $\lambda\rightarrow 0$ in the limit in order for $\Lambda$ to remain finite, showing that in this case the behavior is the reverse of that of the previous case. For $d=4$ we have that $\Lambda=\lambda$ and therefore in this case it is not possible to make any definite statement of this type. Given these scaling relations between $\lambda$ and $\Lambda$ it is reasonable to think that the dimensionfull renormalized coupling constant $\Lambda_{R}$ should be defined in terms of $\lambda_{R}$ in an analogous way.

Our expectation is that, just as it is the constant $\Lambda$ that has physical relevance in the classical theory, the constant $\Lambda_{R}$ should play the same role in the quantum theory. As was already pointed out, the analysis of the block propagator of the free theory in the section in [6] indicated that it is the dimensionfull quantities, based on the dimensionfull field $\phi$, that have direct physical relevance, being directly related to the quantities which are observable in the quantum theory. Although we have only shown this fact in the case of the free theory, we will assume that it is true in general, a working hypothesis that we will only be able to confirm a posteriori by the accumulation of calculational experience, numerical or otherwise. We will see that this relation of scale between $\lambda_{R}$ and $\Lambda_{R}$ will be very useful to enable us to understand heuristically the behavior of the quantum models. Before anything else is done, however, it will be necessary to define in a more precise way the constant $\lambda_{R}$ of the quantum theory, which we will do in the third chapter of this book.

Problems

1. Derive the classical equation of motion for the $\lambda \varphi ^{4}$ model, showing that it is a non-linear equation. Write the equation both on finite lattices, in terms of the dimensionless field, and in the continuum limit, using the dimensionfull field.

2. Calculate the position, the depth and the bottom curvature of the potential wells in the models $\lambda\varphi^{2p}$, $p=3,4,\ldots$, which are defined by the action

   
   

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

   verifying that the spontaneous symmetry breaking situation is qualitatively similar to that of the $\lambda \varphi ^{4}$ model. Sketch qualitatively the phase diagrams of these models and estimate whether the critical curves in these cases are more or less steep than the critical curve of the quartic model.

3. Show that the stochastic variable $\chi$ introduced in the text for the $\lambda \varphi ^{4}$ model is bound within the interval $[0,\chi_{\rm max}]$, where $\chi_{\rm max}$ is a finite quantity which depends on $\alpha$ and $\lambda$.

4. Define a generalization of the stochastic variable $\chi$, which was introduced in the text for the case of the $\lambda \varphi ^{4}$ model, for the case of the $\lambda\varphi^{2p}$, $p=3,4,\ldots$ models.

5. Estimate and sketch the positions, on the phase diagram of the $\lambda \varphi ^{4}$ model, of the curves defined by $\alpha_{R}(\alpha,\lambda)=C$, where $C$ is a constant. Do it on both sides of the critical curve. Remember that $\alpha_{R}=0$ over the critical curve, that $\alpha_{R}=\alpha$ for the free theory, and that $\alpha_{R}\approx -2\alpha$ for negative $\alpha$ and $\lambda\approx 0$.

6. Write the action of the $\lambda \varphi ^{4}$ model in terms of the Fourier transforms $\widetilde\varphi$ of the fields and show that, due to the presence of the quartic term, the Fourier modes in momentum space do not decouple from each other when $\lambda\neq 0$.

7. Show that, in the case $d=1$, the model $\lambda \varphi ^{4}$ is identical to the quantum mechanics of an anharmonic oscillator.

8. Derive the scaling relations between $\lambda$ and $\Lambda$ for the classical theory of the $\lambda\varphi^{2p}$, $p=3,4,\ldots$ models, in dimensions $d$ from $3$ to $5$. Assuming only that $\lambda$ remains finite in the limit, verify in which cases it is possible to have non-trivial $N\rightarrow\infty$ limits, in which $\Lambda$ is finite and non-zero.