Discontinuity of the Configurations

In this section we will discuss one of the most basic properties of quantum field theory, which characterizes its inner workings in a very fundamental way. This is the fact that the field configurations in configuration space that contribute in a dominant way to the expectation values of the observables of the theory are discontinuous, in the continuum limit, as functions of the coordinates that span space-time. This property will have important consequences relating to many aspects of the inner workings of the theory, as we will discuss later, mostly in subsequent volumes of these notes.

Let us cite a few examples, so that the reader can judge the importance of the topic: the discontinuity of the field configurations is the basic phenomenon responsible for the appearance of divergent quantities in the perturbative approach to the theory; it causes different finite-differencing schemes on the lattice, which are equivalent in the classical theory, not to be so in the quantum theory; it changes in a qualitative way our conception of the role played in the theory by mathematical concepts of a topological nature; it leads us in an emphatic way to the idea that only the block variables to be discussed later can actually be physical observables within the theory; it strongly suggests new quantization procedures on the lattice, which are of a very geometrical nature, for theories that have a curved internal symmetry space, as is the case for the very important non-Abelian gauge theories. Hence, this is a central and unifying concept, related to many of the difficulties that appear in the theory, difficulties that usually cause great confusion, specially among those trying to learn it.

In order to examine this important concept we must start by discussing what we mean by continuity of the configurations, because we will be taking the continuum limit from finite lattices, which are discrete mathematical objects in which there is no natural concept of continuity. Usually functions $\varphi(\vec{x})$, mappings from $\mathbb{R}^{d}$ into $\mathbb{R}$, are considered continuous in the direction $\mu$ of the domain if the finite difference

$$\Delta_{\mu}\varphi(\vec{x})= \varphi(\vec{x}+\varepsilon\hat{x}_{\mu})-\varphi(\vec{x})$$

approaches zero when $\varepsilon\rightarrow 0$ both by positive and by negative values. However, in quantum field theories defined by means of Euclidean functional integrals only averages of functionals of the fields on some particular ensemble can be calculated and used to extract the physically relevant properties of the theories. It is clear that, since the fields at sites are random variables that undergo constant fluctuations including changes of sign, the direct expectation value of the difference $\langle\Delta_{\mu}\varphi(\vec{x})\rangle$ will be of no use to determine the character of continuity or discontinuity of the fields because, even if these differences never vanish, they can change sign, causing the average to vanish even for discontinuous fields. In fact, this average can be written as

$$\langle\Delta_{\mu}\varphi(\vec{x})\rangle= \langle\varphi(\... ...repsilon\hat{x}_{\mu})\rangle -\langle\varphi(\vec{x})\rangle,$$

so that its vanishing would only mean that the expectation value $\langle\varphi\rangle$ of the field is independent of position. It is clear that we need a quantity that vanishes only when the field is continuous, that is, of an observable that cannot change sign. In this situation we may use the quantity $\langle[\Delta_{\mu}\varphi(\vec{x})]^{2}\rangle$, which is a measure of the average “jump” and hence of the average discontinuity of the fields in the direction $\mu$, in order to define what we mean by continuity (problem 4.2.1). The fields will be considered typically continuous if this quantity vanishes when we make $\varepsilon$ tend to zero. To be more precise, the configurations of $\varphi(\vec{x})$ are predominantly continuous at $\vec{x}$, in terms of the measure of the action, if and only if

$$\lim_{\varepsilon\rightarrow 0} \langle[\Delta_{\mu}\varphi(\vec{x})]^{2}\rangle=0.$$

However, this definition still does now exhaust the issue, because there is more than one form to take this limit. In the formulation of quantum field theory on the lattice, $\varepsilon$ has to be some multiple of the fundamental lattice spacing $a$. The continuity of the fields may be examined by first taking the lattice spacing to zero, while $\varepsilon$ is kept constant in relation to the correlation length of the theory, and only after that making $\varepsilon$ tend to zero within the continuous space that results from the first limit. This might be the more natural way to take the limit in order to establish a notion of continuity for fields defined on a lattice, but it is not the most relevant way in the context of the definition of the theory in terms of functional integrals on the lattice.

A different notion of continuity appears in this context. The quantum theory is defined by the functional integral of the exponential of the action, and it is the behavior of the derivatives that this action contains which is of far more direct interest to us. When the models are defined on the lattice, the discrete representation of the action contains finite differences of the fields at close-neighboring sites. Therefore, in the continuum limits within this formalism the quantity $\varepsilon$ is is always kept equal to the lattice spacing $a$. Hence, in this case the limits $\varepsilon\rightarrow 0$ and $a\rightarrow 0$ are taken simultaneously, unlike what happened in the other type of limit. In what follows we will stick to this type of limit, which is the one with greater relevance for the definition of the quantum theory of fields by means of the lattice. However, one can show (problem 4.2.2) that the situation does not change qualitatively when one exchanges one type of limit by the other.

As always, we will use the theory of the free scalar field as an example. In order to have a definite case to examine, we will also adopt periodical boundary conditions, but this does not have any fundamental importance for the results. The calculations we must do are not, in fact, very complex. It suffices to use the result obtained before for the propagator of this model in momentum space, which can be written as

$$\langle\widetilde\varphi (\vec{k})\widetilde\varphi (\vec{k}... ...{d}(\vec{k},-\vec{k}')} {N^{d}[\rho^{2}(\vec{k})+\alpha_{0}]}.$$

From this result it is simple to show that $\langle[\Delta_{\mu}\varphi(\vec{n})]^{2}\rangle_{N}$ tends to a finite and non-vanishing value when $a\rightarrow 0$, that is, when $N\rightarrow \infty$. Using the Fourier transforms we may write

$$\begin{eqnarray*}[\Delta_{\mu}\varphi(\vec{n})]^{2} & = & \left[\sum_{\vec{k}}\w... ...\pi}{N}k'_{\mu}} \;e^{\imath\frac{2\pi}{N}\vec{k}'\cdot\vec{n}}, \end{eqnarray*}$$

where we used the fact that the complex exponentials are eigenvectors of the finite-difference operator, as discussed in section 2.7. We also used the quantities $\rho_{\mu}(k_{\mu})$ defined in equation (2.7.1). Taking now the expectation value on an $N$-lattice we obtain

$$\begin{eqnarray*} \langle[\Delta_{\mu}\varphi(\vec{n})]^{2}\rangle_{N} & = & -\s... ...}}\frac{\rho^{2}_{\mu}(k_{\mu})} {\rho^{2}(\vec{k})+\alpha_{0}}, \end{eqnarray*}$$

where we substituted the result for the propagator and used the delta function to eliminate one of the sums over the momenta. Note that there is no sum over $\mu$ in this expression. Since both the lattice and this observable are symmetrical by permutations of the various directions $\mu$, we may write this result in terms of $\rho^{2}(\vec{k})=\sum_{\mu}\rho^{2}_{\mu}(k_{\mu})$ as

$$\langle[\Delta_{\mu}\varphi(\vec{n})]^{2}\rangle_{N}= \frac{... ...ec{k}}\frac{\rho^{2}(\vec{k})} {\rho^{2}(\vec{k})+\alpha_{0}}.$$

It is not difficult to find upper and lower bounds to the sum that appears in this expression, which will hold for any value of $N$. In order to find an upper bound it suffices to take off the positive constant $\alpha_{0}$ that appear in the denominator. It is interesting to note that, since $\alpha_{0}\rightarrow 0$ in the continuum limit, it is reasonable to think that this change does not in fact affect the result in the limit. In order to find a lower bound it suffices to exchange the $\rho^{2}(\vec{k})$ that appears in the denominator by its maximum value, which is $4d$. With this we obtain the relations

$$\begin{array}{*{6}{c}} \displaystyle \frac{1}{dN^{d}}\sum_{\... ...\leq & \displaystyle \frac{1}{d} & \displaystyle . \end{array}$$

The sum that remains in the case of the lower bound can be calculated (problem 4.2.3) by the decomposition of the sines in terms of complex exponentials, followed by the use of the orthogonality and completeness relations. The result is simply the number $2N$, so that we have the final result for our sum,

$$\frac{2}{4d+\alpha_{0}}<\frac{1}{dN^{d}}\sum_{\vec{k}} \frac{\rho^{2}(\vec{k})}{\rho^{2}(\vec{k})+\alpha_{0}}\leq\frac{1}{d}.$$

In the limit $N\rightarrow \infty$ we have that $\alpha_{0}\rightarrow 0$ and therefore we can write, recalling that there is no sum over $\mu$,

$$\frac{1}{2d}<\langle[\Delta_{\mu}\varphi(\vec{n})]^{2}\rangle \leq\frac{1}{d}.$$

This result shows that, for any finite dimension of space-time, on average over the ensemble of configurations, the variation of the field from one site to the next one does not approach zero when the lattice spacing goes to zero. It follows that, on average, in the continuum limit, the configurations of the dimensionless field are not continuous as functions of position. Observe that this is a very violent type of discontinuity, because we are not talking here of a set of isolated discontinuities in a continuous functions. The field is discontinuous in all directions and at all the points where it is defined. Obtaining an exact result for $\langle[\Delta_{\mu}\varphi(\vec{n})]^{2}\rangle$ requires more work. One can show (problem 4.2.4) that in the continuum limit the sum in fact assumes its upper bound, so that we have

$$\langle[\Delta_{\mu}\varphi(\vec{n})]^{2}\rangle=\frac{1}{d}.$$

Having established this important result, we may now examine some of its immediate consequences. If we recall that the dimensionfull field $\phi$ is related to the dimensionless field by $\phi=a^{(2-d)/2}\varphi$, we can determine how the derivatives of the dimensionfull fields behave in the continuum limit. For $a$ small but non-zero we have

$$\langle[\Delta_{\mu}\phi(\vec{n})]^{2}\rangle\sim\frac{a^{2-d}}{d}.$$

The meaning of this relation depends on the dimension $d$. For $d=1$ we have that, in the continuum limit,

$$\langle[\Delta_{\mu}\phi]^{2}\rangle\sim a\rightarrow 0,$$

while for the derivative $\partial_{\mu}\phi=\Delta_{\mu}\phi/a$ itself we have

$$\langle[\partial_{\mu}\phi]^{2}\rangle\sim\frac{1}{a}\rightarrow\infty,$$

which shows that in this case the dimensionfull field is continuous but not differentiable. One can show (problem 3.1.2) without difficulty that the quantum theory of the free scalar field in one dimension is formally identical to the quantum mechanics of the harmonic oscillator, by mapping the quantities that appear in one of these two structures onto corresponding quantities of the other. This result for $d=1$ reproduces, therefore, the well-known quantum-mechanical fact that the one-dimensional configurations involved in the functional integration are in that case random walks, continuous but non-differentiable paths, as those of a Brownian motion. In this case the denomination of the functional integral as a path integral is justified, but this is a characteristic exclusively of the one-dimensional case. Already for $d=2$ we have, for both the dimensionless field and the dimensionfull field, since they are in fact equal in this case,

$$\langle[\Delta_{\mu}\phi]^{2}\rangle\rightarrow\frac{1}{2},$$

showing that in this case the discontinuities exist but are finite. In dimensions $d=3$ and larger the discontinuities of the dimensionfull field diverge with powers that increase with the dimension,

$$\langle[\Delta_{\mu}\phi]^{2}\rangle\sim \frac{1}{a^{d-2}d}\rightarrow\infty.$$

We see here two very important basic facts: first, that there is a qualitative difference between the behavior of quantum mechanics (the case $d=1$) and the behavior of the quantum theory of fields (the cases $d>1$); second, confirming what was already discussed in section 4.1, the fundamental dimensionfull fields display extreme fluctuations in the continuum limit, which puts immediately in great doubt any possibility that they may have any direct physical significance as observables. We will now explore a consequence of these facts that is directly related to the action. Since the action $S[\varphi]$ is itself a functional of the fields, we may consider the calculation of its expectation value $\langle S\rangle$. The result may help us to understand the behavior of the theory from another point of view. We know that the classical solution of the theory is the one that minimizes the action, whose minimum in the case of the action $S_{0}$ of the free scalar field is zero. This is the value that maximizes the relative statistical weight $\exp(-S)$ of the configuration within the ensemble. The average value of $S$ will tell us something about the typical relative weights of the configurations that contribute in a dominant way to the averages. For the time being, we will limit ourselves to the calculation of the expectation value of the kinetic part $S_{K}$ of the action $S_{0}$ of the free theory, the part that involves the derivatives,

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

which also appears in any model involving scalar fields. The expectation value of $S_{K}$ is given by

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

where we already know that the expectation value of the square of the finite difference converges to $1/d$ in the continuum limit. Since this limit does not depend on the link at which it is being calculated and there are $dN^{d}$ links in the lattice, we obtain immediately that

$$\langle S_{K}\rangle\sim\frac{1}{2}N^{d}\rightarrow\infty,$$

that is, the expectation value of the action diverges as $N^{d}$. Note that the action is dimensionless and that this divergence has nothing to do with an increase in the volume of the box within which we are defining the model, the action diverges even if we are within a finite box. This is a property related to the behavior of the high-frequency modes of the theory, which we denominate the ultraviolet regime, which has nothing to do with the infrared aspects connected to the volume of the box.

We may also examine the behavior of the mass term $S_{M}$ of the action $S_{0}$, the one that contains the parameter $\alpha_{0}$,

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

in a way analogous to the analysis of $S_{K}$. This is another term that will also appear in other models involving scalar fields, usually as part of a potential involving the fields, which establishes non-linear relations among them. The expectation value of this part of the action may be written as

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

As we saw in section 4.1, the expectation value $\sigma^{2}=\langle\varphi^{2}\rangle$ has a finite and non-vanishing limit for $d\geq 3$, diverging for $d=1$ and $d=2$. Besides, $\sigma ^{2}$ does not depend on position, and it therefore follow that

$$\langle S_{M}\rangle=\frac{\alpha_{0}}{2}N^{d}\sigma^{2}.$$

Since $\alpha_{0}=(m_{0}L)^{2}/N^{2}$, we have that for $d\geq 3$ this part of the action also diverges, although in a somewhat slower way than the kinetic part,

$$\langle S_{M}\rangle =\frac{(m_{0}L)^{2}}{2}N^{d-2}\sigma^{2}\rightarrow\infty.$$

For $d=2$ the divergence becomes logarithmic, due to the fact that in this case $\sigma^{2}\sim\ln(N)$, while in the case $d=1$ there is no divergence, because in this case $\sigma ^{2}$ diverges with $N$ and cancels the factor of $N^{-1}$, resulting on a finite action. Once more we see that the case $d=1$ is significantly different from the others. In the cases of greater interest for us, $d\geq 3$, both $\langle S_{K}\rangle$ and $\langle S_{M}\rangle$ diverge in the limit, this second term in a somewhat less violent way than the first. We see therefore that the dynamics defined by $S_{0}$ tends to be dominated by the term $S_{K}$ containing the derivatives.

The most important thing is that we arrive at the inevitable conclusion that $S_{0}$ typically diverges and therefore that the statistical weights $\exp(-S_{0})$ typically go to zero very fast in the continuum limit. It is necessary to emphasize that this is the minimum possible value for these weights, very different from the maximum possible value that it assumes for the classical solution, which shows that typically the dominant configurations of the quantum theory are very distant from the classical solution, if we use the action as a measure of this distance. This fact may seem surprising at first sight, but this surprise is due only to the fact that, when we examine these relative statistical weights, we are leaving aside another very important factor, the number of configurations with each value of the weight that exist in configuration space and that, therefore, contribute to the averages. Although the discontinuous configurations have, individually, negligible statistical weights as compared to the weights of continuous configurations such as the classical solution, their number is immensely larger, so that they end up dominating the situation completely. It becomes quite clear that the quantum-mechanical concept of the semi-classical approximation will have to be revised before one considers its application in quantum field theory. In future volumes we will also see that divergences like these are intimately connected to the divergences that appear pervasively in perturbation theory.

We might ask ourselves if this behavior could be just a peculiarity of the theory of the free scalar field. In future volumes we will see that it also applies to other free fields such as, for example, the free vector field of electromagnetism without sources, that is, without charges and currents. For the case of free electrodynamics this can be found in reference [1]. In the case of non-linear models, we do not know how to do a direct analytical verification, being in this case limited to extrapolations from computer simulations on finite lattices. We may, however, argue as follows to the effect that this is a property that must also hold in these models. The non-linear models in general contain a real parameter $\lambda$, the coupling constant, which is such that they converge to the free theory when we make $\lambda$ go to zero. In fact, it is important for the physical interpretation of these models that they have the free theory as a smooth limit for $\lambda\rightarrow 0$. The property of discontinuity of the fields that we examined in this section depends only on the facts that $\langle(\Delta_{\ell}\varphi)^{2}\rangle$ and $\langle\varphi^{2}(s)\rangle$ have non-vanishing finite values in the limit. It is very reasonable to think, then, that they will still be finite and non-vanishing for $\lambda\neq 0$, although their values might vary with $\lambda$, so that the $\lambda\rightarrow 0$ limits of the expectation values of these observables may be smooth ones. In fact, despite the strong fluctuations and discontinuities of the fields, usually the expectation values are smooth or at least continuous functions of the parameters of the models.

Hence we see that the main facts described here are, in all probability, general properties of all quantum field theories. As we progress in our exploration of the subject, we will continue to verify and solidify this important notion.

Problems

1. Show, in the classical theory of fields, that the criterion that the quantity

   
   

   $$[\Delta_{\mu}\varphi(\vec{x})]^{2} =[\varphi(\vec{x}+\varepsilon\hat{x}_{\mu})-\varphi(\vec{x})]^{2}$$
   
   

   goes to zero when $\varepsilon\rightarrow 0$, by either positive or negative values, is equivalent to the condition that $\Delta_{\mu}\varphi(\vec{x})\rightarrow 0$.

2. Calculate, in the case of the quantum theory of the free scalar field, the limit

   
   

   $$\lim_{\varepsilon\rightarrow 0}\left[\lim_{N\rightarrow\inft... ...\hat{x}_{\mu}) -\varphi(\vec{x})]^{2}\right\rangle_{N}\right],$$
   
   

   in the indicated order, at an arbitrary point $\vec{x}$ and for an arbitrary direction $\mu$, showing that it is finite and not zero. In order to do this expand the square and use the values of $\sigma_{0}^{2}=g(\vec{0})$ and $g(\vec{x}-\vec{x}')$ that were calculated in section 4.1. In this way one shows that the fundamental facts associated to the discontinuity of the fields do not depend on the way in which the limit is defined.

3. Calculate the $d$-dimensional sum in momentum space

   
   

   $$\sum_{k_{\mu}}\rho^{2}_{\mu}(k_{\mu}),\mbox{~~where~~} \rho^{2}_{\mu}(k_{\mu})=4\sin^{2}(\pi k_{\mu}/N),$$
   
   

   decomposing the sine into complex exponentials and using the orthogonality and completeness relations that hold for them.

4. ($\star$) Find out a more restrictive lower bound for the sum that appears in the calculation of the square of the finite difference of the fields,

   
   

   $$\frac{1}{dN^{d}}\sum_{\vec{k}} \frac{\rho^{2}(\vec{k})}{\rho^{2}(\vec{k})+\alpha_{0}},$$
   
   

   that has the property that its limit is equal to $1/d$ in the continuum limit, hence demonstrating rigorously that

   
   

   $$\langle[\Delta_{\mu}\varphi(\vec{n})]^{2}\rangle\rightarrow\frac{1}{d}$$
   
   

   in the continuum limit^4.1. Use for the momentum components the interval of values $[0,N-1]$ and the fact that

   
   

   $$\rho^{2}(k_{1},k_{2},\ldots,k_{d})=\rho^{2}(N-k_{1},k_{2},\ldots,k_{d})= \rho^{2}(N-k_{1},N-k_{2},\ldots,k_{d}),\mbox{ etc.}$$
   
   

   The idea is to take off the sum classes of terms that end up not contributing in the limit, in particular the terms with $k_{\mu}<\sqrt{N}$, in addition to decreasing the sum in other ways that simplify it, until one obtains a sum that can be calculated in the limit and that tends to $1/d$.

5. Consider the quantum theory of the free scalar field, whose action is $S_{0}$. Consider a change of variables from the $N^{d}$ field variables $\varphi(s)$ to another set of $N^{d}$ variables, one of which is the action $S_{0}$ itself, which varies from $0$ to $\infty$.
   1. Using the fact that the action is quadratic on the fields, and hence that it is homogeneous of degree $2$ on the fields, show that the distribution of the theory can be written as

      
      

      $$\int_{\Omega}[{\bf d}\varphi]\;e^{-S_{0}} =Z_{0}\;S_{0}^{\frac{N^{d}}{2}-1}\;e^{-S_{0}}\;{\rm d}S_{0},$$
      
      

      where $Z_{0}$ is a constant and we integrate over the manifold $\Omega$, a kind of solid angle in configuration space, described by all the variables except $S_{0}$, which is a kind of radial variable.

   2. Defining the action per site by $s_{0}=S_{0}/N^{d}$ and using the distribution in terms of $S_{0}$ given above, show that we have for its average value $\langle s_{0}\rangle=1/2$, which implies that $\langle S_{0}\rangle=N^{d}/2$ diverges, as was seen in the text. Note that the integrals that appear can be written as $\Gamma$ functions.

   3. Calculate also the width of the distribution of values of $s_{0}$, showing that $\langle s_{0}^{2}\rangle-\langle s_{0}\rangle^{2}=1/(2N^{d})$. This quantity goes to zero in the continuum limit and this fact shows that the distribution of values of $s_{0}$ becomes a delta function centered at the value $1/2$, in that limit. This is another way to see that the continuous configurations, for which $s_{0}=0$, do not contribute to the expectation values of the theory in the continuum limit, being therefore of zero measure.