Structure of the Two-Point Function

As we saw in section 3.4, the complete solution of the theory of the free scalar field is reducible to the calculation of the propagator, with the consequence that all the physics of the theory is contained in the structure of this propagator. We will examine here, in more detail, the properties of this function, which describes how objects propagate in this model. Since propagation is in itself an important physical phenomenon, it is worth wile to focus for some time on a more detailed analysis of the structure of the propagator.

In momentum space this structure is very simple, it is just a function of the momenta that decays quadratically for large momenta. In the context of the acquisition of a better understanding of the inner workings of the quantum model, the exploration that is of most interest to us is that of the propagator in position space. The dimensionless propagator in position space,

$$g(\vec{x}_{1},\vec{x}_{2})= \langle\varphi(\vec{x}_{1})\varphi(\vec{x}_{2})\rangle,$$

is a function only of $\vec{x}_{1}-\vec{x}_{2}$ and may be written as

$$\begin{eqnarray*} g(\vec{x}_{1}-\vec{x}_{2}) & = & \sum_{\vec{k}}e^{-\imath\frac... ...vec{x}_{1}-\vec{x}_{2})}} {N^{d}[\rho^{2}(\vec{k})+\alpha_{0}]}, \end{eqnarray*}$$

as we saw in the derivation of equation (3.4.3). Our initial objective relative to this function is to calculate its value for $\vec{x}_{1}=\vec{x}_{2}=\vec{x}$, in which case we have the quantity

$$\sigma^{2}=g(\vec{x}-\vec{x})=g(\vec{0})= \langle\varphi^{2}(\vec{x})\rangle,$$

a quantity that, by discrete translation invariance on the torus, is independent of position. Since we have that $\langle\varphi(\vec{x})\rangle=0$, it follows that $\sigma ^{2}$ is the square of the width of the distribution of values of the dimensionless field $\varphi$ at a give site, that is, $\sigma=\sqrt{\sigma^{2}}$ is the average size of the fluctuations of the field around its average value of zero. We will refer to $\sigma$ as the local width of the distribution of the fields. We may write for this quantity

  $$\sigma^{2}=\frac{1}{N^{d}}\sum_{\vec{k}} \frac{1}{\rho^{2}(\vec{k})+\alpha_{0}}.$$ (4.1.1)

With the objective of determining whether or not this quantity has a finite limit in the continuum limit, we start by approximating it by an integral. Note that this is a mere approximation, which allows us to acquire a qualitative idea of the behavior of this quantity, and that the expression above for $\sigma ^{2}$ does not converge to an integral in the continuum limit, since we are taking this limit within a finite box, where the Fourier modes and their momenta are always discrete. Since the smallest non-vanishing value for the momentum components inside the box is $2\pi/L$, the “element of volume” in momentum space is given by

$${\rm d}^{d}p=\left(\frac{2\pi}{L}\right)^{d},\mbox{~~that is,~~} \left(\frac{L}{2\pi}\right)^{d}{\rm d}^{d}p=1.$$

Substituting this “$1$” in equation (4.1.1) and approximating the discrete lattice quantity $\rho^{2}(\vec{k})+\alpha_{0}$ by $a^{2}[p^{2}(\vec{k})+m^{2}_{0}]=L^{2}[p^{2}+m^{2}_{0}]/N^{2}$ we obtain

$$\sigma^{2}\sim\frac{1}{N^{d}}\sum_{\vec{k}}{\rm d}^{d}p \lef... ...2\pi}\right)^{d} \frac{N^{2}}{L^{2}}\frac{1}{p^{2}+m^{2}_{0}},$$

so that we may now approximate $\sigma ^{2}$ by the integral

$$\sigma^{2}\sim\frac{1}{(2\pi)^{d}}\left(\frac{L}{N}\right)^{d-2}\int{\rm d}^{d}p\;\frac{1}{p^{2}+m^{2}_{0}}.$$

We must now discuss how to determine the extremes of integration. Note that for $d\geq 3$, due to the factor of $N^{d-2}$ in the denominator, the result can only be non-vanishing in the $N\rightarrow \infty$ limit if the integral diverges in that limit. For $p$ near zero the integrand is finite, so long as $m_{0}$ is not zero, so that the integral cannot diverge at this extreme. For simplicity, let us assume temporarily that $m_{0}\neq 0$, postponing until later the discussion of the case $m_{0}=0$. It follows from these considerations that for $d\geq 3$ the only part of the domain of integration that matters is the one for large absolute values $p$ of the momenta. In any case, we can write the integral in spherical coordinates in momentum space, obtaining

$$\sigma^{2}\sim \frac{1}{(2\pi)^{d}} \left(\frac{L}{N}\right)... ...-1}\int_{0}^{N\pi/L}{\rm d}p\;\frac{p^{d-1}}{p^{2}+m^{2}_{0}},$$

where $\Omega_{d-1}$ is the complete solid angle in $d$ dimensions and $N\pi/L$ is the largest possible value for the components of the momentum on a lattice with $N$ sites in each direction. Since we are not interested in the lower limit of the integral, we may neglect the $m_{0}$ that appears in the denominator and write the integral as

$$\sigma^{2}\sim \frac{1}{(2\pi)^{d}} \left(\frac{L}{N}\right)^{d-2} \Omega_{d-1} \int_{p_{m}}^{N\pi/L}{\rm d}p\;p^{d-3},$$

where $p_{m}$ is some small and finite but non-vanishing value of the momentum, which we could choose to be $m_{0}$ or $2\pi/L$. Recalling that we are discussing all this mostly in the context of the case $d\geq 3$, we see now that for dimensions $d\leq 2$ the factor of $p$ appear in the denominator, so that indeed we will have to examine the cases $d=1$ and $d=2$ separately. We may now do the integration for the case $d\geq 3$, obtaining

$$\begin{eqnarray*} \sigma^{2} & \sim & \left. \frac{1}{(2\pi)^{d}} \left(\frac{L}... ...2} \Omega_{d-1} \frac{1}{d-2} \left(\frac{N\pi}{L}\right)^{d-2}, \end{eqnarray*}$$

Figure 4.1.1: Behavior of the squared local width $\sigma ^{2}$ with $N$ in the case $d=1$.

Figure 4.1.2: Behavior of the squared local width $\sigma ^{2}$ with $N$ in the case $d=2$.

Figure 4.1.3: Behavior of the squared local width $\sigma ^{2}$ with $N$ in the case $d=3$.

Figure 4.1.4: Behavior of the squared local width $\sigma ^{2}$ with $N$ in the case $d=4$.

Figure 4.1.5: Behavior of the squared local width $\sigma ^{2}$ with $N$ in the case $d=5$.

were we neglected the contribution from the lower extreme of the integral, which vanishes in the limit. We see that the factors of $N$ cancel out and that the final result for $\sigma ^{2}$ in the limit $N\rightarrow \infty$ is

$$\sigma^{2}\sim\frac{\Omega_{d-1}}{2^{d}\pi^{2}(d-2)},$$

which is finite, for in our case here $d\geq 3$ and $\Omega_{d-1}$ is always a finite number different from zero, since it is the volume of a compact manifold, the surface of the $d$-dimensional unit sphere. For the case $d=2$ the exponent of $N$ vanishes, so that the factor in front of the integral neither diverges nor vanishes. It follows that in this case both the upper and the lower extremes of integration are in principle important. In this case we have $\Omega_{1}=2\pi$ and the integral results in

$$\sigma^{2}\sim\frac{1}{2\pi}\int_{p_{m}}^{N\pi/L}\frac{{\rm d}p}{p}\sim \frac{1}{2\pi}\ln\left(\frac{N\pi}{Lp_{m}}\right),$$

so that the upper extreme still dominates and in this case $\sigma ^{2}$ diverges logarithmically with $N$, in fact a type of behavior that is very common in the case $d=2$. In the case $d=1$ the factor in front of the integral diverges as $N$, so that in this case we have the opposite of what happens in the other cases, and only the lower extreme of the integral contributes. In this case we have that $\Omega_{0}=2$ and the integral may be written as

$$\sigma^{2}\sim\frac{N}{\pi L}\int_{p_{m}}^{N\pi/L}\frac{{\rm d}p}{p^{2}}\sim\frac{N}{\pi Lp_{m}},$$

so that in this case $\sigma ^{2}$ diverges linearly with $N$. Due to the fact that in these cases the lower extreme contributes significantly to the integral, these are the only cases in which the results depend on $p_{m}$, that is, on the details of the choice of the lower integration limit. Since this limit was introduced only to allow us to eliminate $m_{0}$ from the integrand and hence facilitate the realization of the integral, this means, in fact, that in these cases the results depend on $m_{0}$. In fact, it is possible to improve the analytical calculation in the case $d=1$ (problem 4.1.1), so as to verify this is an explicit way.

One can perform more careful calculations than these to evaluate the sums that appear in the formulas for $\sigma ^{2}$, building integrals that are strict lower bounds and strict upper bounds to the sums (problem 4.1.4), so as to prove that $\sigma ^{2}$ in fact behaves as a function of $N$, for large $N$, in the way given here. In any case, it must be emphasized that these are all just approximations that allow us to determine no more than the type of asymptotic dependency of $\sigma ^{2}$ with $N$. This is still true even if we make the volume $V=L^{d}$ of the box go to infinity, so that the volume element $(2\pi/L)^{d}$ in momentum space goes to zero. The reason is that the quantity $\rho^{2}(\vec{k})/a^{2}$ that appears in the integrand only approaches $p^{2}$ if $N\rightarrow \infty$ with $k_{\mu}$ kept finite, while the sum over the momenta will always include terms in which $k_{\mu}\sim\pm N/2$. Hence, for terms close to the upper limit of the sum it is not true that $(\rho/a)^{2}$ approaches $p^{2}$. Note that for $d\geq 3$, since it is necessary that the integrals diverge in order for $\sigma ^{2}$ not to vanish, the main contribution to the final result comes precisely from such terms. The same is true for $d=2$, while for $d=1$ the situation is reversed, and the main contribution comes from the lower extreme of the integral, so that in this case it is possible that the result of the approximation by the integral in fact becomes exact when we go to infinite space, if we do not make any further approximations.

For a precise calculation of the values of $\sigma ^{2}$ in each dimension it is necessary to write programs to performs the sums on finite lattices with various sizes and then to extrapolate the results to the case $N\rightarrow \infty$. The graphs that can be found in the figures numbered from 4.1.1 to 4.1.5 show the values of $\sigma ^{2}$ for sequences of finite lattices in dimensions $d$ from $1$ to $5$, for the case $m_{0}=1$, obtained by the use of such programs. For $d=1$ one can see clearly the linear divergence with $N$. In the case $d=2$ the logarithmic divergence is also quite clear and it is not difficult to make sure of its nature by simply plotting the graph on an adequate logarithmic scale (problem 4.1.5). Starting with the $d=3$ case the behavior changes radically, the function $\sigma^{2}(N)$ becomes a decreasing rather than increasing function of $N$, approaching a plateau at a finite and non-vanishing value. The flatness of this plateau becomes clearer as the dimension increases, at the same time that its value decreases. In addition to this, as the dimension increases the value of the plateau is approached ever faster, for lattices which are ever smaller in their linear dimensions. Extrapolating these results (problem 4.1.6) to the limit $N\rightarrow \infty$ we obtain for $\sigma$ the final results shown in table 4.1.1.

Table 4.1.1: Results for the local width $\sigma$ for large values of $N$.

$d$	$\sigma(N\rightarrow\infty)$
$1$	$\simeq \sqrt{1/12+1/(m_{0}L)^{2}}\sqrt{N}$
$2$	$\simeq 0.4095\sqrt{\ln\left(N\right)}$
$3$	$\simeq 0.5027$
$4$	$\simeq 0.3936$
$5$	$\simeq 0.3400$

In all this analysis we see, in a very clear way, that the cases $d=1$ and $d=2$ are very special. For $d\geq 3$ we have finite fluctuations of the values of $\varphi$ at the sites, while for $d=1$ and $d=2$ these fluctuations diverge. Observe that in all cases $\sigma ^{2}$ is the maximum value of $g$, because for $\vec{x}_{1}\neq\vec{x}_{2}$ the terms of the sum that defines $g$ are multiplied by numbers with absolute values smaller than $1$. This is consistent with the graphs of these functions that we saw before in section 3.5, which decay when $\vec{x}_{1}$ moves away form $\vec{x}_{2}$.

We will now examine the behavior of the dimensionfull versions of $g$ and $\sigma ^{2}$. Note that we can define a quantity $\Sigma^{2}$ for the dimensionfull field in a way analogous to the definition of $\sigma ^{2}$. Given the scaling relations for the fields, we immediately have that $\Sigma^{2}=a^{2-d}\sigma^{2}$, so that we may immediately deduce from table 4.1.1 the behavior of $\Sigma^{2}$. In $d=1$ this quantity has a finite value proportional to $L$ and for $d=2$ it is equal to $\sigma ^{2}$, and therefore it diverges in the same way, logarithmically. However, for $d\geq 3$ it diverges with some power of $N$, from which it follows that the dimensionfull field undergoes fluctuations of infinite magnitude in the continuum limit. This is the first sign indicating the extremely singular character of the behavior of the theory, and possibly the fact that the fundamental fields are not variables amenable to a direct physical interpretation.

Let us now continue our analysis by the examination of the behavior of the propagators in position space in the case in which $\vec{x}_{1}$ and $\vec{x}_{2}$ are two distinct points. Still from our scaling relations for the fields we have that those of the propagator should be $G=a^{2-d}g$, where $a=L/N$, so that we may write for $G$

$$G(\vec{x}_{1}-\vec{x}_{2})=\frac{N^{d-2}}{L^{d-2}N^{d}}\sum_... ...dot(\vec{x}_{1}-\vec{x}_{2})}} {\rho^{2}(\vec{k})+\alpha_{0}}.$$

This time our objective is to show that the function $G$ is finite in the continuum limit, so long as $\vec{x}_{1}\neq\vec{x}_{2}$. We will once more approximate the sum by an integral,

$$G(\vec{x}_{1}-\vec{x}_{2}) \sim \frac{1}{L^{d-2}N^{2}}\sum_{\vec{k}} \left(\frac{L}{2\pi}\right)^... ...\imath\vec{p}\cdot(\vec{x}_{1}-\vec{x}_{2})}} {a^{2}[p^{2}(\vec{k})+m_{0}^{2}]}$$

$$\sim \frac{1}{(2\pi)^{d}}\int{\rm d}^{d}p\; \frac{e^{-\imath\vec{p}\cdot(\vec{x}_{1}-\vec{x}_{2})}} {p^{2}+m_{0}^{2}}.$$ (4.1.2)

Observe that this time there are no divergent terms in front of the integrals. Once more, we must discuss the extremes of integration. Our intention here is to eventually make $L\rightarrow\infty$, going in this way from the finite box to infinite space, where $G$ has a simpler form. For the time being, however, we are still doing the integration in the context of a finite position-space volume. For simplicity, we will approximate the momentum-space integral doing it over a spherical domain whose radius is the largest possible value of a momentum component on a $d$-dimensional lattice with $N^{d}$ sites. Under these conditions, if we define $\vec{r}=\vec{x}_{1}-\vec{x}_{2}$, we see that $G$ depends only on the modulus $r$ of the vector $\vec{r}$ because, if we make an arbitrary change in the angles of the versor $\hat{r}$, we can make the integral over the momenta return to its previous form doing a corresponding rotation of the integration variables. Hence we can put $\vec{r}$ in the direction of the $d^{\rm th}$ component of $\vec{p}$ and write, for dimensions $d>2$, without loss of generality,

  $$G(r)\sim\frac{1}{(2\pi)^{d}} \int_{\Omega_{d-1}}{\rm d}^{d-1}\Ome... ...}{\rm d}p\;p^{d-1}\;\frac{e^{-\imath pr\cos(\theta_{d-2})}} {p^{2}+m_{0}^{2}},$$ (4.1.3)

where $\theta_{d-2}$ is the angle between the vector $\vec{p}$ and its $d^{\rm th}$ component and the angular integration is over the solid angle $\Omega_{d-1}$ of the $d$-dimensional space, with integration element given by

$${\rm d}^{d-1}\Omega={\rm d}\phi\sin(\theta_{1}){\rm d}\theta... ...}\theta_{2}\ldots \sin^{d-2}(\theta_{d-2}){\rm d}\theta_{d-2},$$

Figure 4.1.6: Integration contours in the complex $p$ plane.

where the azimuthal angle $\phi$ goes from $0$ to $2\pi$ and all the others go from $0$ to $\pi$. Naturally, the total solid angle being a compact integration domain and the integrand a bounded function within it, the integration over the angles in (4.1.3) always gives finite results. In addition to this, as one can verify in detail in each case, the oscillations of the complex exponential cause the integration over $p$ to converge, resulting always in a function that decays quickly for large $r$, as a decreasing exponential. Due to this, in this case it is always the lower integration extreme of the integral over $p$ that dominates and, therefore, the results will depend on $m_{0}$ in any dimension $d$.

We will perform here the integrals in the cases $d=1$ and $d=3$, leaving the others to the reader (problem 4.1.7). For the time being, we restrict the discussion to the case $r\neq 0$. In the simplest case, $d=1$, as well as in the case $d=2$, it is really neither necessary nor useful to write the integral in the form of given in equation (4.1.3). In the case $d=1$, with $x=x_{1}-x_{2}$, we may calculate directly the integral in the form shown in (4.1.2),

$$G(\vec{x}_{1}-\vec{x}_{2}) \sim\frac{1}{2\pi}\int_{-N\pi/L}^... ... d}p\;\frac{e^{-\imath px}}{(p-\imath m_{0})(p+\imath m_{0})}.$$

We may calculate this integral in the complex-$p$ plane without difficulty, in the limit $N\rightarrow \infty$. In this case the integral runs over the real line and, if $x>0$, we should close the circuit with an arc at infinity, of size $\pi$, in the lower half-plane, where the imaginary part of $p$ is negative, so that the argument of the exponential, $-\imath px$, has a negative real part. Figure 4.1.6 illustrates the complex-$p$ plane, with the integration contours and the poles of the integrand at $p=\pm\imath m_{0}$. In this case the integral is equal to $(-2\pi\imath)$ times the value of the residue of the integrand in the lower pole, that is,

$$G(x>0)=\frac{1}{2\pi}(-2\pi\imath)\frac{e^{-m_{0}x}}{-2\imath m_{0}} =\frac{e^{-m_{0}x}}{2m_{0}}.$$

If $x<0$ we should close the contour by the other side, the factor multiplying the residue is $(2\pi\imath)$, and hence we obtain in this case

$$G(x<0)=\frac{1}{2\pi}(2\pi\imath)\frac{e^{m_{0}x}}{2\imath m_{0}} =\frac{e^{m_{0}x}}{2m_{0}}.$$

Defining $r=\vert x\vert$, we can join the two answers in the final result

$$G(r)=\frac{e^{-m_{0}r}}{2m_{0}}.$$

We see that the result is finite for all values of $r$, including $r=0$. If fact, we can verify directly that $G(0)$ is finite in this case. As we saw before, we have for the dimensionless function the behavior $g(0)\sim N/(\pi Lp_{m})$ and, since in the case $d=1$ it holds that $G=ag$, it follows that $G(0)\sim 1/(\pi p_{m})$, which is also finite. The complete equality of the two results corresponds to the choice $p_{m}=2m_{0}/\pi$ for the lower extreme $p_{m}$ on the integral used in the approximate calculation of $\sigma ^{2}$.

Passing now to the case $d=3$, in this case we have from equation (4.1.3) that

$$G(r)\sim\frac{1}{(2\pi)^{3}}\int_{0}^{2\pi}{\rm d}\phi\int_{... ...p\; \frac{p^{2}}{p^{2}+m_{0}^{2}}\;e^{-\imath pr\cos(\theta)}.$$

We can do immediately the integrals over $\phi$ e $\theta$, obtaining

$$\begin{eqnarray*} G(r) & \sim & \frac{1}{4\pi^{2}}\int_{0}^{N\pi/L}{\rm d}p\; \f... ...\int_{0}^{N\pi/L}{\rm d}p\; \frac{p}{p^{2}+m_{0}^{2}}\;\sin(pr). \end{eqnarray*}$$

If we now observe that the integrand is even, we may write this as

$$\begin{eqnarray*} G(r) & \sim & \frac{1}{4\pi^{2}r} \int_{-N\pi/L}^{N\pi/L}{\rm ... ...p+\imath m_{0})} \;\frac{e^{\imath pr}-e^{-\imath pr}}{2\imath}. \end{eqnarray*}$$

where we again wrote the sine in terms of complex exponentials. Each one of these two integrals can be done in the complex-$p$ plane, in the limit $N\rightarrow \infty$, in the same way in which we did the integral in the case $d=1$, closing the circuit in the appropriate way in each case. Doing this we obtain

$$\begin{eqnarray*} \frac{1}{8\pi^{2}\imath r}\int_{-\infty}^{\infty}{\rm d}p\; \f... ...\imath m_{0})(p+\imath m_{0})} & = & \frac{e^{-m_{0}r}}{8\pi r}. \end{eqnarray*}$$

With this, we have the final result

$$G(r)=\frac{e^{-m_{0}r}}{4\pi r}.$$

This function is the Yukawa potential that, in the limit $m_{0}=0$, reduces to the Coulomb potential of electrostatics. Just as in the case $d=1$, for $m_{0}\neq 0$ this function also falls off exponentially for large values of $r$, and it is finite at all points except at $r=0$, where it diverges. Once more this is compatible with our previous calculation for $g(0)$, since we saw that $g(0)$ is finite and, for $d=3$, we have that $G=g/a$, which means that $G(0)$ diverges when $N\rightarrow \infty$ and therefore $a\rightarrow 0$. In fact, one can verify that $G(r)$ is finite at the origin only for $d=1$, and that in all the other cases, starting with $d=2$, it diverges, typically with a negative power of $r$ which is characteristic of each dimension $d$. The functions $G$ calculated as in the examples above, for various dimensions $d$, are given in the table 4.1.2, which also contains the corresponding asymptotic behaviors for $r\rightarrow\infty$, that is, for $r\gg 1/m_{0}$, as well as for $m_{0}\rightarrow 0$. The symbols $K_{0}$ and $K_{1}$ in this table are Bessel functions.

Table 4.1.2: Table of correlation functions in the continuum limit.

$d$	$G(r)$	$m_{0}\rightarrow 0$	$r\gg 1/m_{0}$
$1$	$\frac{\displaystyle 1}{\displaystyle 2m_{0}}\;e^{-m_{0}r}$	$\rightarrow\infty$	$=\frac{\displaystyle 1}{\displaystyle 2m_{0}}\;e^{-m_{0}r}$
$2$	$\frac{\displaystyle 1}{\displaystyle 2\pi}{\rm K}_{0}(m_{0}r)$	$\rightarrow\infty$	$\sim\frac{\displaystyle 1}{\displaystyle (8\pi m_{0}r)^{1/2}}\;e^{-m_{0}r}$
$3$	$\frac{\displaystyle 1}{\displaystyle 4\pi r}\;e^{-m_{0}r}$	$\rightarrow\frac{\displaystyle 1}{\displaystyle 4\pi r}$	$=\frac{\displaystyle 1}{\displaystyle 4\pi r}\;e^{-m_{0}r}$
$4$	$\frac{\displaystyle m_{0}}{\displaystyle 4\pi^{2}r}{\rm K}_{1}(m_{0}r)$	$\rightarrow\frac{\displaystyle 1}{\displaystyle 4\pi^{2}r^{2}}$	$\sim\frac{\displaystyle m_{0}^{1/2}}{\displaystyle 2(2\pi r)^{3/2}}\;e^{-m_{0}r}$
$5$	$\frac{\displaystyle 1+m_{0}r}{\displaystyle 8\pi^{2}r^{3}}\;e^{-m_{0}r}$	$\rightarrow\frac{\displaystyle 1}{\displaystyle 8\pi^{2}r^{3}}$	$\sim\frac{\displaystyle m_{0}}{\displaystyle 8\pi^{2}r^{2}}\;e^{-m_{0}r}$

As one can see, these functions in infinite space have relatively simple forms in terms of known functions. In a finite box the form of the correlation functions is not so simple, and in general cannot be written in a simple way in terms of known functions, but only as infinite series (problem 4.1.8). However, they continue to be finite at all points $\vec{x}_{2}$ different from $\vec{x}_{1}$, so that there are no important qualitative differences between the two cases. One observes that the propagators in the cases $d=1$ and $d=2$ diverge in the limit $m_{0}\rightarrow 0$. These are what one refers to as infrared divergences, a type of behavior which is characteristic of the lower dimensions, in particular of $d=1$ and $d=2$. One can verify that the behavior for $m_{0}\rightarrow 0$ is also problematic in the case of the calculations of $\sigma ^{2}$ which we did before. We will now examine how $\sigma ^{2}$ behaves in this limit, in each dimension. From equation (4.1.1) we see that $\sigma ^{2}$ always diverges if we make $\alpha_{0}\rightarrow 0$, even on finite lattices, because the term of the sum involving the mode $\vec{k}=\vec{0}$ diverges in this limit. We may write for $\sigma ^{2}$

$$\sigma^{2}=\frac{1}{N^{d}\alpha_{0}} +\frac{1}{N^{d}}\sum^{\hfill\prime}_{\vec{k}} \frac{1}{\rho^{2}(\vec{k})+\alpha_{0}},$$

where in the sum $\sum'_{\vec{k}}$ the zero mode is omitted. As we mentioned before, the Gaussian model on the torus indeed has a zero mode in the case $\alpha_{0}=0$, which is what is causing us trouble. This is not a physical problem, but only a mathematical problem that reflects the fact that the periodical boundary conditions are not completely realistic from the physical point of view. If we want to deal with models where $\alpha_{0}=0$ on finite lattices, it will be necessary to change slightly the dynamics of the models in order to eliminate the degree of freedom corresponding to the zero mode, as indeed we will do in future volumes, when we discuss non-linear models of scalar fields.

However, it is not necessary to make $\alpha_{0}=0$ on finite lattices in order to study field theories which are massless in the continuum limit. It suffices to recall that $\alpha_{0}$ is related to the mass by $\alpha_{0}=(m_{0}a)^{2}=(m_{0}L)^{2}/N^{2}$, so that $\alpha_{0}$ goes to zero in the limit no matter what value is given to $m_{0}$. Since we have

$$m_{0}^{2}=\frac{1}{L^{2}}\alpha_{0}N^{2},$$

we can either cause $m_{0}$ to have a finite non-vanishing limit, by means of a decrease in $\alpha_{0}$ given by $1/N^{2}$, or cause $m_{0}$ to vanish in the limit by means of a decrease in $\alpha_{0}$ which is faster than $1/N^{2}$. Hence, there is a way to represent zero-mass theories by means of infinite sequences of finite lattices in which $\alpha_{0}$ is always different from zero, which avoids the divergence of the zero-mode term in the sum that defines $\sigma ^{2}$. It remains to be seen how this term behaves in the continuum limit, in each dimension. For finite masses we have that this term is

$$\frac{1}{N^{d}}\;\frac{N^{2}}{(m_{0}L)^{2}} =\frac{1}{(m_{0}L)^{2}N^{d-2}},$$

so that we see that this term goes to zero for $d\geq 3$, is constant for $d=2$ and diverges with $N$ for $d=1$. In all cases these results do not significantly affect the calculations made before for $m_{0}\neq 0$. For $d=1$ the zero-mode term has exactly the same behavior found for the sum, for $d=2$ it is constant while the sum diverges logarithmically and for $d\geq 3$ it goes to zero, while the sum has a finite non-vanishing limit.

In order to have $m_{0}\rightarrow 0$ in the continuum limit it suffices to make $\alpha_{0}$ vary with $N$ as

$$\alpha_{0}=\frac{C}{N^{2+\varepsilon}},$$

where $C$ is some positive constant and $\varepsilon$ some positive number, which we imagine to be small. In this case the zero-mode term is

$$\frac{1}{N^{d}}\;\frac{N^{2+\varepsilon}}{C} =\frac{1}{C\;N^{d-2-\varepsilon}}.$$

We see that this term continues to go to zero for $d\geq 3$, so long as $\varepsilon$ is smaller than $1$. For $d=2$, however, it becomes divergent as $N^{\varepsilon}$, that is, faster than the sum, while in $d=1$ the situation is similar, since in this case it diverges as $N^{1+\varepsilon}$, also faster than the sum. Therefore, we see that for $d=1$ and $d=2$ $\sigma ^{2}$ presents in fact infrared divergences similar to those of $G(r)$, when we make $m_{0}\rightarrow 0$. These facts are peculiar of these low dimensions and should now worry us. The important thing is that there are no divergences in the expression of $\sigma ^{2}$ in the cases $d\geq 3$, when we make $m_{0}$ go to zero, so that the results we obtained before continue to hold in these dimensions. This is not surprising because, as we discussed before, for $d\geq 3$ one can see that the main contributions to $\sigma ^{2}$ come from the large-momentum terms of the sum, so that the results should not depend on $m_{0}$, which is a quantity characteristic of the low-momentum region. Since $\sigma ^{2}$ cannot depend at all on $m_{0}$ under these conditions, it is clear that its behavior should not change when we make $m_{0}\rightarrow 0$.

The graphs we showed before to illustrate the behavior of $\sigma ^{2}$ as a function of $N$ in each dimension were obtained using the value $1$ for $m_{0}$, but we see now that this is not really a relevant fact. For the cases $d\geq 3$, the only ones in which $\sigma ^{2}$ converges to a finite value in the limit, one can show (problem 4.1.9) that in fact the limits are completely independent of $m_{0}$.

We will end this section using the facts established do far in order to show a rather surprising fact relating to the behavior of the two-point correlation functions in quantum field theory. If we recall the basic definition of the correlation function in the context of statistical mechanics, discussed in section 3.2, we see that it does not really make any difference if we discuss the correlations in terms of the dimensionless function $g$ or in terms of the dimensionfull function $G$, because in any case we should analyze the statistical correlations among the fields at various points by means of the homogeneous correlation function

$$\mathfrak{f}(r)=\frac{g(r)}{g(0)}=\frac{G(r)}{G(0)}.$$

We can calculate this function on finite lattices without any trouble and then take the continuum limit. Let us examine then how $\mathfrak{f}(r)$ behaves in this limit. It is clear that, by definition, $\mathfrak{f}(0)$ is always equal to one, both on finite lattices and in the continuum limit. For other values of $r$ we saw that $G(r)$ is finite in the limit, while $G(0)$ diverges. It follows therefore that in the continuum limit $\mathfrak{f}(r)$ vanishes for all non-vanishing values of $r$. Since for the cases of interest, with $d\geq 3$, $g(0)$ is also finite and non-vanishing in the limit, it follows that for $d\geq 3$ the two-point function $g(r)$ is also zero for $r\neq 0$. In short, we have the continuum-limit results

$$\begin{eqnarray*} \mathfrak{f}(r) & = & \left\{ \begin{array}{lll} 1 & \mbox{if... ...& \mbox{if} & r=0 \\ 0 & \mbox{if} & r\neq 0 \end{array}\right.. \end{eqnarray*}$$

The meaning of these results is that the fundamental fields of quantum field theory become completely uncorrelated in the continuum limit. One can say that when one takes the continuum limit all the structure of the two-point function, including the characteristics of the exponential decay related to the mass, collapse into the origin. The result of the continuum limit looks like utter uncorrelated chaos. This is a surprising result, because the correlation between the fields in Euclidean space is directly related to the propagation of perturbations across space-time in the non-Euclidean version of the theory. Furthermore, the nature of this propagation process is supposedly classifiable according to the value of the physical mass $m_{0}$, which can be either zero or not zero.

In order to better understand the significance of these functions in quantum field theory it is essential that we introduce the concept of block variables, which we shall examine in detail later on. We will see that these block variables are the mechanism by means of which physical order arises out of the utter chaos of the underlying realm of the fundamental field variables. In this way one can say that quantum field theory is an example of a type of self-organizing structure. We will also see that in quantum field theory the only reasonably simple way to deal with the dynamics of the models is to work with the dimensionfull propagator $G(r)$, which is finite at all points but the origin. Only through an analysis involving block variables we may understand why this function is the one that has most physical relevance, despite its divergence at the origin.

The behavior of $g(r)$ shown above will also be useful in the intuitive discussion of the important phenomenon of the triviality of the non-linear models of scalar fields, which we will discuss in future volumes. This triviality means that the models fail to contain physical interactions between particles in the continuum limit, despite their non-linear nature. This is one reason why we must look elsewhere for physically relevant interacting models, and are thus naturally led to the study of gauge theories.

Problems

1. Consider, in the case $d=1$, the quantity $\Sigma'^{2}=L\sigma'^{2}/N$ where $\sigma'^{2}$ is defined like $\sigma ^{2}$ except for the omission from the sum of the zero-mode $k=0$. Calculate $\Sigma'^{2}$ in the particular case $m_{0}=0$. Analyze from which extreme of the integral comes the main contribution in this case, and verify that one can approximate $N^{2}\rho^{2}(k)$ by $(2\pi k)^{2}$ for large $N$. It will be useful to recall the definition of the Riemann $\zeta(z)$ function in terms of a sum of $1/k^{z}$ and look up in a table of integrals its value for $z=2$.

2. From the result of the problem 4.1.1 show that $\Sigma^{2}$, as defined in the text, is bounded by a finite real number so long as $L$ and $m_{0}$ are finite and not zero,

   
   

   $$\Sigma^{2}\leq \frac{L}{12}+\frac{L}{(m_{0}L)^{2}},$$
   
   

   as reported in one of the tables given in the text.

3. Write a program to calculate $\Sigma'^{2}$ numerically for lattices with increasing sizes. From its results produce an extrapolation to the limit $N\rightarrow \infty$ with constant $m_{0}$, and show that the result of problem 4.1.1 is exact within your numerical precision.

4. ($\star$) Consider the quantity $\Sigma^{2}$ in the case $d=1$, in a finite interval of length $L$, for $m_{0}\neq 0$ and an arbitrary $N$, which implies that $\alpha_{0}\neq 0$ on each finite lattice, leaving open, however, the possibility that $\alpha_{0}\rightarrow 0$ when $N\rightarrow \infty$. Build, for each finite $N$, two integrals $I_{M}$ and $I_{m}$ over the coordinates $k$ of momentum space, extending them to real values, so that $I_{M}$ is strictly larger and $I_{m}$ strictly smaller than the sum that defines $\Sigma^{2}$. Calculate the integrals, take after that the limit $N\rightarrow \infty$ and demonstrate in this way that $\Sigma^{2}$ has a finite and non-vanishing limit in the continuum limit, assuming that $m_{0}\neq 0$ in the limit.

5. Write a program to calculate the local width $\sigma ^{2}$ of the fields in the case $d=2$. Use the program to calculate $\sigma^{2}(N)$ for the values of $N$ shown in the graph given in the text and plot the results as a function of $\ln(N)$, with constant $m_{0}$, verifying in this way that the results display indeed a logarithmic divergence in the continuum limit.

6. ($\star$) Write a program to calculate the local width $\sigma ^{2}$ of the fields in the case $d=3$. Use the program to calculate $\sigma^{2}(N)$ for as many values of $N$ as possible with a reasonable amount of computer time, and make a numerical fitting of the resulting function to an expression of the form

   
   

   $$f(N)=f_{0}+\frac{f_{1}}{N^{p}},$$
   
   

   where $f_{0}$, $f_{1}$ and $p$ are unknown constants. Repeat the fitting for several subsets of the data, each with an increasing maximum value of $N$, in some convenient way, thus obtaining successive estimates for these three quantities, for increasing values of $N$. In this fashion, obtain an extrapolation of the result for these three quantities to the continuum limit $N\rightarrow \infty$, with constant $m_{0}$. Hint: try to start your fitting with $p\sim 1$, $f_{0}\sim 1/4$ and $f_{1}\sim 1$ and remember that the important thing is to adjust the function for large values of $N$.

7. Obtain the dimensionfull correlation functions $G(r)$ in position space for the cases $d=2$, $d=4$ and $d=5$, calculating the corresponding integrals over momentum space, as discussed in the text.

8. Calculate the functions $G(\vec{x}_{1}-\vec{x}_{2})$ in the continuum, for $d=1$ and $d=2$, within finite cubical boxes of side $L$, using fixed boundary conditions with vanishing fields at the border, starting from the results presented in the text for infinite space. Remember that these functions, besides being the propagators of the quantum models in position space, are also the Green functions (fields of unit point-like external sources) of the corresponding classical models. For simplicity make $\vec{x}_{2}=\vec{0}$ and calculate the functions in terms of $\vec{x}_{1}$. Remember also that the Gaussian models are linear and that a principle of superposition holds for them, which enables one to use a version of the method of images, which is very popular in electrostatics, to solve the problem. Your answers should be written in the form of infinite series. In general these series are not absolutely convergent, but they are Borel-sumable, that is, it is possible to find a specific order of summation for which they converge. See if you can figure out what is the special order for these sums.

9. Consider $\sigma^{2}(m_{0})$ as defined in equation (4.1.1), where $\alpha_{0}=(m_{0}L)^{2}/N^{2}$. Consider the cases for which $d\geq 3$, in which both $\sigma ^{2}$ and $m_{0}$ have finite and non-vanishing limits in the continuum limit. Differentiate $\sigma ^{2}$ as a function of $m_{0}$ and show that the resulting sum goes to zero in the limit, thus showing that $\sigma ^{2}$ becomes independent of $m_{0}$ in the limit. In order to do this, approximate the sums by integrals over the momenta, as was done in the text for the calculation of $\sigma ^{2}$.