Block Variables and Observables

Block variables are defined as averages of some type, over the fields contained within finite regions of space-time, which we call blocks. The name comes from the realization of statistical-mechanic models on the lattice, where the regions consist of block of sites, having been introduced by Kadanoff in the study of statistical systems involving spins. In the example we will study here the average at issue will be a simple arithmetic average of the fields within the blocks but, in general, some other type of linear or even non-linear superposition may be involved. The block variables and the corresponding systems of linear superposition that define them play a central role in the definition of quantum field theories, because they determine the types of physical observables that can, in fact, be measured.

If we think about how we would go about measuring the instantaneous value of the field at a certain point in space-time, in the continuum limit, it will immediately become clear that we would not be able to do it at all. In order to do this it would be necessary to use as the instrument of measurement some object that could be completely localized at the point in question, so that we may detect exclusively the field at that point. However, the wave-like nature of all objects existing in nature, added to the uncertainty principle of quantum mechanics (or, equivalently, to the simple uncertainty principle of classical wave physics [2]), implies that this object should have vanishing wavelength and, therefore, consist of “quantums” of infinite energy. We are forced to conclude that, as refined and developed as the measurement apparatus may be, it will always end up measuring the field over some region of space-time with finite and non-vanishing dimensions, however small they may be.

Since we are interested in quantum field theory in the continuum limit from the lattice, we will always end up having an infinite number of points within any finite region, when we take the limit. Therefore any measurement instrument will always, in practice, be measuring the field over an infinite number of points, never at a single point. The actual result of the measurement by the apparatus depends not only on the values of the fields at the points involved, but also on the nature of the superposition rule for the fields, generating from the values of the field at all the points a single resulting value to be associated to the region as a whole. This rule is not arbitrary, of course, it is related to the way in which the measurement apparatus interacts with the fields. Note that, so far as we are able to define the structure of the theory with our current knowledge, the measurement apparatus is an object which remains external do the structure. Hence, any model intended to represent all possible physical measurements based on fields distributed over a continuous space-time should not only discriminate the number and type of fields involved and define a dynamics for them, but should also discriminate a superposition process to be used for each type of field.

In this way we see that, at the stage of development in which the theory is being built here, the discrimination of the superposition process is an integral part of the definition of a quantum field theory, in addition to the discrimination of the fields involved and the definition of their dynamics, by means of an action functional $S$. It is possible that this will cease to be so in a more complete future theory, in which the notion of the physical measurement is included in the structure of the theory from the start, but for the time being we must be content with this state of affairs.

The introduction of block variables in a given model may be understood as a kind of change of variables in the model. However, such a change from point variables to block variables is not invertible, because it involves loss of information about the behavior of the point variables above a certain energy. One should not, however, get the impression that there will be a smaller number of block variables than of the original point variables, that they will always be a finite or even a discrete set. This is due to the fact that it is not necessary to separate space-time into disjoint and exclusive blocks; quite to the contrary, the blocks may very well overlap each other. In fact, we may associate to each single point of space-time a corresponding block variable $\bar\varphi$ which is the result of the superposition of the fields $\varphi$ inside a region of volume $V_{r}$ centered at that point. In this way we produce a block field $\bar\varphi (\vec{n})$ defined over all of space-time, just like the fundamental field $\varphi(\vec{n})$. These new fields are sometimes denominated block-renormalized fields for blocks of volume $V_{r}$. Note that this is a specific definition using the term “renormalized”, which is used in many different ways, not necessarily clear or even consistent, in the usual formalism.

Given a certain energy limit, we may define the corresponding block-renormalized field, by means of a judicious choice of the volume $V_{r}$ of the blocks, and then discuss the theory in terms of these block variables, for all phenomena in the theory which are below that energy limit. In fact, one seldom does this, because usually the theory acquires a much more complex form when written in terms of these block variables, but in principle we may discuss the $n$-point correlation functions for these variables, in a way similar to the discussion of the correlation functions of the fundamental field,

$$g_{r}(\vec{n}_{1},\ldots,\vec{n}_{n})= \langle\bar\varphi (\vec{n}_{1})\ldots\bar\varphi (\vec{n}_{n})\rangle.$$

We will examine here in detail only the case of the two-point function in the free theory, that is, the block propagator. This will be useful to develop our understanding of the roles played by the dimensionfull and dimensionless versions of the fields, as well as to illustrate the reasons due to which we use, most of the time, the point variables rather than the block variables for the development of the theory, although the two-point function of these variables in position space has a singular behavior at the origin, in the continuum limit.

Let us consider then the calculation of the two-point function for block variables. Our objective here will be to verify how the block variables are correlated for short and long distances, relative to the size of the blocks. We will do the calculation both in position space and in momentum space, and in this second case we will be particularly interested in verifying whether or not the block propagator has a pole at the position of the renormalized mass, as is the case for the usual renormalized propagator that appears in perturbation theory. Observe that the usual renormalized propagator, written in terms of the fundamental field, is not the same as the block-renormalized propagator. The fact that both are referred to as “renormalized propagators” is just an example of the use of the term “renormalized” for multiple different ends. As we shall see, these two propagators have similar behavior for large distances and small momenta, but their behavior for short distances and large momenta is very different.

Figure 4.3.1: The problem of the calculation of the gravitational potential of a homogeneous spherical body.

Figure 4.3.2: The gravitational potentials of the point body and of the extended spherical body, showing the smoothing of the singularity.

The calculations related to the block propagator will be done in the spirit of the Newtonian gravitation problem in which one calculates the gravitational potential of a spherical homogeneous body of radius $a$. In fact, the Newtonian gravitational potential of a point mass is just the Green function of the three-dimensional Laplacian, that is, the Green function of the free theory with $m_{0}=0$ in three dimensions. The geometrical situation is described in figure 4.3.1. Solving this elementary problem of classical mechanics one verifies that, outside the sphere, what is seen is exactly the potential of a point particle located at the origin, with mass equal to the total mass of the body. However, inside the sphere the situation is very different, instead of the singularity of the Green function at the origin, we have a finite and smooth potential over all the interior of the sphere. In figure 4.3.2 a comparison of the two potentials can be found. At distances that are large compared to the radius $a$ of the sphere we have the simple potential of a point mass, while a distances that are small compared to the radius we have a finite field, smoothed out by a mechanism we may call the “smearing” of the point source. This situation is similar to the one we will find for our block variables, with the size of the block playing the role of the volume of the spherical body.

Figure 4.3.3: A cubical lattice with identical cubical blocks.

Consider then the free scalar field in a cubical $d$-dimensional lattice with $N^{d}$ sites and periodical boundary conditions. Consider also in this lattice cubical blocks of sites, each with $N_{r}^{d}$ sites and $N_{r}=N/r$ sites along each direction, for some number $r$ such that $1\leq r\leq N$. The geometrical situation is illustrated in figure 4.3.3. For simplicity of notation, in this section the vectors $\vec{n}$ and $\vec{k}$ will be represented by $\mathbf{n}$ and $\mathbf{k}$, while an upper bar will denote average over a block. The block variables are defined by arithmetical averages over the blocks. In terms of the dimensionless fields, for a block $B$ centered at the position $\mathbf{n}$, we have

$$\bar\varphi (\bar{\mathbf{n}})=\frac{1}{N_{r}^{d}}\sum_{\mathbf{n}\in B}\varphi(\mathbf{n}),$$

where the position $\bar{\mathbf{n}}$ of the block is given by

$$\bar{\mathbf{n}}=\frac{1}{N_{r}^{d}}\sum_{\mathbf{n}\in B}\mathbf{n}.$$

Note that, unlike $\mathbf{n}$, $\bar{\mathbf{n}}$ does not necessarily have integer components, depending on how the blocks are chosen. If we so wish, we may simplify this situation choosing the blocks in a more symmetrical way, with odd $N_{r}$ and center at a site of the original lattice, but this is not actually important or necessary. The dimensionless block propagator in position space is given by

$$g_{r}(\bar{\mathbf{n}}_{1},\bar{\mathbf{n}}_{2})=\langle\bar... ...bar{\mathbf{n}}_{1})\bar\varphi (\bar{\mathbf{n}}_{2})\rangle,$$

relating two blocks, a block $B_{1}$ at $\bar{\mathbf{n}}_{1}$ and another block $B_{2}$ at $\bar{\mathbf{n}}_{2}$. We may write this propagator in terms of the fundamental field as

$$\begin{eqnarray*} g_{r}(\bar{\mathbf{n}}_{1},\bar{\mathbf{n}}_{2}) & = & \frac{1... ...}^{2d}}\sum_{B_{1}}\sum_{B_{2}}g(\mathbf{n}_{1},\mathbf{n}_{2}). \end{eqnarray*}$$

We may now write the Green function $g(\mathbf{n}_{1},\mathbf{n}_{2})$ in terms of the Fourier modes of the lattice as usual,

$$g(\mathbf{n}_{1},\mathbf{n}_{2})=\sum_{\mathbf{k}} e^{\imath... ...\cdot(\mathbf{n}_{2}-\mathbf{n}_{1})}\widetilde g(\mathbf{k}),$$

where it becomes explicit that $g(\mathbf{n}_{1},\mathbf{n}_{2})$ is a function only of $\mathbf{n}_{2}-\mathbf{n}_{1}$. For the free theory we know that

$$\widetilde g(\mathbf{k})=\frac{1}{N^{d}}\frac{1}{\rho^{2}(\mathbf{k})+\alpha_{0}},$$

so that we obtain for the block propagator

$$g_{r}(\bar{\mathbf{n}}_{1},\bar{\mathbf{n}}_{2})= \frac{1}{N... ...2}-\mathbf{n}_{1})} \frac{1}{\rho^{2}(\mathbf{k})+\alpha_{0}}.$$

If we now define coordinates $\mathbf{n}_{1}'$ and $\mathbf{n}_{2}'$ internal to the blocks, which give the position of the sites with respect, respectively, to $\bar{\mathbf{n}}_{1}$ and $\bar{\mathbf{n}}_{2}$, then we have for the sites the relations

$$\mathbf{n}_{1}=\bar{\mathbf{n}}_{1}+\mathbf{n}_{1}',\mbox{ }... ...on to~~} \mathbf{R}=\bar{\mathbf{n}}_{2}-\bar{\mathbf{n}}_{1},$$

from which it follows that

  $$\mathbf{n}_{2}-\mathbf{n}_{1}=\bar{\mathbf{n}}_{2}-\bar{\mathbf{n... ...1}+\mathbf{n}_{2}'-\mathbf{n}_{1}'=\mathbf{R}+\mathbf{n}_{2}'-\mathbf{n}_{1}',$$ (4.3.1)

so that we may write for the block propagator

$$\hspace{-3em}g_{r}(\bar{\mathbf{n}}_{1},\bar{\mathbf{n}}_{2}) = \frac{1}{N_{r}^{2d}}\frac{1}{N^{d}}\sum_{B_{1}}\sum_{B_{2}}\sum_{... ...}\cdot\mathbf{n}_{1}'} \;e^{\imath\frac{2\pi}{N}\mathbf{k}\cdot\mathbf{n}_{2}'}$$

$$= \frac{1}{N^{d}}\sum_{\mathbf{k}} \frac{e^{-\imath\frac{2\pi}{N}\m... ...{d}}\sum_{B_{2}} e^{\imath\frac{2\pi}{N}\mathbf{k}\cdot\mathbf{n}_{2}'}\right).$$ (4.3.2)

We now see that the block propagator depends only on $\mathbf{R}=\bar{\mathbf{n}}_{2}-\bar{\mathbf{n}}_{1}$. The two sums in parenthesis are now internal sums within each block, they do not depend on the position of the blocks, but only on the momenta. Since the blocks are all equal by hypothesis, these two parenthesis are the complex conjugates of each other. They are in fact a form factor $f_{r}^{(d)}(\mathbf{k})$, in terms of which we may write

  $$g_{r}(\mathbf{R})=\frac{1}{N^{d}}\sum_{\mathbf{k}} \frac{e^{\imat... ...{2}(\mathbf{k})+\alpha_{0}} \left\vert f_{r}^{(d)}(\mathbf{k})\right\vert^{2},$$ (4.3.3)

where the form factor is defined within an arbitrary block $B$ as

$$f_{r}^{(d)}(\mathbf{k})=\frac{1}{N_{r}^{d}}\sum_{\mathbf{n}'\in B} e^{-\imath\frac{2\pi}{N}\mathbf{k}\cdot\mathbf{n}'}.$$

From the expression above for $g_{r}$ we may read immediately its Fourier transform,

$$\widetilde g_{r}(\mathbf{k})=\frac{1}{N^{d}} \frac{\left\ver... ...(\mathbf{k})\right\vert^{2}}{\rho^{2}(\mathbf{k})+\alpha_{0}},$$

which is, therefore, the block propagator in momentum space. Note that, unlike what happens in position space, in momentum space no change is needed in the coordinates $\mathbf{k}$.

We have, then, the block propagator dully calculated, in terms of $f_{r}^{(d)}$, both in momentum space and in position space. In order to examine the properties of this propagator, we must first examine the properties of the form factor $f_{r}^{(d)}$. Observe, in the first place, that since $f_{r}^{(d)}$ is an average of complex phases it is always true that $\vert f_{r}^{(d)}(\mathbf{k})\vert\leq 1$, from which it follows that, in momentum space, the block propagator is always smaller than or equal to the propagator of the fundamental field. Observe also that, for simple cubical blocks like the ones we are using here,

Figure 4.3.4: Logarithms of the fundamental and block propagators in momentum space.

Figure 4.3.5: Behavior of the local block width as a function of $N$, for $d=1$.

Figure 4.3.6: Behavior of the local block width as a function of $N$, for $d=2$.

$$\begin{eqnarray*} f_{r}^{(d)}(\mathbf{k}) & = & \frac{1}{N_{r}^{d}}\sum_{\mathbf... ...n_{\mu}'}\right) \\ & = & \prod_{\mu=1}^{d}f_{r}^{(1)}(k_{\mu}), \end{eqnarray*}$$

so that it is enough to calculate $f_{r}^{(1)}(k)=f_{r}(k)$ in order to find out how the form factor behaves. Given the system of internal coordinates that we adopted for the blocks, we may write explicitly

$$f_{r}(k)=\frac{1}{N_{r}}\sum_{n'=-(N_{r}-1)/2}^{(N_{r}-1)/2} e^{-\imath\frac{2\pi}{N}kn'},$$

where $n'$ spans $N_{r}$ consecutive values, being half-integer if $N_{r}$ is even and integer if $N_{r}$ is odd, as is the case for a symmetrical choice of the blocks around the sites. We may execute the sum using the formula for the sum of a geometrical progression,

$$\begin{eqnarray*} f_{r}(k) & = & \frac{1}{N_{r}} \;\frac{e^{-\imath\frac{2\pi}{N... ...\left(\frac{k\pi}{r}\right)} {N\sin\left(\frac{k\pi}{N}\right)}. \end{eqnarray*}$$

This is true if $k\neq 0$, and if $k=0$ we have immediately that $f_{r}(0)=1$. Finally observe that, for any mode $\mathbf{k}$ of the lattice which coincides with an internal mode of the block, $f_{r}$ vanishes because the sum that defines it coincides in this case with the sum that appears in the orthogonality and completeness relations of the block itself. We may see from the expression above that this will be true for modes $\mathbf{k}$ such that $k_{\mu}N_{r}/N=k_{\mu}/r$ is an integer for at least one value of $\mu$. In other words, $f_{r}$ tends to suppress the modes of the lattice whose wavelengths fit an integer number of times within the block, getting, so to say, in resonance with it. If the wavelength does not fit exactly an integer number of times within the block, the mode will be partially suppressed, only a little if the number of times it fits inside is small, more if that number is large. In short, one perceives that $f_{r}$ tends to suppress preferentially the high-frequency modes in the momentum space of the lattice. In position space this suppression has the effect of eliminating the singularity of the propagator at the origin, corresponding to a smearing of the fields by the blocking process. Figure 4.3.4 shows the fundamental propagator and the block propagator, on a logarithmic scale, along one of the directions in momentum space. The normalizations of the two propagators are arbitrary but consistent with each other. The graph is the same for any dimension $d$. One can clearly see the strong suppression of the block propagator for large momenta, as well as its oscillations due to resonances with the internal modes of the blocks.

Figure 4.3.7: Behavior of the local block width as a function of $N$, for $d=3$.

Figure 4.3.8: Behavior of the local block width as a function of $N$, for $d=4$.

Figure 4.3.9: Behavior of the local block width as a function of $N$, for $d=5$.

We would like to discuss the results for $g_{r}$ and $\widetilde g_{r}$ in two different limits, for $R\gg N_{r}$ and for $R=0$. The first case turns out to be much simpler and we may discuss it directly from equation (4.3.1). If $R\gg N_{r}$ then it follows that $R\gg n_{1}'$ and $R\gg n_{2}'$ and we may neglect $\mathbf{n}_{1}'$ and $\mathbf{n}_{2}'$ in that equation, which makes $f_{r}=1$ in equation (4.3.2) and therefore results in

$$g_{r}(\mathbf{R})=\frac{1}{N^{d}}\sum_{\mathbf{k}} \frac{e^{... ...}\mathbf{k}\cdot\mathbf{R}}}{\rho^{2}(\mathbf{k})+\alpha_{0}},$$

which is exactly the expression of the propagator of the fundamental dimensionless field in momentum space. We see therefore that for the correlations at large distances, much larger than the size of the blocks, the propagators of the fundamental field and of the blocks do in fact coincide, displaying the same long-range behavior. Examining their expressions in momentum space we see that the two propagators have the same simple pole at the position $-\alpha_{0}$, characterizing this long-range behavior. In fact, since for $k\rightarrow 0$ we have that $f_{r}\rightarrow 1$, we can see that the two propagators have the same behavior for small values of the momentum.

As we saw in section 4.1, the dimensionless propagator of the fundamental field goes to zero away from the origin, hence the same will happen with the dimensionless version of the block propagator. It follows that it is more convenient to use the dimensionfull version of the block propagator which, starting from equation (4.3.3), is given by

  $$G_{r}(\mathbf{R})=\frac{1}{N^{2}L^{d-2}}\sum_{\mathbf{k}} \frac{e... ...{2}(\mathbf{k})+\alpha_{0}} \left\vert f_{r}^{(d)}(\mathbf{k})\right\vert^{2}.$$ (4.3.4)

As we also saw in section 4.1, the dimensionfull propagator of the fundamental field diverges at the origin. We should now verify how this block propagator behaves at the origin. Its value $G_{r}(\mathbf{0})=\langle\bar\phi ^{2}\rangle$ at the origin is related to the average value of the fluctuations of the block variables, because since $\langle\phi\rangle=0$ and the block field is a simple arithmetical average of the fundamental field over the block, it follows that we also have $\langle\bar\phi \rangle=0$. The block propagator at the origin is given by

$$G_{r}(\mathbf{0})=\Sigma_{r}^{2}=\frac{1}{N^{2}L^{d-2}}\sum_... ...^{2}(\mathbf{k})+\alpha_{0}} =\frac{1}{L^{d-2}}\sigma_{r}^{2},$$

where $\Sigma_{r}^{2}$ is the local block width. In the graphs contained in the figures from 4.3.5 to 4.3.9 we present the corresponding dimensionless quantity $\sigma_{r}^{2}$, which is proportional to $\Sigma_{r}^{2}$ in finite boxes, with values from $3$ to $7$ for the ratio $r$ between the sizes of the lattice and the blocks, for a mass $m_{0}=N^{2}\alpha_{0}=1$, in dimensions from $d=1$ to $d=5$, for sequences of lattices of increasing sizes.

We see that, in all cases, $G_{r}(\mathbf{0})$ converges to a finite value for each value of $r$, there being therefore no divergence at the origin. These values increase with $r$, meaning that, the smaller the blocks, the larger the fluctuations of the block variables. Note that for $d=1$ and $d=2$ the curves seem to tend to eventually accumulate near some finite value as $r$ increases. For $d=3$ we see that the width $\sigma_{r}^{2}$ seems to increase linearly with $r$, while for $d=4$ and $d=5$ it seems to increase faster than linearly with $r$. These facts are related to the fact that, at first sight, $G_{r}(\mathbf{R})$ seems to go to zero for $d\geq 3$ when we make the size $L$ of the box tend to infinity, due to the factors of $L$ that appear in its definition. Of course, in this case keeping the blocks at a constant size corresponds to making $r$ increase linearly with $L$, so that the size $L_{r}=L/r$ of the blocks remains finite. One can verify (problem 4.3.1) that this increase of $\sigma_{r}^{2}$ with $r$ exactly compensates the factor $1/L^{d-2}$ that appears in the expression for $G_{r}(\mathbf{R})$, resulting therefore in a finite and non-vanishing block propagator also in the case of the theory in infinite space, that is, in the limit $L\rightarrow\infty$.

Figure 4.3.10: Block propagator in position space, for $d=1$.

Figure 4.3.11: Block propagator in position space, for $d=2$.

We see therefore that the dimensionfull block propagator in position space is a finite and non-vanishing function at all points. We may therefore define for the block variables a correlation function like the one we discussed in section 3.2, normalized to be equal to $1$ at the origin, which, unlike the corresponding function in the case of the fundamental field, will not be a singular function,

$$\mathfrak{f}_{r}(\mathbf{R})=\frac{g_{r}(\mathbf{R})}{g_{r}(\mathbf{0})}=\frac{G_{r}(\mathbf{R})}{G_{r}(\mathbf{0})}.$$

The graphs contained in figures from 4.3.10 to 4.3.13 show the homogeneous correlation functions $\mathfrak{f}(\mathbf{R})$ calculated along one direction of the lattice, for dimensions $d$ from $1$ to $4$, with $L=1$, $m_{0}=N^{2}\alpha_{0}=3$, $r=5$ and different values of $N$ in each case. We also put in these graphs parts of the corresponding dimensionfull propagators of the fundamental field in position space, which are divergent at the origin for $d\geq 2$, calculated for the same values of the parameters $d$, $L$, $m_{0}$ and $N$, and normalized consistently with $\mathfrak{f}(\mathbf{R})$, so as to permit the comparison of the results.

Figure 4.3.12: Block propagator in position space, for $d=3$.

Figure 4.3.13: Block propagator in position space, for $d=4$.

We see that, in general, the block propagator is smaller than the fundamental propagator, particularly near the origin, although this relation can be reversed in a slight way for larger values or $R$, specially for low dimensions, in which the infrared effects are more important. We see also that in the continuum limit $G_{r}(\mathbf{R})$ tends to a finite, continuous and differentiable function, at all points. One might be led to ask how can this happen, how is it possible that there are variables correlated in a finite and non-vanishing way in the continuum limit of a theory in which the fundamental variables become completely uncorrelated in the same limit. We may see that the mechanism which causes this behavior is in fact very simple, if we examine what happens during the process of taking the limit. The fundamental field becomes completely uncorrelated in the limit, but while we are taking the limit there is a very large and ever increasing number of values of the fundamental field which are superposed within each block, with the consequence that a very large number of correlations between the fields inside one of the blocks and the fields inside the other block also superpose. During the limit the increase in the correlation function of the superposition, due to the increase in the number of superposed values, exactly compensates the decrease in the correlation function between each pair of fundamental field values, one in each block. The result is a finite and non-vanishing correlation function between the blocks, despite the complete lack of correlation of the fundamental field in the continuum limit.

It becomes clear, then, how to go about the physical interpretation of the theory. Usually we deal with it in terms of the fundamental field because it is simpler to act in this way, but the physical interpretation must be always in terms of block variables. For distances which are large compared to the size of the blocks, the propagator of the fundamental field coincides with the block propagator and we may use it directly to extract the physical results from the theory. We may say that the fundamental field encodes the information about how the models behave in all energy scales, from zero to infinity, but that in order to measure physical quantities it is necessary to define beforehand what is the proper energy scale of the physical situation one is dealing with, and then to choose block variables of appropriate size.

Note that the block variables do not necessarily have to be associated to cubical blocks centered at each point. Of course it is not essential that the blocks have any given form, they may be cubical, spherical, or of any other form adequate to each situation. Going even beyond that, the block variables may even be objects of a quite different nature such as, for example, the Fourier components of the fundamental field themselves. What matters is that they involve the superposition of an ever increasing number of values of the field as we take the continuum limit, as is the case for the Fourier components, and that there exist for them a fixed maximum limit for the momenta, as is the case for the Fourier transforms for fixed given momenta, which do not increase in the limit. Hence we see that the description in terms of the momentum space is intrinsically more realistic, from the physical point of view, than that in terms of the position space. This is associated to the fact that the concept of wave is more natural and more elementary than the concept of particle in quantum field theory.

It is interesting to observe that, since the fundamental field becomes completely uncorrelated in the continuum limit, which is a very singular limit, it is not really possible to first take the limit and only afterward extract the physical consequences predicted by the theory. It is only possible to extract these physical consequences by taking the limit from finite lattices directly of the relevant physical quantities, in blocks of appropriate size. Hence, we see that the taking of the continuum limit from the lattice is an integral part of the manipulation of the theory for obtaining physical results.

Problems

1. 1. Write a program to calculate $\sigma_{r}^{2}$ in dimensions $d$ from $1$ to $5$, for a given lattice size $N$. Use the program to calculate $\sigma_{r}^{2}$ for the largest fixed value of $N$ that it is possible to use in each dimension within a reasonable amount of computer time, varying $r$ from $1$ to $N$.

   2. Plot graphs of $\sigma_{r}^{2}$ as a function of $r$ for this fixed value of $N$ and make curve fittings in order to try to discover the dependency of this quantity on $r$.

   3. Use these results to find out how $\Sigma_{r}^{2}$ behaves in continuum limits in which both $L$ and $r$ increase with $N$ as $\sqrt{N}$, so that the size $L_{r}=L/r$ of the blocks remains finite. Show in this way that $\Sigma_{r}^{2}$ has a finite and non-vanishing limit under these conditions, in the cases $d=3$, $d=4$ and $d=5$.

   4. Find out the behavior of $\Sigma_{r}^{2}$ under these same conditions, in the cases $d=1$ and $d=2$. Remember that it is quite possible that the behavior is logarithmic in the case $d=2$, in fact, this is likely to happen^4.2.

2. 1. Starting from the formula in equation (4.3.4) for the dimensionfull block propagator, calculate its derivative $\partial_{\mathbf{R}}G_{r}(\mathbf{R})$ and show that it is always negative, indicating that the propagator in a monotonically decreasing function of $\mathbf{R}$.

   2. Calculate also the second derivative and solve the equation $\partial^{2}_{\mathbf{R}}G_{r}(\mathbf{R})=0$, in order to find the position of the inflection point of $G_{r}(\mathbf{R})$. Do this numerically if necessary. Find out the physical meaning of your answer.