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,
is a function only of and may be written as
as we saw in the derivation of equation (3.4.3). Our initial objective relative to this function is to calculate its value for , in which case we have the quantity
a quantity that, by discrete translation invariance on the torus, is independent of position. Since we have that , it follows that is the square of the width of the distribution of values of the dimensionless field at a give site, that is, is the average size of the fluctuations of the field around its average value of zero. We will refer to as the local width of the distribution of the fields. We may write for this quantity
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 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 , the “element of volume” in momentum space is given by
Substituting this “” in equation (4.1.1) and approximating the discrete lattice quantity by we obtain
so that we may now approximate by the integral
We must now discuss how to determine the extremes of integration. Note that for , due to the factor of in the denominator, the result can only be non-vanishing in the limit if the integral diverges in that limit. For near zero the integrand is finite, so long as is not zero, so that the integral cannot diverge at this extreme. For simplicity, let us assume temporarily that , postponing until later the discussion of the case . It follows from these considerations that for the only part of the domain of integration that matters is the one for large absolute values of the momenta. In any case, we can write the integral in spherical coordinates in momentum space, obtaining
where is the complete solid angle in dimensions and is the largest possible value for the components of the momentum on a lattice with sites in each direction. Since we are not interested in the lower limit of the integral, we may neglect the that appears in the denominator and write the integral as
where is some small and finite but non-vanishing value of the momentum, which we could choose to be or . Recalling that we are discussing all this mostly in the context of the case , we see now that for dimensions the factor of appear in the denominator, so that indeed we will have to examine the cases and separately. We may now do the integration for the case , obtaining
were we neglected the contribution from the lower extreme of the integral, which vanishes in the limit. We see that the factors of cancel out and that the final result for in the limit is
which is finite, for in our case here and is always a finite number different from zero, since it is the volume of a compact manifold, the surface of the -dimensional unit sphere. For the case the exponent of 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 and the integral results in
so that the upper extreme still dominates and in this case diverges logarithmically with , in fact a type of behavior that is very common in the case . In the case the factor in front of the integral diverges as , 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 and the integral may be written as
so that in this case diverges linearly with . 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 , that is, on the details of the choice of the lower integration limit. Since this limit was introduced only to allow us to eliminate from the integrand and hence facilitate the realization of the integral, this means, in fact, that in these cases the results depend on . In fact, it is possible to improve the analytical calculation in the case (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 , building integrals that are strict lower bounds and strict upper bounds to the sums (problem 4.1.4), so as to prove that in fact behaves as a function of , for large , 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 with . This is still true even if we make the volume of the box go to infinity, so that the volume element in momentum space goes to zero. The reason is that the quantity that appears in the integrand only approaches if with kept finite, while the sum over the momenta will always include terms in which . Hence, for terms close to the upper limit of the sum it is not true that approaches . Note that for , since it is necessary that the integrals diverge in order for not to vanish, the main contribution to the final result comes precisely from such terms. The same is true for , while for 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 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 . The graphs that can be found in the figures numbered from 4.1.1 to 4.1.5 show the values of for sequences of finite lattices in dimensions from to , for the case , obtained by the use of such programs. For one can see clearly the linear divergence with . In the case 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 case the behavior changes radically, the function becomes a decreasing rather than increasing function of , 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 we obtain for the final results shown in table 4.1.1.
In all this analysis we see, in a very clear way, that the cases and are very special. For we have finite fluctuations of the values of at the sites, while for and these fluctuations diverge. Observe that in all cases is the maximum value of , because for the terms of the sum that defines are multiplied by numbers with absolute values smaller than . This is consistent with the graphs of these functions that we saw before in section 3.5, which decay when moves away form .
We will now examine the behavior of the dimensionfull versions of and . Note that we can define a quantity for the dimensionfull field in a way analogous to the definition of . Given the scaling relations for the fields, we immediately have that , so that we may immediately deduce from table 4.1.1 the behavior of . In this quantity has a finite value proportional to and for it is equal to , and therefore it diverges in the same way, logarithmically. However, for it diverges with some power of , 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 and are two distinct points. Still from our scaling relations for the fields we have that those of the propagator should be , where , so that we may write for
This time our objective is to show that the function is finite in the continuum limit, so long as . We will once more approximate the sum by an integral,
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 , going in this way from the finite box to infinite space, where 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 -dimensional lattice with sites. Under these conditions, if we define , we see that depends only on the modulus of the vector because, if we make an arbitrary change in the angles of the versor , 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 in the direction of the component of and write, for dimensions , without loss of generality,
where is the angle between the vector and its component and the angular integration is over the solid angle of the -dimensional space, with integration element given by
where the azimuthal angle goes from to and all the others go from to . 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 to converge, resulting always in a function that decays quickly for large , as a decreasing exponential. Due to this, in this case it is always the lower integration extreme of the integral over that dominates and, therefore, the results will depend on in any dimension .
We will perform here the integrals in the cases and , leaving the others to the reader (problem 4.1.7). For the time being, we restrict the discussion to the case . In the simplest case, , as well as in the case , it is really neither necessary nor useful to write the integral in the form of given in equation (4.1.3). In the case , with , we may calculate directly the integral in the form shown in (4.1.2),
We may calculate this integral in the complex- plane without difficulty, in the limit . In this case the integral runs over the real line and, if , we should close the circuit with an arc at infinity, of size , in the lower half-plane, where the imaginary part of is negative, so that the argument of the exponential, , has a negative real part. Figure 4.1.6 illustrates the complex- plane, with the integration contours and the poles of the integrand at . In this case the integral is equal to times the value of the residue of the integrand in the lower pole, that is,
If we should close the contour by the other side, the factor multiplying the residue is , and hence we obtain in this case
Defining , we can join the two answers in the final result
We see that the result is finite for all values of , including . If fact, we can verify directly that is finite in this case. As we saw before, we have for the dimensionless function the behavior and, since in the case it holds that , it follows that , which is also finite. The complete equality of the two results corresponds to the choice for the lower extreme on the integral used in the approximate calculation of .
Passing now to the case , in this case we have from equation (4.1.3) that
We can do immediately the integrals over e , obtaining
If we now observe that the integrand is even, we may write this as
where we again wrote the sine in terms of complex exponentials. Each one of these two integrals can be done in the complex- plane, in the limit , in the same way in which we did the integral in the case , closing the circuit in the appropriate way in each case. Doing this we obtain
With this, we have the final result
This function is the Yukawa potential that, in the limit , reduces to the Coulomb potential of electrostatics. Just as in the case , for this function also falls off exponentially for large values of , and it is finite at all points except at , where it diverges. Once more this is compatible with our previous calculation for , since we saw that is finite and, for , we have that , which means that diverges when and therefore . In fact, one can verify that is finite at the origin only for , and that in all the other cases, starting with , it diverges, typically with a negative power of which is characteristic of each dimension . The functions calculated as in the examples above, for various dimensions , are given in the table 4.1.2, which also contains the corresponding asymptotic behaviors for , that is, for , as well as for . The symbols and in this table are Bessel functions.
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 different from , so that there are no important qualitative differences between the two cases. One observes that the propagators in the cases and diverge in the limit . These are what one refers to as infrared divergences, a type of behavior which is characteristic of the lower dimensions, in particular of and . One can verify that the behavior for is also problematic in the case of the calculations of which we did before. We will now examine how behaves in this limit, in each dimension. From equation (4.1.1) we see that always diverges if we make , even on finite lattices, because the term of the sum involving the mode diverges in this limit. We may write for
where in the sum the zero mode is omitted. As we mentioned before, the Gaussian model on the torus indeed has a zero mode in the case , 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 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 on finite lattices in order to study field theories which are massless in the continuum limit. It suffices to recall that is related to the mass by , so that goes to zero in the limit no matter what value is given to . Since we have
we can either cause to have a finite non-vanishing limit, by means of a decrease in given by , or cause to vanish in the limit by means of a decrease in which is faster than . Hence, there is a way to represent zero-mass theories by means of infinite sequences of finite lattices in which is always different from zero, which avoids the divergence of the zero-mode term in the sum that defines . 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
so that we see that this term goes to zero for , is constant for and diverges with for . In all cases these results do not significantly affect the calculations made before for . For the zero-mode term has exactly the same behavior found for the sum, for it is constant while the sum diverges logarithmically and for it goes to zero, while the sum has a finite non-vanishing limit.
In order to have in the continuum limit it suffices to make vary with as
where is some positive constant and some positive number, which we imagine to be small. In this case the zero-mode term is
We see that this term continues to go to zero for , so long as is smaller than . For , however, it becomes divergent as , that is, faster than the sum, while in the situation is similar, since in this case it diverges as , also faster than the sum. Therefore, we see that for and presents in fact infrared divergences similar to those of , when we make . 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 in the cases , when we make 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 one can see that the main contributions to come from the large-momentum terms of the sum, so that the results should not depend on , which is a quantity characteristic of the low-momentum region. Since cannot depend at all on under these conditions, it is clear that its behavior should not change when we make .
The graphs we showed before to illustrate the behavior of as a function of in each dimension were obtained using the value for , but we see now that this is not really a relevant fact. For the cases , the only ones in which converges to a finite value in the limit, one can show (problem 4.1.9) that in fact the limits are completely independent of .
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 or in terms of the dimensionfull function , because in any case we should analyze the statistical correlations among the fields at various points by means of the homogeneous correlation function
We can calculate this function on finite lattices without any trouble and then take the continuum limit. Let us examine then how behaves in this limit. It is clear that, by definition, is always equal to one, both on finite lattices and in the continuum limit. For other values of we saw that is finite in the limit, while diverges. It follows therefore that in the continuum limit vanishes for all non-vanishing values of . Since for the cases of interest, with , is also finite and non-vanishing in the limit, it follows that for the two-point function is also zero for . In short, we have the continuum-limit results
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 , 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 , 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 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.
as reported in one of the tables given in the text.
where , and are unknown constants. Repeat the fitting for several subsets of the data, each with an increasing maximum value of , in some convenient way, thus obtaining successive estimates for these three quantities, for increasing values of . In this fashion, obtain an extrapolation of the result for these three quantities to the continuum limit , with constant . Hint: try to start your fitting with , and and remember that the important thing is to adjust the function for large values of .