Analysis of the Situation

Let us consider what has been successfully obtained within the formalism presented in the previous chapters. First of all, in chapter 2 we obtained a complete and correct realization of the classical theory of fields using the lattice and the continuum limit from it. In addition to this, as was shown along chapters from 3 to 5, the same formalism in the quantum case for dimension $d=1$ succeeded in producing a complete and correct realization of non-relativistic quantum mechanics. In the case of dimensions $d\geq 2$ we can go so far as to say that the formalism can be used to produce a fairly complete and constructive mathematical definition of the quantum theory of fields.

The definition of the Gaussian model on the Euclidean lattice results in the correct set of correlation functions for that simple model. A closer examination of the structure of the two-point function, which is the only non-trivial function of the model, revealed some rather surprising aspects of its behavior, but these issues were completely resolved by the introduction of block variables, leading to a completely satisfactory physical interpretation of the correlation functions of the theory. While it is a widely accepted position that all the observables in quantum field theory must be averages over spacial regions or blocks, the same does not seem to be so clear with respect to averages over the temporal direction. However, in a relativistic theory it is essential that the averages be over both the spacial and the temporal dimensions, because relativistic transformations mix the spacial and temporal coordinates, so that there can be no invariant meaning to a purely spacial average.

The introduction of external sources provided us with a solid handle to probe into the behavior of the models, both in the classical case and in the quantum case. It leads in the usual way to the introduction of the functional generators of the correlation functions, which we developed directly on the Euclidean lattice, and ultimately to the concept of the effective action. We managed to establish a rather complete physical interpretation of the effective action, not only as the functional generator of irreducible correlation functions, but also as a shorthand for the response of the models to the introduction of external sources. In either capacity the effective action can be seen as a useful condensation of the complete physical content of the model. Due to all this the effective action helps significantly with the interpretation of the classical limit of the quantum theory.

An exploration of the mathematical character of the dimensionless field configurations that contribute in a dominant way to the most important observables of the theory resulted in the unexpected and even surprising conclusion that these functions are typically discontinuous at all the points of their domains. One may say that the set of all continuous configurations is of zero measure within the ensemble of the theory, in the sense that their exclusion from the ensemble would not affect the expectation values which are physically relevant. However, most of the more direct consequences of the discontinuity of the field configurations will be found only in the second volume of this series.

Although this situation leads to the usual non-differentiable but still continuous behavior of the paths in the path-integral approach to non-relativistic quantum mechanics [5], in the case of quantum field theory with $d\geq 3$ it leads to infinite discontinuities of the dimensionfull field. This means that, while in the quantum-mechanical case it is possible to represent the configurations by random walks, any such representation in the case of quantum field theory is incorrect unless one takes the continuum limit in one of the asymmetrical ways described in section 5.3, in which case one looses in one fell blow both relativistic invariance and the complete structure of particle states, as discussed in that section.

Finally, a quite complete realization on the lattice of the concept of energy was also obtained. The situation regarding the energy of the vacuum state is qualitatively similar to the corresponding situation in the traditional approach. We also managed to define a complete set of particle states, which have the correct energy and momenta. In the continuum limit there are both virtual particles and real physical particles, which are clearly identified by the all-important on-shell condition, which must be satisfied by states representing relativistic particles. States that correspond to virtual particles can be shown to become energetically degenerate with the vacuum in the limit.

In this formalism the particles are closely associated to the normal modes of oscillation of the cavity represented by the lattice, and are thus more readily represented in momentum space than in position space. Elementary particles are therefore extended objects, not point-like objects. From a conceptual point of view the particles should really be identified with exchanges of packets of energy between external sources and the quantum field within the cavity. These exchanges do not happen in a sharply localized way, but over the whole extent of the cavity. In the non-relativistic limit, in which one makes $T\rightarrow \infty$ while keeping $L$ finite, this association of relativistic particles with the $d$-dimensional cavity is mapped onto a corresponding association of the physical particles with the modes of the $(d-1)$-dimensional cavity that is left after the limit. This is a direct consequence of the non-relativistic limit of the on-shell condition, which establishes the expected values for the energy of the particles in terms of the momenta of the modes of the $(d-1)$-dimensional cavity.

We see, therefore, that the construction of the theory on the lattice is successful in many respects. There remains, however, one main issue to be dealt with, because we arrived at the unexpected result that neither the vacuum state nor any of the particle states are eigenstates of the blocked Hamiltonian observable, which is in sharp contrast with the situation in the case of non-relativistic quantum mechanics, in which we do find that the vacuum is an eigenstate of the blocked Hamiltonian. Due to this the construction does not lead to the usual structure of states and operators in a Hilbert space, as one might have expected it would do. Since there is a well-known formalism due to Osterwalder and Schrader [6] dealing precisely with the construction of a Hilbert space structure starting from the lattice structure, we must now compare our results with those of that formalism.

It is the examination of the asymmetrical continuum limits discussed in section 5.3, which first reduce the structure of the theory to the quantum mechanics of a finite number of coupled degrees of freedom, and only after that may let the number of degrees of freedom tend to infinity, that leads us to make contact with the Osterwalder-Schrader formalism. In this formalism the authors establish necessary and sufficient conditions for the construction of a positive-norm Hilbert space from the discrete structure defined on a lattice. When one examines the development of the argument in that formalism, one observes that it implicitly assumes that $N_{T}=\infty$ from the very beginning, so that the applicability of its conclusions to lattice systems such as the ones discussed in this book is limited to those that result from the asymmetrical limits.

Although the condition that $N_{T}=\infty$ is not explicit within the hypothesis of the formalism, it is implied by the operations that are performed during the development of the argument. One of the basic hypothesis of that formalism, which is given explicitly, is that the lattice must be separable into two disjoint sets by means of the definition of a $(d-1)$-dimensional boundary surface, which defines a moment in time. This eliminates the possibility of the use of periodical boundary conditions in the temporal direction of the lattice, as we have done regularly in this book. In addition to this, during the development one requires the possibility of performing temporal translations under which the system should be invariant. Without periodical boundary conditions this is only possible if the lattice is infinite in the temporal direction from the very beginning.

One perceives that the formalism of Osterwalder and Schrader is built around the idea that the Hamiltonian is to be the generator of time translations, and assumes that states and operators are to be defined at completely sharp instants of time. The formalism assumes that there is a Hilbert space and that there is a Hamiltonian, both with the usual properties found in non-relativistic quantum mechanics, and proceeds to construct them. In order to do this it must require that $N_{T}=\infty$ from the start. One can have either a finite or an infinite $N_{L}$, but one absolutely must have an infinite $N_{T}$. By contrast, the states defined here are intrinsically $d$-dimensional objects, not $(d-1)$-dimensional objects existing at a sharply defined time. On the same token, observables can only be measured on the extent of $d$-dimensional boxes, not at sharply defined times or spacial positions.

It is important to emphasize that there is in fact no conflict between the results that we found here and those of the Osterwalder-Schrader formalism, because if we assume that the limits are to be taken in the asymmetrical way, then it is in fact possible to adjust things so that the vacuum becomes an eigenstate of the Hamiltonian, as we have shown in section 5.3. What is at issue here, due to the nature of the results we found, is not simply the existence or not of Hilbert spaces that can be associated to the structure of the theory, but their usefulness in representing systems of fundamental quantum fields of physical interest, having relativistic invariance and that contain relativistic particles with finite and non-vanishing additional energies above the energy of the vacuum.

One is inevitably led, then, to consider how the definition of the theory could possibly be changed in order to recover the usual Hilbert-space structure, without violating the basic precepts relating to the definition of the mathematical structure of a physical theory, that were discussed in the first chapter. However, it seems that any trial at this leads to some physically unacceptable loss. Taking the asymmetrical limit does the job, but leads to loss of relativistic invariance, and to the collapse of the whole structure of particle states into the vacuum. This is similar to trying to redefine states and observables at sharply defined times, as is done in the Osterwalder-Schrader formalism, and that leads to the loss of the connection between the particles and the modes of the $d$-dimensional cavity. Since this connection and the on-shell condition lead naturally, in the non-relativistic limit, to a corresponding connection between physical particles and the modes of the remaining $(d-1)$-dimensional cavities, the loss is a serious one.

It is a well-known experimental fact that physical particles are closely connected to the modes of oscillation of the corresponding fields when they are within a cavity. This can be shown experimentally by the introduction of excited atomic states into high-quality electromagnetic cavities [7]. If the cavity is tuned so that none of its modes has the frequency of the photon that the atom must emit in order to decay, then its spontaneous decay can be very effectively delayed or prevented. If, on the other hand, the cavity is tuned to the frequency of the photon, then the spontaneous decay can be stimulated, or a certain mode of decay can be stimulated at the expense of others. This shows in a decisive way that the photons, the particles of the electromagnetic field, are excitations of the modes of oscillation of the electromagnetic field within the cavity. The photons are clearly identified with packets of energy that are exchanged between an external source, in this case the atom, and the modes of oscillation of the field within the cavity.

The only way in which it seems possible to keep the symmetry between the temporal and spacial directions and still be within the hypothesis of the Osterwalder-Schrader formalism is to start with lattices that are infinite in all directions. Starting with both $N_{T}$ and $N_{L}$ infinite would be compatible with the aforementioned hypothesis of that formalism, but is equivalent to giving up the constructive definition of the theory by a limiting process from finite and discrete mathematical systems. We regard this as philosophically unacceptable, since adopting such a definition would rule out any truly constructive analysis of the structure of the theory. Note that it would also rule out any type of finite computational simulation as a calculational tool for the theory.

It seems to us that there is no reasonable way out of this situation, and that we must accept the fact that the traditional Hilbert-space formalism is not an appropriate tool for the description of relativistic quantum theory at the most fundamental level. A constructive definition which could include such a formalism at the fundamental level is, it would seem, still to be exhibited. As we will see in our continued explorations in the next volume, the usual perturbative theory can be formulated entirely on the lattice, without any reference to Hilbert spaces, and hence all the calculations that can be made in that formalism can also be performed within the lattice formalism, possibly with some quantitative differences, so that not much is lost in the perturbative front. In fact, something is gained, due to a much clearer and more solid insight into the mathematical structure of the theory.

Note that the loss of the usual Hilbert-space formalism is not really a physical loss, but rather a mathematical one. The fundamental physical principle underlying the quantum theory, the principle of uncertainty, is not lost. In fact, one can say that the exact opposite is true, and that quantum field theory contains a higher degree of uncertainty than non-relativistic quantum mechanics, as was argued by Landau and Peierls a long time ago [8]. This is reflected in the violent fluctuations undergone by the fundamental field, leading to its being typically a completely discontinuous function, and also causing the Hilbert-space formalism to cease to be an appropriate tool for the description of the structure of the theory. It is possible, however, that the Hilbert-space structure can be recovered as an approximation, under certain conditions. For example, this is certainly to be expected in the non-relativistic limit of the theory.

One situation in which one would expect that an approximate Hilbert-space structure can be implemented would be for an effective theory using block variables and an energy cutoff. This is made reasonable due to the fact that the behavior we see in quantum field theory is clearly related to the large fluctuations of the fields, and these become much smaller for the block variables. As we saw in section 4.3, the larger the blocks, the smaller the fluctuations undergone by the block variables, so that large blocks are associated to the classical limit of the theory. If in some specific circumstance we have phenomena involving only long wavelengths and long-range correlations, then we may use large blocks in order to analyze that situation, and hence the block variables will fluctuate very little, leading to a semi-classical or even to a classical limit, as the case may be.

However, we do loose something with the Hilbert-space formalism, namely its description of the temporal evolution process, in the usual way that works so well for non-relativistic quantum mechanics. We are faced therefore with the challenge of finding out how to define and handle the evolution in time of $d$-dimensional objects, which do not correspond to sharply-defined moments of time.