The Canonical of a priori and a posteriori Variational Calculus as Phenomenologically Driven. Note Quote.


The expression variational calculus usually identifies two different but related branches in Mathematics. The first aimed to produce theorems on the existence of solutions of (partial or ordinary) differential equations generated by a variational principle and it is a branch of local analysis (usually in Rn); the second uses techniques of differential geometry to deal with the so-called variational calculus on manifolds.

The local-analytic paradigm is often aimed to deal with particular situations, when it is necessary to pay attention to the exact definition of the functional space which needs to be considered. That functional space is very sensitive to boundary conditions. Moreover, minimal requirements on data are investigated in order to allow the existence of (weak) solutions of the equations.

On the contrary, the global-geometric paradigm investigates the minimal structures which allow to pose the variational problems on manifolds, extending what is done in Rn but usually being quite generous about regularity hypotheses (e.g. hardly ever one considers less than C-objects). Since, even on manifolds, the search for solutions starts with a local problem (for which one can use local analysis) the global-geometric paradigm hardly ever deals with exact solutions, unless the global geometric structure of the manifold strongly constrains the existence of solutions.



A further a priori different approach is the one of Physics. In Physics one usually has field equations which are locally given on a portion of an unknown manifold. One thence starts to solve field equations locally in order to find a local solution and only afterwards one tries to find the maximal analytical extension (if any) of that local solution. The maximal extension can be regarded as a global solution on a suitable manifold M, in the sense that the extension defines M as well. In fact, one first proceeds to solve field equations in a coordinate neighbourhood; afterwards, one changes coordinates and tries to extend the found solution out of the patches as long as it is possible. The coordinate changes are the cocycle of transition functions with respect to the atlas and they define the base manifold M. This approach is essential to physical applications when the base manifold is a priori unknown, as in General Relativity, and it has to be determined by physical inputs.

Luckily enough, that approach does not disagree with the standard variational calculus approach in which the base manifold M is instead fixed from the very beginning. One can regard the variational problem as the search for a solution on that particular base manifold. Global solutions on other manifolds may be found using other variational principles on different base manifolds. Even for this reason, the variational principle should be universal, i.e. one defines a family of variational principles: one for each base manifold, or at least one for any base manifold in a “reasonably” wide class of manifolds. The strong requirement, which is physically motivated by the belief that Physics should work more or less in the same way regardless of the particular spacetime which is actually realized in Nature. Of course, a scenario would be conceivable in which everything works because of the particular (topological, differentiable, etc.) structure of the spacetime. This position, however, is not desirable from a physical viewpoint since, in this case, one has to explain why that particular spacetime is realized (a priori or a posteriori).

In spite of the aforementioned strong regularity requirements, the spectrum of situations one can encounter is unexpectedly wide, covering the whole of fundamental physics. Moreover, it is surprising how the geometric formalism is effectual for what concerns identifications of basic structures of field theories. In fact, just requiring the theory to be globally well-defined and to depend on physical data only, it often constrains very strongly the choice of the local theories to be globalized. These constraints are one of the strongest motivations in choosing a variational approach in physical applications. Another motivation is a well formulated framework for conserved quantities. A global- geometric framework is a priori necessary to deal with conserved quantities being non-local.

In the modem perspective of Quantum Field Theory (QFT) the basic object encoding the properties of any quantum system is the action functional. From a quantum viewpoint the action functional is more fundamental than field equations which are obtained in the classical limit. The geometric framework provides drastic simplifications of some key issues, such as the definition of the variation operator. The variation is deeply geometric though, in practice, it coincides with the definition given in the local-analytic paradigm. In the latter case, the functional derivative is usually the directional derivative of the action functional which is a function on the infinite-dimensional space of fields defined on a region D together with some boundary conditions on the boundary ∂D. To be able to define it one should first define the functional space, then define some notion of deformation which preserves the boundary conditions (or equivalently topologize the functional space), define a variation operator on the chosen space, and, finally, prove the most commonly used properties of derivatives. Once one has done it, one finds in principle the same results that would be found when using the geometric definition of variation (for which no infinite dimensional space is needed). In fact, in any case of interest for fundamental physics, the functional derivative is simply defined by means of the derivative of a real function of one real variable. The Lagrangian formalism is a shortcut which translates the variation of (infinite dimensional) action functionals into the variation of the (finite dimensional) Lagrangian structure.

Another feature of the geometric framework is the possibility of dealing with non-local properties of field theories. There are, in fact, phenomena, such as monopoles or instantons, which are described by means of non-trivial bundles. Their properties are tightly related to the non-triviality of the configuration bundle; and they are relatively obscure when regarded by any local paradigm. In some sense, a local paradigm hides global properties in the boundary conditions and in the symmetries of the field equations, which are in turn reflected in the functional space we choose and about which, it being infinite dimensional, we do not know almost anything a priori. We could say that the existence of these phenomena is a further hint that field theories have to be stated on bundles rather than on Cartesian products. This statement, if anything, is phenomenologically driven.

When a non-trivial bundle is involved in a field theory, from a physical viewpoint it has to be regarded as an unknown object. As for the base manifold, it has then to be constructed out of physical inputs. One can do that in (at least) two ways which are both actually used in applications. First of all, one can assume the bundle to be a natural bundle which is thence canonically constructed out of its base manifold. Since the base manifold is identified by the (maximal) extension of the local solutions, then the bundle itself is identified too. This approach is the one used in General Relativity. In these applications, bundles are gauge natural and they are therefore constructed out of a structure bundle P, which, usually, contains extra information which is not directly encoded into the spacetime manifolds. In physical applications the structure bundle P has also to be constructed out of physical observables. This can be achieved by using gauge invariance of field equations. In fact, two local solutions differing by a (pure) gauge transformation describe the same physical system. Then while extending from one patch to another we feel free both to change coordinates on M and to perform a (pure) gauge transformation before glueing two local solutions. Then coordinate changes define the base manifold M, while the (pure) gauge transformations form a cocycle (valued in the gauge group) which defines, in fact, the structure bundle P. Once again solutions with different structure bundles can be found in different variational principles. Accordingly, the variational principle should be universal with respect to the structure bundle.

Local results are by no means less important. They are often the foundations on which the geometric framework is based on. More explicitly, Variational Calculus is perhaps the branch of mathematics that possibilizes the strongest interaction between Analysis and Geometry.