Geometry and Localization: An Unholy Alliance? Thought of the Day 95.0

There are many misleading metaphors obtained from naively identifying geometry with localization. One which is very close to that of String Theory is the idea that one can embed a lower dimensional Quantum Field Theory (QFT) into a higher dimensional one. This is not possible, but what one can do is restrict a QFT on a spacetime manifold to a submanifold. However if the submanifold contains the time axis (a ”brane”), the restricted theory has too many degrees of freedom in order to merit the name ”physical”, namely it contains as many as the unrestricted; the naive idea that by using a subspace one only gets a fraction of phase space degrees of freedom is a delusion, this can only happen if the subspace does not contain a timelike line as for a null-surface (holographic projection onto a horizon).

The geometric picture of a string in terms of a multi-component conformal field theory is that of an embedding of an n-component chiral theory into its n-dimensional component space (referred to as a target space), which is certainly a string. But this is not what modular localization reveals, rather those oscillatory degrees of freedom of the multicomponent chiral current go into an infinite dimensional Hilbert space over one localization point and do not arrange themselves according according to the geometric source-target idea. A theory of this kind is of course consistent but String Theory is certainly a very misleading terminology for this state of affairs. Any attempt to imitate Feynman rules by replacing word lines by word sheets (of strings) may produce prescriptions for cooking up some mathematically interesting functions, but those results can not be brought into the only form which counts in a quantum theory, namely a perturbative approach in terms of operators and states.

String Theory is by no means the only area in particle theory where geometry and modular localization are at loggerheads. Closely related is the interpretation of the Riemann surfaces, which result from the analytic continuation of chiral theories on the lightray/circle, as the ”living space” in the sense of localization. The mathematical theory of Riemann surfaces does not specify how it should be realized; if its refers to surfaces in an ambient space, a distinguished subgroup of Fuchsian group or any other of the many possible realizations is of no concern for a mathematician. But in the context of chiral models it is important not to confuse the living space of a QFT with its analytic continuation.

Whereas geometry as a mathematical discipline does not care about how it is concretely realized the geometrical aspects of modular localization in spacetime has a very specific geometric content namely that which can be encoded in subspaces (Reeh-Schlieder spaces) generated by operator subalgebras acting onto the vacuum reference state. In other words the physically relevant spacetime geometry and the symmetry group of the vacuum is contained in the abstract positioning of certain subalgebras in a common Hilbert space and not that which comes with classical theories.

Catastrophe

Since, natural phenomena are continuously battered by perturbations, any thematic of the fundament that classifies critical points of smooth functions becomes the defining parameter of catastrophe. If a natural system is defined by a function of state variables, then the perturbations are represented by control parameters on which the function depends. An unfolding of a function is such a family: it is a smooth function of the state variables with the parameters satisfying a specific condition. Catastrophe’s aim is then to detect properties of a function by studying its unfoldings.

Thom studied the continuous crossing from a variety (space) to another, the connections through common boundaries and points between spaces even endowed with different dimensions (a research on the so called “cobordism” (1) which yielded him the Field Medal in 1958), until he singled out few universal forms, that is mathematical objects representing catastrophes or abrupt, although continuous, transitions of forms: specific singularities appearing when an object is submitted to bonds, such as restrictions with regard to its ordinary dimensions, that it accepts except in particular points where it offers resistance by concentrating there, so to say, its structure. The theory is used to classify how stable equilibria change when parameters are varied, with points in parameter space at which qualitative changes affects behavior termed catastrophe points. Catastrophe theory should apply to any gradient system where the force can be written as the negative gradient of a potential, and the points where the gradient vanishes are what the theory prefers degenerate points. There are seven elementary types of catastrophes or generic singularities of an application and Thom decided to study their applications in caustics, surfaces lit according to different angle shots, reflections and refractions. Initially catastrophe theory was of use just to explain caustic formation and only afterwards many other phenomena, but without yielding quantitative solutions and exact predictions, rather qualitatively framing situations that were uncontrollable by only reductionistic quantitative methods summing up elementary units. The study of forms in irregular, accidental and even chaotic situations had truly in advance led scientists like Poincaré and Hadamard, to single out structurally invariable catastrophic evolutions in the most disparate phenomena, in terms of divergences due to sensitive dependence on little variations of the initial conditions. In such cases there were not exact laws, rather evolutionary asymptotic tendencies, which did not allow exact predictions, in case only statistic ones. While when exact predictions are possible, in terms of strict laws and explicit equations, the catastrophe ceases.

For Thom, catastrophe was a methodology. He says,

Mathematicians should see catastrophe theory as just a part of the theory of local singularities of smooth morphisms, or, if they are interested in the wider ambitions of this theory, as a dubious methodology concerning the stability or instability of natural systems….the whole of qualitative dynamics, all the ‘chaos’ theories talked about so much today, depend more or less on it.

Thom gets more philosophical when it comes to the question of morphogenesis. Stability for Thom is a natural condition to place upon mathematical models for processes in nature because the conditions under which such processes take place can never be duplicated and therefore must be invariant under small perturbations and hence stable. what makes morphogenesis interesting for Thom is the fact that locally, as the transition proceeds, the parameter varies, from a stable state of a vector field to an unstable state and back to a stable state by means of a process which locally models system’s morphogenesis. Furthermore, what is observed in a process undergoing morphogenesis is precisely the shock wave and resulting configuration of chreods (2) separated by strata of the shockwave, at each interval of time and over intervals of observation time. It then follows “that to classify an observed phenomenon or to support a hypothesis about the local underlying dynamic, we need in principle only observe the process, study the observed catastrophe or discontinuity set and try to relate it to one of the finitely many universal catastrophe sets, which would become then our main object of interest. Even if a process depends on a large number of physical parameters, as long as it is described by the gradient model, its description would involve one of seven elementary catastrophes; in particular one can give a relatively simple mathematical description of such apparently complicated processes even if one does not know what the relevant physical parameters are or what the physical mechanism of the process is. According to Thom, “if we consider an unfolding, we can obtain a qualitative intelligence about the behaviors of a system in the neighborhood of an unstable equilibrium point. this idea was not accepted widely and was criticized by applied mathematicians because for them only numerical exactness allowed prediction and therefore efficient action. After the work of Grothendieck, it is known that the theory of singularity unfolding is a particular case of a general category, the theory of flat deformations of an analytic set and for flat local deformations of an analytic set only the hyper surface case has a smooth unfolding of finite dimension. For Thom, this meant the if we wanted to continue the scientific domain of calculable exact laws, we would be justified in considering the instance where an analytic process leads to a singularity of codimension one in internal variables. Might we then not expect that the process be diffused and subsequently propagated in the unfolding according to a mode that is to be defined? Such an argument allows one to think that the Wignerian domain of exact laws can be extended into a region where physical processes are no longer calculable but where analytic continuation remains qualitatively valid.

Anyway, catastrophe theory studies forms as qualitative discontinuities though on a continuous substrate. In any case forms as mental facts are immersed in a matter which is still a thought object. The more you try to analyze it the more it appears as a fog, revealing a more and more complex and inexhaustible weaving the more it refines itself through the forms it assumes. In fact complexity is more and more ascertained until a true enigma is reached when you once for all want to define reality as a universe endowed with a high number of dimensions and then object of mental experiences to which even objective phenomena are at the end concretely reduced. Concrete reality is yet more evident than a scientific explanation and naïve ontology appears more concrete than the scientific one. It is steady and universal, while the latter is always problematic and revisable. Besides, according to Bachelard, while naïve explanation is immediately reflected into the ordinary language which is accessible to everybody, the claimed scientific explanation goes with its jargon beyond immediate experience, away from the life world which only we can know immediately.

As for example, the continuous character of reality, which Thom entrusts to a world intuition as a frame of the phenomenological discontinuities themselves, is instead contradicted by the present tendency to reduce all to discrete units of information (bits) of modern computing. Of course it has a practical value: an animal individuating a prey perceives it as an entity which is absolutely distinct from its environment, just as we discretize linguistic phonemata to learn speaking without confounding them. Yet a continuous background remains, notwithstanding the tendency of our brains to discretize. Such background is for example constituted by space and time. Continuum is said an illusion as exemplified by a film which appears continuous to us, while it is made of discrete frames. Really it is an illusion but with a true mental base, otherwise it would not arise at all, and such base is just the existence of continuum. Really we perceive continuum but need discreteness, finiteness in order to keep things under control. Anyway quantum mechanics seems to introduce discreteness in absolute terms, something we do not understand but which is operatively valid, as is shown by the possibility to localize or delocalize a wave packet by simply varying the value distributions of complementary variables as position and momentum or time and energy, according to Heisenberg’s principle of indetermination. Anyway, also the apparent quantum discontinuity hides a continuity which, always according to Heisenberg’s principle, may be only obscured and not cancelled in several phenomena. It is difficult to conceive but not monstrous. The hypothesis according to which we are finite and discrete in our internal structure is afterwards false for we are more than that. We have hundreds billions of neurons, which are in continuous movement, as they are constituted by molecules continuously vibrating in the space, so giving place to infinite possible variations in a considerable dimensions number, even though we are reduced to the smallest possible number of states and dimensions to deal with the system under study, according to a technical and algorithmic thought which is operatively effective, certainly practically motivated but unidentifiable with reality.

(1) Two manifolds M and N are said to be cobordant if their disjoint union is the boundary of some other manifold. Given the extreme difficulty of the classification of manifolds it would seem very unlikely that much progress could be made in classifying manifolds up to cobordism. However, René Thom, in his remarkable, if unreadable, 1954 paper (French), gave the full solution to this problem for unoriented manifolds, as well as many powerful insights into the methods for solving it in the cases of manifolds with additional structure. The key step was the reduction of the cobordism problem to a homotopy problem, although the homotopy problem is still far from trivial. This was later generalized by Lev Pontrjagin, and this result is now known as the Thom-Pontrjagin theorem.

(2) Every natural process decomposes into structurally stable islands, the chreods. The set of chreods and the multidimensional syntax controlling their positions constitute the semantic model. When the chreod is considered as a word of this multidimensional language, the meaning (signification) of this word is precisely that of the global topology of the associated attractor (or attractors) and of the catastrophes that it (or they) undergo. In particular, the signification of a given attractor is defined by the geometry of its domain of existence on the space of external variables and the topology of the regulation catastrophes bounding that domain. One result of this is that the signification of a form (chreod) manifests itself only by the catastrophes that create or destroy it. This gives the axiom dear to the formal linguists: that the meaning of a word is nothing more than the use of the word; this is also the axiom of the “bootstrap” physicists, according to whom a particle is completely defined by the set of interactions in which it participates.