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Manifold Theory: (W)holes and Boundaries

Luce Irigaray, in her famous article ``Is the Subject of Science Sexed?'', pointed out that

the mathematical sciences, in the theory of wholes [théorie des ensembles], concern themselves with closed and open spaces ... They concern themselves very little with the question of the partially open, with wholes that are not clearly delineated [ensembles flous], with any analysis of the problem of borders [bords] ...gif
In 1982, when Irigaray's essay first appeared, this was an incisive criticism: differential topology has traditionally privileged the study of what are known technically as ``manifolds without boundary''. However, in the past decade, under the impetus of the feminist critique, some mathematicians have given renewed attention to the theory of ``manifolds with boundary'' [Fr. variétés à bord]. gif Perhaps not coincidentally, it is precisely these manifolds that arise in the new physics of conformal field theory, superstring theory and quantum gravity.

In string theory, the quantum-mechanical amplitude for the interaction of n closed or open strings is represented by a functional integral (basically, a sum) over fields living on a two-dimensional manifold with boundary. gif In quantum gravity, we may expect that a similar representation will hold, except that the two-dimensional manifold with boundary will be replaced by a multidimensional one. Unfortunately, multidimensionality goes against the grain of conventional linear mathematical thought, and despite a recent broadening of attitudes (notably associated with the study of multidimensional nonlinear phenomena in chaos theory), the theory of multidimensional manifolds with boundary remains somewhat underdeveloped. Nevertheless, physicists' work on the functional-integral approach to quantum gravity continues apacegif, and this work is likely to stimulate the attention of mathematicians. gif

As Irigaray anticipated, an important question in all of these theories is: Can the boundary be transgressed (crossed), and if so, what happens then? Technically this is known as the problem of ``boundary conditions''. At a purely mathematical level, the most salient aspect of boundary conditions is the great diversity of possibilities: for example, ``free b.c.'' (no obstacle to crossing), ``reflecting b.c.'' (specular reflection as in a mirror), ``periodic b.c.'' (re-entrance in another part of the manifold), and ``antiperiodic b.c.'' (re-entrance with tex2html_wrap_inline1413 twist). The question posed by physicists is: Of all these conceivable boundary conditions, which ones actually occur in the representation of quantum gravity? Or perhaps, do all of them occur simultaneously and on an equal footing, as suggested by the complementarity principle?gif

At this point my summary of developments in physics must stop, for the simple reason that the answers to these questions -- if indeed they have univocal answers -- are not yet known. In the remainder of this essay, I propose to take as my starting point those features of the theory of quantum gravity which are relatively well established (at least by the standards of conventional science), and attempt to draw out their philosophical and political implications.



next up previous
Next: Transgressing the Boundaries: Towards Up: Transgressing the Boundaries: Towards Previous: Differential Topology and Homology



Daniel Sleator
Thu Jun 6 15:34:37 EDT 1996