by Alan Sokal
We can no longer speak of the behaviour of the particle independently of the process of observation. As a final consequence, the natural laws formulated mathematically in quantum theory no longer deal with the elementary particles themselves but with our knowledge of them. Nor is it any longer possible to ask whether or not these particles exist in space and time objectively ...
When we speak of the picture of nature in the exact science of our age, we do not mean a picture of nature so much as a picture of our relationships with nature.... Science no longer confronts nature as an objective observer, but sees itself as an actor in this interplay between man [sic] and nature. The scientific method of analysing, explaining and classifying has become conscious of its limitations, which arise out of the fact that by its intervention science alters and refashions the object of investigation. In other words, method and object can no longer be separated.9, 10
Along the same lines, Niels Bohr wrote:
An independent reality in the ordinary physical sense can ... neither be ascribed to the phenomena nor to the agencies of observation.11
Stanley Aronowitz has convincingly traced this worldview to the crisis of liberal hegemony in Central Europe in the years prior and subsequent to World War I.12,13
A second important aspect of quantum mechanics is its principle of complementarity or dialecticism. Is light a particle or a wave? Complementarity “is the realization that particle and wave behavior are mutually exclusive, yet that both are necessary for a complete description of all phenomena.”14 More generally, notes Heisenberg,
the different intuitive pictures which we use to describe atomic systems, although fully adequate for given experiments, are nevertheless mutually exclusive. Thus, for instance, the Bohr atom can be described as a small-scale planetary system, having a central atomic nucleus about which the external electrons revolve. For other experiments, however, it might be more convenient to imagine that the atomic nucleus is surrounded by a system of stationary waves whose frequency is characteristic of the radiation emanating from the atom. Finally, we can consider the atom chemically. ... Each picture is legitimate when used in the right place, but the different pictures are contradictory and therefore we call them mutually complementary.15
And once again Bohr:
A complete elucidation of one and the same object may require diverse points of view which defy a unique description. Indeed, strictly speaking, the conscious analysis of any concept stands in a relation of exclusion to its immediate application.16
This foreshadowing of postmodernist epistemology is by no means coincidental. The profound connections between complementarity and deconstruction have recently been elucidated by Froula17 and Honner,18 and, in great depth, by Plotnitsky.19, 20, 21
A third aspect of quantum physics is discontinuity or rupture: as Bohr explained,
[the] essence [of the quantum theory] may be expressed in the so-called quantum postulate, which attributes to any atomic process an essential discontinuity, or rather individuality, completely foreign to the classical theories and symbolized by Planck’s quantum of action.22
A half-century later, the expression “quantum leap” has so entered our everyday vocabulary that we are likely to use it without any consciousness of its origins in physical theory.
Finally, Bell’s theorem23 and its recent generalizations24 show that an act of observation here and now can affect not only the object being observed – as Heisenberg told us – but also an object arbitrarily far away (say, on Andromeda galaxy). This phenomenon – which Einstein termed “spooky” – imposes a radical reevaluation of the traditional mechanistic concepts of space, object and causality,25 and suggests an alternative worldview in which the universe is characterized by interconnectedness and (w)holism: what physicist David Bohm has called “implicate order”.26 New Age interpretations of these insights from quantum physics have often gone overboard in unwarranted speculation, but the general soundness of the argument is undeniable.27 In Bohr’s words, “Planck’s discovery of the elementary quantum of action ... revealed a feature of wholeness inherent in atomic physics, going far beyond the ancient idea of the limited divisibility of matter.”28
Hermeneutics of Classical General Relativity
In the Newtonian mechanistic worldview, space and time are distinct and absolute.29 In Einstein’s special theory of relativity (1905), the distinction between space and time dissolves: there is only a new unity, four-dimensional space-time, and the observer’s perception of “space” and “time” depends on her state of motion.30 In Hermann Minkowski’s famous words (1908):
Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.31
Nevertheless, the underlying geometry of Minkowskian space-time remains absolute.32
It is in Einstein’s general theory of relativity (1915) that the radical conceptual break occurs: the space-time geometry becomes contingent and dynamical, encoding in itself the gravitational field. Mathematically, Einstein breaks with the tradition dating back to Euclid (and which is inflicted on high-school students even today!), and employs instead the non-Euclidean geometry developed by Riemann. Einstein’s equations are highly nonlinear, which is why traditionally-trained mathematicians find them so difficult to solve.33 Newton’s gravitational theory corresponds to the crude (and conceptually misleading) truncation of Einstein’s equations in which the nonlinearity is simply ignored. Einstein’s general relativity therefore subsumes all the putative successes of Newton’s theory, while going beyond Newton to predict radically new phenomena that arise directly from the nonlinearity: the bending of starlight by the sun, the precession of the perihelion of Mercury, and the gravitational collapse of stars into black holes.
General relativity is so weird that some of its consequences – deduced by impeccable mathematics, and increasingly confirmed by astrophysical observation – read like science fiction. Black holes are by now well known, and wormholes are beginning to make the charts. Perhaps less familiar is Gödel’s construction of an Einstein space-time admitting closed timelike curves: that is, a universe in which it is possible to travel into one’s own past!34
Thus, general relativity forces upon us radically new and counterintuitive notions of space, time and causality;35, 36, 37, 38 so it is not surprising that it has had a profound impact not only on the natural sciences but also on philosophy, literary criticism, and the human sciences. For example, in a celebrated symposium three decades ago on Les Langages Critiques et les Sciences de l’Homme, Jean Hyppolite raised an incisive question about Jacques Derrida’s theory of structure and sign in scientific discourse:
When I take, for example, the structure of certain algebraic constructions [ensembles], where is the center? Is the center the knowledge of general rules which, after a fashion, allow us to understand the interplay of the elements? Or is the center certain elements which enjoy a particular privilege within the ensemble? ... With Einstein, for example, we see the end of a kind of privilege of empiric evidence. And in that connection we see a constant appear, a constant which is a combination of space-time, which does not belong to any of the experimenters who live the experience, but which, in a way, dominates the whole construct; and this notion of the constant – is this the center?39
Derrida’s perceptive reply went to the heart of classical general relativity:
The Einsteinian constant is not a constant, is not a center. It is the very concept of variability – it is, finally, the concept of the game. In other words, it is not the concept of something – of a center starting from which an observer could master the field – but the very concept of the game...40
In mathematical terms, Derrida’s observation relates to the invariance of the Einstein field equation Gµυ= 8πGTµυ under nonlinear space-time diffeomorphisms (self-mappings of the space-time manifold which are infinitely differentiable but not necessarily analytic). The key point is tha
t this invariance group “acts transitively”: this means that any space-time point, if it exists at all, can be transformed into any other. In this way the infinite-dimensional invariance group erodes the distinction between observer and observed; the π of Euclid and the G of Newton, formerly thought to be constant and universal, are now perceived in their ineluctable historicity; and the putative observer becomes fatally decentered, disconnected from any epistemic link to a space-time point that can no longer be defined by geometry alone.
Quantum Gravity: String, Weave or Morphogenetic Field?
However, this interpretation, while adequate within classical general relativity, becomes incomplete within the emerging postmodern view of quantum gravity. When even the gravitational field – geometry incarnate – becomes a non-commuting (and hence nonlinear) operator, how can the classical interpretation of Gµυ as a geometric entity be sustained? Now not only the observer, but the very concept of geometry, becomes relational and contextual.
The synthesis of quantum theory and general relativity is thus the central unsolved problem of theoretical physics;41 no one today can predict with confidence what will be the language and ontology, much less the content, of this synthesis, when and if it comes. It is, nevertheless, useful to examine historically the metaphors and imagery that theoretical physicists have employed in their attempts to understand quantum gravity.
The earliest attempts – dating back to the early 1960s – to visualize geometry on the Planck scale (about 10–33 centimeters) portrayed it as “space-time foam”: bubbles of space-time curvature, sharing a complex and ever-changing topology of interconnections.42 But physicists were unable to carry this approach farther, perhaps due to the inadequate development at that time of topology and manifold theory (see below).
In the 1970s physicists tried an even more conventional approach: simplify the Einstein equations by pretending that they are almost linear, and then apply the standard methods of quantum field theory to the thus-oversimplified equations. But this method, too, failed: it turned out that Einstein’s general relativity is, in technical language, “perturbatively nonrenormalizable”.43 This means that the strong nonlinearities of Einstein’s general relativity are intrinsic to the theory; any attempt to pretend that the nonlinearities are weak is simply self-contradictory. (This is not surprising: the almost-linear approach destroys the most characteristic features of general relativity, such as black holes.)
In the 1980s a very different approach, known as string theory, became popular: here the fundamental constituents of matter are not point-like particles but rather tiny (Planck-scale) closed and open strings.44 In this theory, the space-time manifold does not exist as an objective physical reality; rather, space-time is a derived concept, an approximation valid only on large length scales (where “large” means “much larger than 10–33 centimeters”!). For a while many enthusiasts of string theory thought they were closing in on a Theory of Everything – modesty is not one of their virtues – and some still think so. But the mathematical difficulties in string theory are formidable, and it is far from clear that they will be resolved any time soon.
More recently, a small group of physicists has returned to the full nonlinearities of Einstein’s general relativity, and – using a new mathematical symbolism invented by Abhay Ashtekar – they have attempted to visualize the structure of the corresponding quantum theory.45 The picture they obtain is intriguing: As in string theory, the space-time manifold is only an approximation valid at large distances, not an objective reality. At small (Planck-scale) distances, the geometry of space-time is a weave: a complex interconnection of threads.
Finally, an exciting proposal has been taking shape over the past few years in the hands of an interdisciplinary collaboration of mathematicians, astrophysicists and biologists: this is the theory of the morphogenetic field.46 Since the mid-1980s evidence has been accumulating that this field, first conceptualized by developmental biologists,47 is in fact closely linked to the quantum gravitational field:48 (a) it pervades all space; (b) it interacts with all matter and energy, irrespective of whether or not that matter/energy is magnetically charged; and, most significantly, (c) it is what is known mathematically as a “symmetric second-rank tensor”. All three properties are characteristic of gravity; and it was proven some years ago that the only self-consistent nonlinear theory of a symmetric second-rank tensor field is, at least at low energies, precisely Einstein’s general relativity.49 Thus, if the evidence for (a), (b) and (c) holds up, we can infer that the morphogenetic field is the quantum counterpart of Einstein’s gravitational field. Until recently this theory has been ignored or even scorned by the high-energy-physics establishment, who have traditionally resented the encroachment of biologists (not to mention humanists) on their “turf”.50 However, some theoretical physicists have recently begun to give this theory a second look, and there are good prospects for progress in the near future.51
It is still too soon to say whether string theory, the space-time weave or morphogenetic fields will be confirmed in the laboratory: the experiments are not easy to perform. But it is intriguing that all three theories have similar conceptual characteristics: strong nonlinearity, subjective space-time, inexorable flux, and a stress on the topology of interconnectedness.
Differential Topology and Homology
Unbeknownst to most outsiders, theoretical physics underwent a significant transformation – albeit not yet a true Kuhnian paradigm shift – in the 1970s and 80s: the traditional tools of mathematical physics (real and complex analysis), which deal with the space-time manifold only locally, were supplemented by topological approaches (more precisely, methods from differential topology52) that account for the global (holistic) structure of the universe. This trend was seen in the analysis of anomalies in gauge theories;53 in the theory of vortex-mediated phase transitions;54 and in string and superstring theories.55 Numerous books and review articles on “topology for physicists” were published during these years.56
At about the same time, in the social and psychological sciences Jacques Lacan pointed out the key role played by differential topology:
This diagram [the Möbius strip] can be considered the basis of a sort of essential inscription at the origin, in the knot which constitutes the subject. This goes much further than you may think at first, because you can search for the sort of surface able to receive such inscriptions. You can perhaps see that the sphere, that old symbol for totality, is unsuitable. A torus, a Klein bottle, a cross-cut surface, are able to receive such a cut. And this diversity is very important as it explains many things about the structure of mental disease. If one can symbolize the subject by this fundamental cut, in the same way one can show that a cut on a torus corresponds to the neurotic subject, and on a cross-cut surface to another sort of mental disease.57, 58
As Althusser rightly commented, “Lacan finally gives Freud’s thinking the scientific concepts that it requires”.59 More recently, Lacan’s topologie du sujet has been applied fruitfully to cinema criticism60 and to the psychoanalysis of AIDS.61 In mathematical terms, Lacan is here pointing out that the first homology group62 of the sphere is trivial, while those of the other surfaces are profound; and this homology is linked with the connectedness or disconnectedness of the surface after one or more cuts.63 Furthermore, as Lacan suspected, there is an intimate connection between the external structure of the physical world and its inner psychological representation qua knot theory: this hypothesis has recently been confirmed by Witten’s derivation of knot invariants (in particular the Jones polynomial64) from three-dimensional Chern-Simons quantum field theory.65
Analogous topological structures arise in quantum gravity, but inasmuch as the manifolds involved are multidimensional rather than two-dimensional, higher homology groups play a role as well. These multidimensional manifolds are no longer amenable to visualization in conventional three-dimensional Cartesian space: for example, the projective space RP3, which arises from the ordinary 3-sphe
re by identification of antipodes, would require a Euclidean embedding space of dimension at least 5.66 Nevertheless, the higher homology groups can be perceived, at least approximately, via a suitable multidimensional (nonlinear) logic.67, 68
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] ...69
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].70 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.71 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 apace,72 and this work is likely to stimulate the attention of mathematicians.73