Physicists believe there to be four fundamental forces. Three of these—the electromagnetic, the strong force, and the weak force—are amalgamated in the standard model of elementary particle physics, a family of quantum field theories that has enjoyed stupendous empirical success. Gravity, the fourth and feeblest fundamental force, is the subject of a stupendously successful nonquantum field theory, Einstein’s general theory of relativity (GTR).
Desiring to fit all of fundamental theoretical physics into a quantum mechanical framework, and suspecting that GTR would break down at tiny (“Planck scale,” i.e., 10-33 cm) distances where quantum effects become significant, physicists have been searching for a quantum theory of gravity since the 1930s.
In the last quarter of the twentieth century, string theory became the predominant approach to quantizing gravity, as well as to forging a unified picture of the four fundamental forces. A minority approach to quantizing gravity is the program of loop quantum gravity, which promises no grand unification.
Both attempts to quantize gravity portend a science of nature radically different from the Newtonian one that frames much of classical philosophical discourse. They also present gratifying instances of working physicists actively concerned with recognizably philosophical questions about space, time, and theoretical virtue.
The Standard Model
String theory would quantize gravity by treating the gravitational force as other forces are treated. In the standard model, pointlike elementary particles, quarks, and leptons constitute matter. Each particle is characterized by invariants, such as mass, spin, charge, and the like.
The matter-constituting particles have half-integer multiples of spin, which makes them fermions. Beside fermions, the standard model posits gauge bosons, “messenger particles” or carriers of the interaction, for each force in its ambit. Bosons are distinguished from fermions by having whole-integer multiples of spin.
As early as 1934, preliminary work on the sort of coupling with matter required by a quantum theory suggested that, if the gravitational force had a gauge boson, it must be a mass 0 spin 2 particle, dubbed the graviton. No such particle is predicted by the standard model.
According to string theory, the elementary particles of the standard model are not the ultimate constituents of nature. Filamentary objects—strings—are. Different vibrational modes of these strings correspond to the different masses (charges, spins) of elementary particles.
The standard model is recovered, and fundamental physics unified, in a string theory incorporating vibrational modes corresponding to every species of particle in the standard particle zoo (and so incorporating the strong, weak, and electromagnetic forces), as well as to the graviton (and so incorporating gravity).
The Early Years of String Theory
String theory evolved from attempts, undertaken within the standard model in the 1970s, to model the strong nuclear force in terms of a band between particles. As a theory of the strong nuclear force, these attempts suffered in comparison to quantum chromodynamics. They also predicted the existence of a particle that had never been detected: a mass 0 spin 2 particle.
In 1974, John Schwartz and Joël Scherk proposed to promote this empirical embarrassment to a theoretical resource: The undetected particle, they suggested, was in fact the graviton! (Further evidence that string theory encompasses gravity comes in the form of a consistency constraint on the background spacetime in which string theoretic calculations are carried out, which consistency constraint resembles the equations of GTR).
String theory evolved piecemeal in the 1970s and 1980s, roughly by adapting perturbative approximation techniques developed for the standard model’s point particles to stringy entities. One benefit of the adaptation was the suppression of infinities that arise in perturbative calculations for point particles.
In the standard model, these infinities call for the expedient of renormalization, the barelyprincipled subtraction of other infinities to yield finite outcomes. Perturbative string theories require no such expedient.
Worries that they harbored inconsistencies all their own, called anomalies, were allayed by Schwartz’s and Michael Green’s 1984 argument that string theories were anomaly-free—a result that galvanized research in the field.
By the early 1990s there were five different consistent realizations of perturbative string theory. These realizations shared some noteworthy features. First, their equations were consistent only in ten space-time dimensions.
To accord with the appearance that space is three-dimensional, the extra six dimensions are supposed to be Planck-scale and compactified (“rolled up”). (The usual analogy invokes the surface of a cylinder, which is a twodimensional object: one dimension runs along the length of the cylinder; the other is “rolled up” around its circumference.
Supposing the rolled-up dimension to be small enough, a cylinder looks like a one-dimensional object, a line.) Details of the geometries of these extra dimensions influence the physics string theory predicts. These details are adjustable; only with certain choices of the geometries can string theory mimic the standard model.
The initial string theories dealt only with bosons. So that they might incorporate fermions as well, supersymmetry was imposed. That is, the equations of string theory were required to be invariant under half-integer changes in spin. Thus the theory predicts for every particle in the standard zoo that it has a supersymmetric partner.
For the (spin 1/2) electron, a spin 0 “selectron;” for the (spin-1) photon, a spin 1/2 “photino,” and so on. Of these supersymmetric partners, none are observable using present technologies. But there is hope of detecting the lightest, the neutralino, with the Large Hadron Collider, slated to come on-line at CERN in 2007.
|changes in spin|
Parameters describing, for example, coupling strengths or the volume of the compactified extra dimensions appear in string theories. This means that each string theory can be thought of as a member of a family of related string theories, obtained from the first by varying the values of these parameters. A duality is said to obtain between theories so related.
In the mid-1990s, Ed Witten and others uncovered evidence of dualities connecting pairs in the set of five consistent perturbative string theories. This embolded Witten to propose that the existing, approximate, string theories were all approximations to a single underlying exact theory he dubbed “Mtheory.”
Although the equations of M-theory are unknown, it is believed that they hold in an elevendimensional spacetime, and have eleven-dimensional supergravity (ironically enough, a leading contender for the title “theory of everything” which string theory dislodged in the early 1980s) as their low-energy limit.
In addition to strings, M-theory boasts higher-dimensional supersymmetric objects—membranes—some theorists have put to cosmological use, for example, by maintaining that the three spatial dimensions of this world are a three-brane moving through an eleven-dimensional universe harboring other worlds such as this one.
Most predictions of fledgling programs in quantum gravity are experimentally inaccessible, and liable to stay that way. But a nonempirical circumstance is widely believed to confirm string theory. In black hole thermodynamics (developed by Stephen Hawking, Jacob Bekenstein, and others), black holes are attributed properties, such as temperature and entropy, that obey thermodynamic laws. (For instance, entropy, identified as the surface area of a black hole’s future event horizon, never decreases.)
For certain black holes known as extremal black holes, string theoretic calculations exactly reproduce the Bekenstein entropy formula. Although there has never been an observation confirming (or disconfirming) black hole thermodynamics, the recovery of the black hole entropy formula is widely held to be evidence that string theory is on the right track.
More empirical tests have been proposed, none strong. For example, if the extra dimensions posited by string theory are large enough, new mechanisms for the production of microscopic black holes could be unleashed at energies attainable in the Large Hadron Collider. But string theory is not required to posit large extradimensions. So the failure of microscopic black holes to appear would not force the abandonment of string theory.
Despite its successes, there are causes for complaint about string theory. It is not an exact theory yet. Its predictions might seem unduly sensitive to the discretionary matter of the geometry of the extra dimensions.
In addition to predicting the existence of the standard particles and the graviton, it predicts the existence of infinitely many particles, including supersymmetric particles, humans have not seen (yet). It requires seven extra spatial dimensions humans have not seen (yet). And as formulated at present, it takes place in a fixed space-time background.
String Theory and Loop Quantum Gravity
The game of background-independent M-theory is afoot; some (e.g., Smolin 2001) hope that its pursuit will reveal connections between string theory and its main rival, loop quantum gravity. Background-independence is the rallying cry of the (much less populated) loop quantum gravity camp.
Largely trained as general relativists, adherents of this approach take the fundamental moral of GTR to be that space-time is not a setting in which physics happens but is itself a dynamical object, malleable in response to the matter and energy filling it.
Whereas string theory seeks a quantum theory of gravity on the model of early twenty-first century quantum theory of other forces—a model that adds a graviton to a particle zoo revealed by approximations carried out in a fixed spacetime background—loop quantum gravity seeks a quantum theory of gravity by quantizing gravity: that is, by casting GTR as a classical theory in Hamiltonian form, and following a canonical procedure for quantizing such theories. Insofar as GTR’s variables determine the geometry of space-time, should the quantization procedure succeed, space-time itself would be the commodity quantized.
The quantization procedure is complicated by the fact that GTR is a constrained Hamiltonian system: its canonical momenta are not independent. Instead, they satisfy constraint equations that must be reflected in the final quantum theory.
The origin of these constraint equations is the diffeomorphism invariance of GTR, that is, if one starts with a solution to the equations of GTR and smoothly reassigns the dynamical fields comprising that solution to the manifold on which they are defined, one winds up with a solution to the equations of GTR. Adherents of loop quantum gravity take diffeomorphism invariance to express the background independence of GTR.
Loop quantum gravity exploits a Hamiltonian formulation of GTR due to Abhay Ashtekar, a physicist at Syracuse University. Its quantization is set in a Hilbert space spanned by spin-network states: graphs whose edges are labeled by integer multiples of 1/2.
Not set in some background space, these spin-network states are supposed to be the constituents from which space is built. Defined on their Hilbert space are area and volume (but not length) operators that have discrete spectra. A free parameter in the theory can be adjusted so that this quantization occurs at the Planck scale.
On these grounds, its adherents claim loop quantum gravity to be a background-independent exact theory that quantizes space. Like string theory, loop quantum gravity finds quasi-confirmation in its accord with black hole thermodynamics: for all black holes, loop quantum gravity reproduces the Bekenstein entropy within a factor of 4.
Despite its successes, there are causes for complaint about loop quantum gravity. It does not incorporate the predictions of the standard model. So whereas it may be a quantum theory of gravity, it is not a theory of everything.
More telling, loop quantum gravity as yet fails to reflect the full diffeomorphism invariance of GTR in a way that is both consistent and has GTR as its classical limit. The sticking point is the classical Hamiltonian constraint, related to diffeomorphisms that can be interpreted as time translations.
Until this constraint is wrangled, loop quantum gravity lacks a dynamics: it consists of a space of possible instantaneous spacetime geometries, without an account of their time development. Given loop quantum gravity’s ideology of background-independence, this is disappointing.
There is no established philosophy of quantum gravity. But there is much to provoke the philosopher.What, according to string theory or loop quantum gravity, is the nature of space(-time)? How many dimensions has it? (These questions are complicated by dualities between string theories revealed by varying the volumes of their compactified geometries, as well as by the holographic hypothesis, according to which physics in the interior of a region—an n-dimensional space—is dual to physics on that region’s boundary—an (n-1)-dimensional space.) The search for quantum gravity was set off by no glaring empirical shortcoming in existing theories, and has reached theories for which no empirical evidence is readily forthcoming.
In the absence of empirical adequacy, other theoretical virtues occupy center stage: the ideal of unification, the capacity to reproduce the results, or preserve the insights, of other theories (even unconfirmed ones); the susceptibility of puzzles posed in one theoretical framework to solution techniques available in another. The nature of these virtues, and how best to pursue them, are often live questions for quantum gravity researchers. Their work holds interest for the methodologist and the metaphysician alike.