Quantum gravity predicts piecemeal space.

Just as electrons in atoms can have only certain energies, space itself may come packaged in discrete units.

This possibility arises from an effort to integrate quantum theory, which applies to the behavior of matter on a microscopic scale, with the general theory of relativity, which holds that gravity is a geometric effect dependent on the curvature of space.

Using a particular formulation of quantum gravity, physicists Carlo Rovelli of the University of Pittsburgh and Lee Smolin of Pennsylvania State University in University Park show that measurements of the volumes and areas of regions of space are quantized. They can also calculate the range, or spectrum, of sizes that pieces of space can have.

"Many different approaches to quantum gravity incorporate a fundamental length," the researchers say. "The discreteness of areas and volumes found [in our work] supports and strengthens the evidence for the existence of a discrete, short-scale structure of space." They will report their findings in the May 29 Nuclear Physics B.

The quantum gravity theory developed by Rovelli and Smolin evolved out of the work of Abhay Ashtekar, now at Penn State. In the mid-1980s, Ashtekar discovered that he could reformulate and drastically simplify the equations of general relativity by replacing a single variable, representing a unified, four-dimensional spacetime continuum, with two distinct spacetime variables. Such a transformation enabled him and others to find new solutions to the equations and suggested various possibilities for quantizing the theory.

In 1988, Rovelli and Smolin developed a way of interpreting solutions to the quantized theory as patterns of closed loops -- lines of force of the gravitational field somewhat analogous to the lines of magnetic force around a bar magnet. Expressed in terms of loops, the quantum states of the gravitational field depend on how these loops are knotted and linked.

In their new work, Rovelli and Smolin construct two mathematical expressions, called operators, to represent measurements of the volumes and areas of regions of space. Applying these operators, they find that the measurements correspond to particular loop patterns described by spin networks (see diagram).

"Spin networks are the quantum states of gravity, just as electron shells are the quantum states of the atom," Rovelli notes.

In other words, the Smolin-Rovelli model of quantum gravity predicts that measurements of volumes and areas can't have just any value. A measured value must belong to a certain set of numbers.

"One way to describe these predictions is that the geometry of space itself is made out of discrete quanta analogous to the photons of light or the electron shells of atoms," Smolin says.

These pieces of space are extremely small, having length scales on the order of 10-35 meters, compared to 10-15 m for the diameter of a typical atomic nucleus. If volume and area could be measured to this level of precision, "the answers would have to fit into the discrete spectra that we calculated," Smolin says.

The loop picture of quantum gravity may prove useful for studying the problem of what quantum theory has to say about singularities, Smolin says.

Arising in general relativity and many other physical theories, singularities are situations in which attempts to make predictions result in infinite quantities. When networks of loops define the structure of space, nothing can be smaller, and no infinities come up in the theory.

The same model could also provide a reasonable description of the nascent universe, particularly in order to establish the spectrum of primordial gravitational waves generated during the universe's earliest moments.

The researchers are interested in exploring the relationship between their results and those coming out of alternative approaches to quantum gravity, especially string theory (SN: 2/27/93, p.136). In string theory, the point particles of relativity and quantum mechanics are replaced by extended objects called strings, which can be visualized as either closed loops or segments with two free ends. But this theory says nothing directly about the background space in which the strings vibrate and move.

"I believe that our results and those of string theory are complementary," Smolin says. "They explore different regimes of quantum gravity and could easily be both true."