We perceive space and little more than 100 years ago most people— and most scientists—thought of matter as time to be continuous, continuous. Although since ancient times some philosophers and scientists had specu- but if the amazing lated that if matter were broken up into small enough bits, it might turn out to be made up theory of loop quantum of very tiny atoms, few thought the existence of atoms could ever be proved. Today we have imaged individual gravity is correct, they atoms and have studied the particles that compose them. The granularity of matter is old news. actually come in In recent decades, physicists and mathematicians have asked if space is also made of discrete pieces. Is it continuous, as we learn in school, or is it more like a piece of cloth, woven out of discrete pieces individual fibers? If we could probe to size scales that were small enough, would we see “atoms” of space, irreducible pieces of volume that cannot be broken into anything smaller? And what By Lee Smolin about time: Does nature change continuously, or does the world DUSAN PETRICIC 66 SCIENTIFIC AMERICAN JANUARY 2004 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. Atoms of Space and Time evolve in series of very tiny steps, acting more like a digital To explain why this is an important question—and what it has computer? to do with the granularity of space and time—I must first say a The past 16 years have seen great progress on these ques- bit about quantum theory and the theory of gravity. tions. A theory with the strange name of “loop quantum gravi- The theory of quantum mechanics was formulated in the ty” predicts that space and time are indeed made of discrete first quarter of the 20th century, a development that was close- pieces. The picture revealed by calculations carried out within ly connected with the confirmation that matter is made of atoms. the framework of this theory is both simple and beautiful. The The equations of quantum mechanics require that certain quan- theory has deepened our understanding of puzzling phenome- tities, such as the energy of an atom, can come only in specific, na having to do with black holes and the big bang. Best of all, it discrete units. Quantum theory successfully predicts the prop- is testable; it makes predictions for experiments that can be done erties and behavior of atoms and the elementary particles and in the near future that will enable us to detect the atoms of space, forces that compose them. No theory in the history of science if they are really there. has been more successful than quantum theory. It underlies our understanding of chemistry, atomic and subatomic physics, elec- Quanta tronics and even biology. MY COLLEAGUES AND I developed the theory of loop quan- In the same decades that quantum mechanics was being for- tum gravity while struggling with a long-standing problem in mulated, Albert Einstein constructed his general theory of rela- physics: Is it possible to develop a quantum theory of gravity? tivity, which is a theory of gravity. In his theory, the gravitational www.sciam.com SCIENTIFIC AMERICAN 67 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. force arises as a consequence of space and quantum theory and general relativity, or time (which together form “spacetime”) new particles or fields, or new entities of being curved by the presence of matter. A some kind. Perhaps with the right addi- loose analogy is that of a bowling ball tions or a new mathematical structure, a placed on a rubber sheet along with a quantumlike theory could be developed marble that is rolling around nearby. The that would successfully approximate gen- balls could represent the sun and the eral relativity in the nonquantum regime. earth, and the sheet is space. The bowling To avoid spoiling the successful predic- ball creates a deep indentation in the rub- tions of quantum theory and general rel- ber sheet, and the slope of this indentation ativity, the exotica contained in the full causes the marble to be deflected toward theory would remain hidden from exper- the larger ball, as if some force—gravity— iment except in the extraordinary cir- were pulling it in that direction. Similar- cumstances where both quantum theory ly, any piece of matter or concentration of and general relativity are expected to have energy distorts the geometry of spacetime, large effects. Many different approaches causing other particles and light rays to be along these lines have been tried, with deflected toward it, a phenomenon we names such as twistor theory, noncom- call gravity. mutative geometry and supergravity. Quantum theory and Einstein’s theo- An approach that is very popular with SPACE IS WOVEN out of distinct threads. ry of general relativity separately have physicists is string theory, which postu- each been fantastically well confirmed by general relativity deals in the geometry of lates that space has six or seven dimen- experiment—but no experiment has ex- spacetime, a quantum theory of gravity sions—all so far completely unobserved— plored the regime where both theories will in addition be a quantum theory of in addition to the three that we are famil- predict significant effects. The problem is spacetime. iar with. String theory also predicts the that quantum effects are most prominent Physicists have developed a consider- existence of a great many new elementary at small size scales, whereas general rela- able collection of mathematical proce- particles and forces, for which there is so tivistic effects require large masses, so it dures for turning a classical theory into a far no observable evidence. Some re- takes extraordinary circumstances to quantum one. Many theoretical physicists searchers believe that string theory is sub- combine both conditions. and mathematicians have worked on ap- sumed in a theory called M-theory [see Allied with this hole in the experi- plying those standard techniques to gen- “The Theory Formerly Known as Strings,” mental data is a huge conceptual prob- eral relativity. Early results were discour- by Michael J. Duff; Scientific Ameri- lem: Einstein’s theory of general relativi- aging. Calculations carried out in the can, February 1998], but unfortunately ty is thoroughly classical, or nonquan- 1960s and 1970s seemed to show that no precise definition of this conjectured tum. For physics as a whole to be logically quantum theory and general relativity theory has ever been given. Thus, many consistent, there has to be a theory that could not be successfully combined. Con- physicists and mathematicians are con- somehow unites quantum mechanics and sequently, something fundamentally new vinced that alternatives must be studied. general relativity. This long-sought-after seemed to be required, such as addition- Our loop quantum gravity theory is the theory is called quantum gravity. Because al postulates or principles not included in best-developed alternative. Overview/Quantum Spacetime A Big Loophole IN THE MID-1980S a few of us—in- ■ To understand the structure of space on the very smallest size scale, we must cluding Abhay Ashtekar, now at Penn- turn to a quantum theory of gravity. Gravity is involved because Einstein’s sylvania State University, Ted Jacobson of general theory of relativity reveals that gravity is caused by the warping of the University of Maryland and Carlo space and time. Rovelli, now at the University of the Med- ■ By carefully combining the fundamental principles of quantum mechanics and iterranean in Marseille—decided to reex- general relativity, physicists are led to the theory of “loop quantum gravity.” amine the question of whether quantum In this theory, the allowed quantum states of space turn out to be related to mechanics could be combined consis- diagrams of lines and nodes called spin networks. Quantum spacetime tently with general relativity using the corresponds to similar diagrams called spin foams. standard techniques. We knew that the ■ Loop quantum gravity predicts that space comes in discrete lumps, the smallest negative results from the 1970s had an of which is about a cubic Planck length, or 10–99 cubic centimeter. Time proceeds in important loophole. Those calculations discrete ticks of about a Planck time, or 10–43 second. The effects of this discrete assumed that the geometry of space is structure might be seen in experiments in the near future. continuous and smooth, no matter how minutely we examine it, just as people DUSAN PETRICIC 68 SCIENTIFIC AMERICAN JANUARY 2004 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. had expected matter to be before the dis- are set up in a predetermined classical dations of our theory of loop quantum covery of atoms. Some of our teachers (that is, nonquantum) spacetime. gravity. The term “loop,” by the way, and mentors had pointed out that if this The second principle, known by the arises from how some computations in assumption was wrong, the old calcula- imposing name diffeomorphism invari- the theory involve small loops marked tions would not be reliable. ance, is closely related to background in- out in spacetime. So we began searching for a way to dependence. This principle implies that, The calculations have been redone by do calculations without assuming that unlike theories prior to general relativity, a number of physicists and mathemati- space is smooth and continuous. We in- one is free to choose any set of coordi- cians using a range of methods. Over the sisted on not making any assumptions nates to map spacetime and express the years since, the study of loop quantum beyond the experimentally well tested equations. A point in spacetime is defined gravity has grown into a healthy field of principles of general relativity and quan- only by what physically happens at it, not research, with many contributors around tum theory.