The claim is remarkable: Forget all about geometry and the fascinating notions about space and time that seem to be at the basis of general relativity. If one considers the exchange of a massless particle (just as a photon is a massless particle that conveys the electromagnetic force), then if the massless particle in question has quantized spin 2 instead of spin 1 as a photon does, the only self-consistent theory that results will, in the classical limit, essentially be Einstein’s general relativity.
This is a truly amazing claim because it suggests that general relativity is not that different from the theories describing the other forces in nature. It can be described by the exchange of fundamental particles just like the rest. All the geometric baggage comes out after the fact, for free. In fact, there are subtleties in the actual true statement of the claim, coming from what is meant by “self-consistent,” but these are really just subtleties. And Weinberg, as I have indicated, was able to prove a more general version of this claim, relying simply on the properties of the interactions of a massless spin 2 particle and the symmetries of space that arise in special relativity.
But these subtleties aside, this new picture of gravity and general relativity created a completely novel bridge between general relativity and the rest of physics that was not there before. It suggested, just as Feynman had hoped it would, that one might use the tools of quantum field theory not only to understand general relativity but also to unify it with the other forces in nature.
First, what are these massless spin 2 particles and what do they correspond to in nature? Well, recall that photons, the quanta of the electromagnetic field, are just quantized versions of classical electromagnetic waves, the waves of electric and magnetic fields that James Clerk Maxwell first showed result from jiggling an electric charge, which is the source of the electromagnetic field. These fields we experience with our eyes as light, our skin as heat from the sun, as radio waves with our radios, or microwaves with our cell phones.
Einstein had shown, shortly after he developed general relativity, that mass, which is the source of gravity, could produce a similar effect. If a mass is moved in just the right way, a new type of wave will be emitted—a gravitational wave, which is literally a wave in which space compresses and expands along the wave, and will travel out at the speed of light, just as photons do. In 1957, when Feynman first discussed his ideas at a physics meeting, many in the audience were dubious that gravitational waves even existed. (In fact, Einstein himself was earlier deterred by H. P. Robertson from publishing a paper denying their existence.) However, in 1993 Joseph Taylor and his former student Russell Hulse received the Nobel Prize for convincingly demonstrating that a pair of orbiting neutron stars was losing energy at the exact rate predicted by general relativity for the emission of gravitational waves from this system. While scientists have yet to directly detect gravitational waves, because gravity is so weak, large terrestrial experiments have been designed to do so, and plans are underway to build a very sensitive detector in space.
Gravitational waves are emitted only from objects in which the distribution of mass is changing in a nonspherically symmetric way. Physicists call the kind of radiation emitted by such a distribution quadrupole radiation. If one wanted to encode this kind of directional anisotropy by associating particles with the emitted waves, these primary “quanta” would have to have a spin 2, which is precisely why Feynman first explored this option. The quantum of gravitational waves is called a graviton, in analogy to a photon.
Having demonstrated that gravity can result simply from the exchange of gravitons between masses, just as electric and magnetic forces result from the exchange of photons between charges, Feynman then proceeded to use precisely the kind of analysis that had stood him in such good stead with QED to calculate quantum corrections to gravitational processes. The effort was not so simple however. General relativity is a far more complex theory than QED because while photons interact with charges in QED, they do not interact directly with each other. However, because gravitons interact with any distribution of mass or energy, and since gravitons carry energy, gravitons interact with other gravitons as well. This additional complexity changes almost everything, or at least makes almost everything harder to calculate.
Needless to say, Feynman did not find that a consistent quantum theory of gravity interacting with matter, without any nasty infinities, could be derived by simply treating general relativity as he had electrodynamics. There still is not such a definitive theory, though candidates have been proposed, including string theory. Nevertheless, every major development that has taken place in the fifty-odd years since Feynman began his work in this area, involving a line of scientists from Feynman to Weinberg to Stephen Hawking and beyond, has built on his approach and on the specific tools he developed along the way.
Here are a few examples:
(1) Black Holes and Hawking Radiation: Black holes have remained perhaps the biggest theoretical challenge to physicists trying to understand the nature of gravity, and they have produced the biggest surprises. While suggestive observational evidence has accumulated in the past forty years of the existence of massive black hole–like objects in the cosmos, from the engines of energetic quasars to million and billion solar mass objects at the centers of galaxies, including our own, the detailed nature of quantum processes that operate in the final stages of black hole collapse has produced surprises and controversy. The biggest surprise came in 1972, when Stephen Hawking explored the detailed quantum mechanical processes that might occur near the event horizon of a black hole, and discovered that these would cause black holes to radiate energy in the form of all types of elementary particles, including gravitons, as if the black hole were hot, at a temperature inversely proportional to its mass. The form of this thermal radiation would be essentially independent of the identity of whatever collapsed to form the black hole, and would cause the black hole to lose mass and perhaps eventually evaporate completely. This result, which is based on the type of approximation Feynman first used to explore the quantum mechanics of gravity—namely, approximating the background space as fixed and approximately flat, and considering quantum fields, including gravitons, propagating in this space—not only flew in the face of commonsense classical thinking but also presented major challenges to our understanding of quantum mechanics in the presence of gravity. What is the source of this finite temperature? What happens to the information that falls down the black hole if the black hole eventually radiates away? What about the singularity at the center of the black hole, where conventional quantum field theory breaks down? These major conceptual and mathematical problems have driven the work of the greatest theoretical minds in physics over the past forty years.
(2) String Theory and Beyond: In an effort to tame the infinities of quantum gravity, scientists discovered in the 1960s that if one considers the quantum mechanics of a loop of vibrating string, there is naturally one type of vibration that would be appropriate to describe a massless spin 2 excitation. This led to the recognition, using precisely the results of Feynman described earlier, that Einstein’s general relativity might naturally arise in a fundamental quantum theory that incorporated such stringlike excitations. This recognition, in turn, suggested the possibility that such a theory might be a true quantum theory of gravity, in which all of the infinities that Feynman had exposed in his exploration of gravity as a quantum field theory might be tamed. In 1984 several candidate string theories in which all such infinities might disappear were proposed, producing the biggest explosion of theoretical excitement that physics had witnessed since perhaps the development of quantum mechanics itself.
As exciting as this possibility was, however, there was also a minor complication. In order to allow the mathematical possibility of a self-consistent quantum theory of gravity without infinities, the underlying stringlike excitations cannot exist in merely four dimensions. They must “vibrate” in at least ten or eleven dimensions. How co
uld such a theory be consistent with the four-dimensional world we experience? What would happen to the six or seven extra dimensions? How could one develop mathematical techniques to treat them consistently and still explore phenomena in the world we experience? How could one develop physical mechanisms to hide the extra dimensions? Finally, and perhaps most important, if gravity arose naturally in these theories, in the spirit of Feynman, could the other particles and forces we experience also naturally arise within the same framework?
These became the central theoretical issues that have been explored in the past twenty-five years, and the results have been mixed at best. Fascinating mathematical theorems that have been developed have given exciting new insights into how to understand seemingly different quantum theories as manifestations of the same underlying physics—something that falls precisely within what Feynman described as the central goal of science—and interesting mathematical results that have been obtained may provide insights into how black holes can radiate thermally, appearing to lose information, and still not violate the central tenets of quantum theory. And finally, string theory, which is based on a new type of Feynman diagram to calculate processes involving the behavior of strings, has allowed theorists to discover new ways to classify Feynman diagrams for normal quantum fields, and allowed physicists to derive analytical results in closed form for processes that would have otherwise involved summing an impossibly large number of Feynman diagrams were the calculations performed directly.
But with the good comes the bad. As our understanding of string theory developed, it became clear that it was much more complicated than previously imagined, and that strings themselves are probably not the key objects in the theory, but rather higher-dimensional objects called branes, making the possible range of predictions of the theory far more complicated to derive. Moreover, while early hopes had sided with the possibility that a single underlying string theory would make unique and unambiguous predictions yielding all of the fundamental physics measured in laboratories today, precisely the opposite has occurred. Almost any possible four-dimensional universe, with any set of laws of physics, might arise in these theories. If this remains true, then rather than producing “theories of everything,” they could produce “theories of anything,” which, in the spirit of Feynman, would not be theories at all.
Indeed, Feynman lived long enough to witness the major string revolution of the 1980s and the hype that went along with it. His natural skepticism of grand claims was not swayed. As he put it at the time, “My feeling has been—and I could be wrong—that there’s more than one way to skin a cat. I don’t think that there’s only one way to get rid of the infinities. The fact that a theory gets rid of infinities is to me not a sufficient reason to believe its uniqueness.” He also understood, as he had so clearly expressed at the beginning of his 1963 paper on the subject, that any effort to understand quantum gravity suffered from the handicap that any predictions, even in a theory that made clear predictions, might be well beyond the range of experimentation. The lack of predictiveness, combined with the remarkable hubris of string theorists, even with a manifest lack of empirical evidence, motivated him to say, in exasperation, “String theorists don’t make predictions, they make excuses!” Or, expressing his frustration in terms of the other key factor that for him defined a successful scientific theory,
I don’t like that they’re not calculating anything. I don’t like that they don’t check their ideas. I don’t like that for anything that disagrees with an experiment, they cook up an explanation—a fix-up to say, “Well, it might be true.” For example, the theory requires ten dimensions. Well, maybe there’s a way of wrapping up six of the dimensions. Yes, that’s all possible mathematically, but why not seven? When they write their equation, the equation should decide how many of these things get wrapped up, not the desire to agree with experiment. In other words, there’s no reason whatsoever in superstring theory that it isn’t eight out of the ten dimensions that get wrapped up and that the result is only two dimensions, which would be completely in disagreement with experience. So the fact that it might disagree with experience is very tenuous, it doesn’t produce anything; it has to be excused most of the time. It doesn’t look right.
The very issues that aroused Feynman’s concerns, expressed more than twenty years ago, have, if anything, been magnified since then. Of course, Feynman was skeptical of all new proposals, including some that turned out to be right. Only time, and a lot more theoretical work, or some new experimental results, will determine whether in this case his intuition was correct.
(3) Path Integrals in Quantum Gravity and “Quantum Cosmology”: The conventional picture of quantum mechanics suffers, as I have described, from the problem that it treats space and time differently. It defines the wave function of a system at a specific time and then gives rules for evolving the wave function with time.
However, a basic tenet of general relativity is that such a distinction between space and time is, in some sense, arbitrary. One can choose different coordinate systems, where one person’s space is another’s time, and the physical results one derives should be independent of this arbitrary separation. This issue becomes particularly important in cases where space is strongly curved—that is, where the gravitational field is strong. As long as gravity is so weak that one can approximate space as being flat, then one can follow the prescription Feynman developed for treating gravity as a small perturbation, and gravitational effects as being primarily due to the exchange of single gravitons moving in a fixed background space. But in the case where gravity is strong, space and time become smeared-out quantum variables, and a rigid separation into a background space and time in which phenomena can evolve becomes problematic, to say the least.
The path-integral formulation of quantum mechanics does not require such a separation. One sums over all of the possibilities for all of the relevant physical quantities, and over all of the paths without requiring a separation of space and time. Moreover, in the case of gravity, where the relevant quantity involves the geometry of space, then one must sum over all of the possible geometries. Feynman’s method gives a prescription for doing this, but it is not at all clear that the remaining picture could be handled by the conventional formulation of quantum mechanics.
The path-integral approach has already been applied, most strongly by Stephen Hawking (and later Sidney Coleman and others), to develop a quantum mechanics of the entire universe, where in the path integral one sums over various possible intermediate universes in which strange new topologies are possible, involving baby universes and wormholes. This approach to treating an entire universe quantum mechanically is called quantum cosmology, and involves a host of new and difficult issues, including how to interpret a quantum system with no external observers, and whether the dynamics of the system can determine its own initial conditions, rather than have them imposed by an outside experimenter.
Clearly the field is in its infancy, especially without a well-defined understanding of quantum gravity. But as Murray Gell-Mann lovingly hoped in an essay written after Feynman’s death—knowing of Feynman’s great desire to discover new laws and not merely reformulate existing ones, as he had feared his approach to QED had done—it could be that Feynman’s path-integral formalism is not just a different but equivalent way of formulating quantum mechanics, but rather the only truly fundamental way. As Gell-Mann put it, “Thus, it would have pleased Richard to know that there are now some indications that his PhD dissertation may have involved a really basic advance in physical theory and not just a formal development. The path integral formulation of quantum mechanics may be more fundamental than the conventional one, in that there is a crucial domain where it may apply and the conventional formulation may fail. That domain is quantum cosmology. . . . For Richard’s sake (and Dirac’s too), I would rather like it to turn out that the path integral method is the real foundation of quantum mechanics, and thus of physical theory.�
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(4) Cosmology, Flatness, and Gravitational Waves: I have saved for last the most concrete, and perhaps least philosophically profound, implication of Feynman’s work, because it allows for the possibility of calculations that might be directly compared to experimental data—without which he viewed theoretical efforts as impotent.
Amazingly, Feynman did his work at a time when almost everything scientists now know about the universe on its largest scales was not yet known. Yet his intuition in a number of key areas was right, with one exception, and experiments at the forefront of observational cosmology may soon provide the first direct evidence that his picture of gravitons as the fundamental quanta of the gravitational field is correct.
Feynman realized early on the possibility that the total energy of a system of particles might be precisely zero. As strange as this may sound, it is possible because while it takes positive energy to create particles from nothing, their net gravitational attraction afterward can imply that they have a negative “gravitational potential energy”—namely, that because it takes work to pull them apart to overcome their gravitational attraction, the net energy lost after they are created and are then attracted together might exactly compensate for the positive energy it took to create them. As Feynman put it in his lectures on gravitation, “It is exciting to think that it costs nothing to create a new particle.”
Quantum Man: Richard Feynman's Life in Science Page 22
