Quantum Man: Richard Feynman's Life in Science (22 page)

(2)
String Theory and Beyon
d
: 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 could 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.

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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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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.”

It is a small step from this, perhaps, to suggest that the total energy of the entire universe might be precisely zero. Such a universe with total energy equal to zero is attractive, because it allows for a universe that began from nothing. All matter and energy we might see could have arisen from a quantum mechanical fluctuation (including a gravitational quantum mechanical fluctuation in space itself). While Feynman speculated on this possibility, the current best model for the evolution of the universe, called inflation, is based on this very idea. The originator of the idea of inflation, Alan Guth, has said that in this case the universe is the ultimate example of a “free lunch.”

Interestingly, a universe with zero total gravitational energy is spatially flat—that is, on large scales it behaves like a normal Euclidean space where light travels in straight lines. There is now very good evidence that the universe is flat by direct measurements of its geometry on large scales, one of the most exciting developments in cosmology in recent times. As early as 1963, however, Feynman suggested this was likely to be the case because the fact that gravitationally bound galaxies and clusters of galaxies—the largest bound objects in the universe, tens of millions of light-years across—do exist implied that the positive kinetic energy of the expansion of the universe was roughly balanced by the negative gravitational potential energy in these systems. Hewas right.

There was one application of his quantum field theory arguments to gravity where he seemed to have departed from his normal sensible physical intuition, however. In his work on QED he, as well as others, had shown that virtual particles not only exist but also are necessary in order to understand the properties of atoms. Thus, empty space is not empty but is a boiling brew of virtual particles. The laws of quantum mechanics tell us that the smaller the scale one wants to consider, the higher the energy the virtual particles that can briefly exist can have. Feynman once referenced this by saying that in the space in the closed palm of a hand, virtual particles existed with enough energy to power our entire civilization. Unfortunately advocates and crackpots have used this statement to express their desire to develop devices that exploit the energy of the vacuum to do precisely this, and solve our energy problems.

What Feynman somehow forgot, and what the Russian physicist Yakov Zel’dovich made clear in 1967, is that all energy gravitates, even the energy of empty space. If empty space had as much energy as Feynman argued, the gravitational forces would be so great as to blow up the earth, because according to general relativity, when energy is put into empty space, the resulting gravitational force is
repulsive
, not attractive. Therefore, the energy of empty space cannot be, on average, orders of magnitude larger than the energy of all matter, or the resulting repulsive force would be so large that galaxies would never have formed.

Nevertheless, Feynman was not completely wrong. The most astounding discovery in the last fifty years, if not longer, has been the discovery that empty space
does
contain energy—far less than Feynman imagined, but enough so that the energy of empty space is currently dominating the expansion of the universe, causing it to accelerate. We currently have no understanding of why this is the case, and why empty space possesses both energy and an amount of energy that is comparable to the total energy contained in all galaxies and matter in the universe. It is probably the biggest mystery in physics, if not all of science.