So if we account for all the types of fermions and bosons in nature—photons, gravitons, gluons, W-bosons, Z-bosons, and Higgs particles on the boson side; neutrinos, electrons, muons, quarks on the fermion side—do they cancel? Not even approximately! The truth is that we have no idea why the vacuum energy is not enormous, why it is not big enough to tear apart the atoms, protons, and neutrons and all other known objects.
Nevertheless, physicists have been able to construct mathematical theories of imaginary worlds in which the positive contributions of bosons exactly cancel the negative vacuum energy of fermions. It’s simple. All you have to do is make sure that the particles come in matched pairs: a fermion for each boson, a boson for each fermion, with each having exactly the same mass. In other words, the electron would have a twin, a boson, with precisely the same mass and charge as the electron. The photon would also have a twin, a massless fermion. In the arcane language of theoretical physics, a matching of that kind, between one thing and another, is called a symmetry. The matching between things and their mirror images is called reflection symmetry. The matching between particles and their antiparticles is called charge conjugation symmetry. It is in keeping with tradition to refer to the fermion-boson matching (in this fictitious world) of elementary particles as a symmetry. The most overworked word in the physicist’s vocabulary is super: superconductors, superfluids, supercollider, supersaturated, superString Theory. Physicists are not often verbally challenged, but the only term they could think of for fermion-boson twinning was supersymmetry. Supersymmetric theories have no vacuum energy because the fermions and bosons exactly cancel.
But super or not, fermi-bose symmetry is not a feature of the real world. There is no superpartner of the electron or for any other elementary particle. The vacuum energies of fermions and bosons do not cancel, and the bottom line is that our best theory of elementary particles predicts vacuum energy whose gravitational effects would be vastly too large. We don’t know what to make of it. Let me put the magnitude of the problem in perspective. Let’s invent units in which 10116 joules per cubic centimeter is called one Unit. Then each kind of particle gives a vacuum energy of roughly a Unit. The exact value depends on the mass and other properties of the particle. Some of the particles give a positive number of Units, and some negative. They must all add up to some incredibly small energy density in Units. In fact a vacuum energy density bigger than. 0000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000001 Units would conflict with astronomical data. For a bunch of numbers, none of them particularly small, to cancel one another to such precision would be a numerical coincidence so incredibly absurd that there must be some other answer.
Theoretical physicists and observational cosmologists have regarded this problem differently. The traditional cosmologists have generally kept an open mind about the possibility that there may be a tiny cosmological constant. In the spirit of experimental scientists, they have regarded it as a parameter to be measured. The physicists, myself included, looked at the absurdity of the required coincidence and said to themselves (and each other) that there must be some deep hidden mathematical reason why the cosmological constant must be exactly zero. This seemed more likely than a numerical cancellation of 119 decimal places for no good reason. We have sought after such an explanation for almost half a century with no luck. String theorists are a special breed of theoretical physicist with very strong opinions about this problem. The theory that they work on has often produced unexpected mathematical miracles, perfect cancellations for deep and mysterious reasons. Their view (and it was, until not too long ago, also my view) has been that String Theory is such a special theory that it must be the one true theory of nature. And being true, it must have some profound mathematical reason for the supposed fact that the vacuum energy is exactly zero. Finding the reason has been regarded as the biggest, most important, and most difficult problem of modern physics. No other phenomenon has puzzled physicists for as long as this one. Every attempt, be it in quantum field theory or in String Theory, has failed. It truly is the mother of all physics problems.
Weinberg Says the A Word
By the mid-1980s physicists had wracked their collective brains for decades about the cosmological constant and had come up completely empty-handed. Desperate situations require desperate measures, and in 1987 Steven Weinberg, one of the most eminent scientists in the world, acted in desperation. Throwing all caution to the wind, he suggested the unthinkable: perhaps the cosmological constant is so small for reasons having nothing to do with special properties of String Theory or any other mathematical theory. Maybe the reason is that if λ were any larger, our own existence would be in jeopardy. This kind of logic went by the name of the Anthropic Principle: some property of the universe or Laws of Physics must be true because if it wasn’t, we couldn’t exist. There are plenty of candidates for anthropic explanations:
Q: Why is the universe big?
Q: Why does the electron exist?
Q: Why is space three-dimensional?
A: At the very least the universe must be as big as the solar system in order for an earthlike planet warmed by a sunlike star to exist.
A: Without electrons there would be no atoms and no organic chemistry.
A: There are many special things that happen in three dimensions that don’t happen in other dimensions. One example is that the stability of the solar system would be compromised in other dimensions. Solar systems in a world of four or more directions would be very chaotic and would not provide stable environments for the billions of years needed for biological evolution to do its work. Even worse is that the forces between electrons and nuclei would suck the electrons into the nucleus, ruining chemistry.
A small universe, a universe with no electrons, or a universe with some other number of dimensions would be a barren universe that could not sustain intelligent creatures to even ask these questions.
No doubt some legitimate applications of anthropic reasoning are justified. We live on the surface of a planet and not the surface of a star because life couldn’t exist at 10,000-degree temperatures. But to use it to explain a fundamental constant of physics? The idea that a fundamental constant was determined by appeal to our own existence was anathema to most physicists. What possible mechanism could adjust a law of nature just so that the human race could exist? What mechanism, other than an appeal to supernatural forces? Physicists often refer to the Anthropic Principle as religion or superstition or “the A word” and claim that it is “giving up.”
Steve Weinberg has been a friend of mine for longer than I care to remember. I first heard his booming baritone voice in a Mexican café in Berkeley. It was 1965: the heyday of Mario Savio’s Free Speech Movement, Jefferson Poland and the Sexual Freedom Movement, LSD and the Vietnam peace protests. I tried all four, plus a few other things. My hair was long, and I was usually dressed in jeans and a tight black T-shirt. Twenty-five years old, I had just arrived there with a freshly minted PhD from Cornell University in upstate New York. Steve was in his early thirties. We both had grown up in the Bronx and gone to the same high school, but there the resemblance ended. When I met Steve, he was already a distinguished academic, the very model of a Berkeley professor. He even dressed like a Cambridge don.
That day in the café he held center stage, pontificating about something or other French and historical. Needless to say I was prepared to dislike him. But once I got to know him, I realized that Steve had that most winning of attributes, the capacity to laugh at himself. He loved being an important man but knew that his own self-importance had its ridiculous side. As you can probably tell, despite our difference in style, I like Weinberg very much.
I have always admired the clarity and depth of Steven Weinberg’s physics. In my opinion he, more than anyone, can lay claim to being the father of the Standard Model. But in recent years I have come to admire him even more for his courage and intellectual integrity. He is one
of the leading voices against creationism and other forms of antiscientific thinking. But on one occasion he was brave enough to express an opinion that went against the scientific prejudices of his colleagues. In fact it was evident from his own writings that he himself strongly disliked the Anthropic Principle. I imagine that it sounded much too close to what some people now call intelligent design. Nevertheless, given the state of desperation concerning the cosmological constant, he felt he could not ignore the possibility of an anthropic explanation. Characteristically, he took a practical course, asking if a cosmological constant bigger than the observed limit of 10–120 Units might be catastrophic for the development of life. If there was no way that a bigger l could inhibit life, then the existence of life would not be significant, and string theorists could go on trying to find an elegant mathematical solution to the problem. But if a reason could be found why a slightly larger cosmological constant would prevent life, then the Anthropic Principle would have to be taken seriously. I’ve always wondered which way Weinberg wanted it to turn out.
To be fair many cosmologists were not only open-minded about the Anthropic Principle but even advocated it. The conjecture that the smallness of the cosmological constant might be anthropic had already appeared in a pioneering book by two cosmologists, John Barrow and Frank Tipler.4 Among others who advocated at minimum an open mind were Sir Martin Rees, the British “Astronomer Royal,” and Andrei Linde and Alex Vilenkin, both famous Russian cosmologists living in the United States. Perhaps cosmologists were more receptive to the idea than physicists because a look at the real universe, instead of abstract equations, is less suggestive of simplicity and elegance than of random and arbitrary numerical coincidences.
In any case Weinberg set out to see if he could find a reason why a cosmological constant much bigger than 10–120 Units would prevent life. To give some idea of the challenge he faced, we can ask how big the effects of such a cosmological constant would be on ordinary terrestrial phenomena. Remember that the cosmological constant manifests itself as a universal repulsion. A repulsive force between the electrons and the nuclei of an atom would change the properties of atoms. But if you plug the numbers in, the repulsion due to such a small cosmological constant would be far smaller than anything that can ever be detected from the properties of atoms or molecules. A cosmological constant many orders of magnitude bigger than 10–120 Units would still be far too small to have any effect on molecular chemistry. Might a small cosmological constant affect the stability of the solar system? Again, the effects are too small for it to do so by many orders of magnitude. There does not seem to be any way that a small cosmological constant could affect life.
Nevertheless, Weinberg found his quarry. It didn’t have to do with physics, chemistry, or astronomy today but rather at the time when galaxies were first beginning to form from the primordial stuff of the early universe. At that time the hydrogen and helium that made up the mass of the universe were spread out in an almost perfectly smooth or homogeneous distribution. The variations of density from one point to another were almost nonexistent.
Today, the universe is full of lumps of many different sizes: everything from small planets and asteroids to giant superclusters of galaxies. If conditions in the past were perfectly homogeneous, then no clumps could ever have formed. The perfect symmetry of an exactly spherical universe would be maintained for all time. But the universe was not exactly homogeneous. At the earliest times that we can see, slight variations in the density and pressure amounted to a few parts in 100,000. In other words, the variations in density were 100,000 times smaller than the density itself. The tendency for gravity to cause clumping is not measured by the overall density of matter but by these small variations.
Even those infinitesimal irregularities were enough to get the process of galaxy formation started. As time progressed, regions with a slight overdensity attracted the matter from the less dense regions. This had the effect of magnifying the slight density contrasts. Eventually the process speeded up, and the galaxies were formed.
But because these density contrasts were initially so small, even a very tiny amount of repulsion could reverse the tendency to cluster. Weinberg found that if the cosmological constant were just an order of magnitude or two bigger than the empirical bound, no galaxies, stars, or planets would ever have formed!
The Case of Negative λ
So far I have told you about the repulsive effects that accompany positive vacuum energy. But suppose that the contribution of fermions outweighed that of bosons: then the net vacuum energy would be a negative number. Is this possible? If so, how does it affect Weinberg’s arguments?
The answer to the first question is yes, it can happen very easily. All you need is a few more fermion-type particles than bosons, and the cosmological constant can be made negative. The second question has an equally simple answer—changing the sign of λ switches the repulsive effects of a cosmological constant to a universal attraction: not the usual gravitational attractive force but a force that increases with distance. To argue convincingly that a large cosmological constant would automatically render the universe uninhabitable, we need to show that life could not form if the cosmological constant were large and negative.
What would the universe be like if the laws of nature were unaltered except for a negative cosmological constant? The answer is even easier than the case of positive λ. The additional attractive force would eventually overwhelm the outward motion of the Hubble expansion: the universe would reverse its motion and start to collapse like a punctured balloon. Galaxies, stars, planets, and all life would be crushed in an ultimate “big crunch.” If the negative cosmological constant were too large, the crunch would not allow the billions of years necessary for life like ours to evolve. Thus, there is an anthropic bound on negative λ to match Weinberg’s positive bound. In fact the numbers are fairly similar. If the cosmological constant is negative, it must also not be much bigger than 10–120 Units if life is to have any possibility of evolving.
Nothing we have said precludes there being pocket universes far from our own with either a large positive or large negative cosmological constant. But they are not places where life is possible. In the ones with large positive λ, everything flies apart so quickly that there is no chance for matter to assemble itself into structures like galaxies, stars, planets, atoms, or even nuclei. In the pockets with large negative λ, the expanding universe quickly turns around and crushes any hope of life.
The Anthropic Principle had passed the first test. Nevertheless, the general attitude of theoretical physicists to Weinberg’s work was to ignore it. Traditional theoretical physicists wanted no part of the Anthropic Principle. Part of this negative attitude stemmed from lack of any agreement about what the principle meant. To some it smacked of creationism and the need for a supernatural agent to fine-tune the laws of nature for man’s benefit: a threatening, antiscientific idea. But even more, theorists’ discomfort with the idea had to do with their hopes for a unique consistent system of physical laws in which every constant of nature, including the cosmological constant, was predictable from some elegant mathematical principle.
But Weinberg took the practical route a little further. He said that whatever the meaning of the Anthropic Principle and the mechanism that enforces it, one thing was clear. The principle may tell us that λ is small enough to not kill us, but there is no reason why it should be exactly zero. In fact there is no reason for it to be very much smaller than what is needed to ensure life. Without worrying about the deeper meaning of the principle, Weinberg was, in effect, making a prediction. If the Anthropic Principle is correct, then astronomers would discover that the vacuum energy was nonzero and probably not much smaller than 10–120 Units.
The Planck Length
The process of discovery has always fascinated me. I’m referring to the mental process; what was the line of reasoning—the insight—that led to the “eureka” moment? One of my favorite daydreams is to put myself in
the mind of a great scientist and imagine how I might have made a crucial discovery.
Let me share with you how I would have made the first great contribution to the quantum theory of gravity. It was a full sixteen years before young Einstein would invent the modern theory of gravity and twenty-six years before those upstarts Werner Heisenberg and Schrödinger invented modern quantum mechanics. As a matter of fact I, Max Planck, did it without even realizing it.
Berlin 1900, The Kaiser Wilhelm Institute
Recently I made the most wonderful discovery of a completely new fundamental constant of nature. People are calling it my constant, Planck’s constant. I was sitting in my office thinking to myself: why is it that the fundamental constants like the speed of light, Newton’s gravitational constant, and my new constant have such awkward values? The speed of light is 2.99 × 108 meters per second. Newton’s constant is 6.7 × 1011 square meters per second-kilogram. And my constant is even worse, 6.626 × 10–34 kilogram-square meters per second. Why are they always so big or so small? Life for a physicist would be so much easier if they were ordinary-size numbers.
Then it hit me! There are three basic units describing length, mass, and time: the meter, kilogram, and second. There are also three fundamental constants. If I change the units, say, to centimeters, grams, and hours, the numerical value of all three constants will change. For example, the speed of light will get worse. It will become 1.08 × 1014 centimeters per hour. But if I use years for time and light-years for distance, then the speed of light will be exactly one since light travels one light-year per year. Doesn’t that mean that I can invent some new units and make the three fundamental constants anything I want? I can even find units in which all three fundamental constants of physics are equal to one! That will simplify so many formulas. I’ll call the new units natural units since they’re based on the constants of nature. Maybe if I’m lucky, people will start calling them Planck units.
The Cosmic Landscape Page 9