If we could create a region where the Higgs field was zero, the most singular thing we would notice (assuming our own survival) is that the electron mass would be zero. The effects on atoms would be devastating. The electron would be so light that it could not be contained within the atom. Neither atoms nor molecules would exist. Life of our kind would almost certainly not exist in such a region of space.
It would be very interesting to test these predictions the same way that we can test physics in a magnetic field. But manipulating the Higgs field is vastly more difficult than manipulating the magnetic field. Creating a region of space where the Higgs field is zero would cost an enormous quantity of energy. Just a single cubic centimeter of Higgs-free space would require energy of about 1040 joules. That’s about the total amount of energy that the sun radiates in a million years. This experiment will have to wait a while.
Why is the Higgs field so different from the magnetic field? The answer lies in the Landscape. Let’s simplify the Landscape to one dimension by ignoring the electric and magnetic fields and include only the Higgs field. The resulting “Higgs-scape” would be more interesting than the simple parabola that represents the magnetic field Landscape. It has two deep valleys separated by an extremely high mountain.
Don’t worry if you don’t understand why the Higgs-scape looks so different. No one completely understands it. It is another empirical fact that we have to accept for now. The top of the hill is the point on the Landscape where the Higgs field is zero. Imagine that some superpowerful cosmic vacuum cleaner has sucked the vacuum clean of Higgs field. Here is the place in the Higgs-scape where all the particles of the Standard Model are massless and move with the speed of light. From the graph you can see that the top of the mountain represents an environment with a large amount of energy. It is also a deadly environment.
By contrast, our corner of the universe is safely nestled in one of the valleys where the energy is lowest. In these valleys the Higgs field is not zero, the vacuum is full of Higgs fluid, and the particles are massive. Atoms behave like atoms, and life is possible. The full Landscape of String Theory is much like these examples but infinitely richer in mostly unpleasant possibilities. Friendly, habitable valleys are very rare exceptions. But that’s a later story.
Why, in each example, do we live at the bottom of a valley? Is it a general principle? Indeed, it is.
Rolling along the Landscape
Hermann Minkowski was a physicist with a flare for words. Here is what he had to say about space and time: “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent identity.” Minkowski was talking about Einstein’s two-year-old child, the Special Theory of Relativity. It was Minkowski who announced to the world that space and time must be joined together into a single, four-dimensional space-time. It follows from the four-dimensional perspective that if the Laws of Physics can vary from one point of space to another, then it must also be possible for them to vary with time. There are things that can make all the normal rules—even the law of gravity—change, suddenly or gradually.
Imagine a very long-wavelength radio wave passing through a physics laboratory. A radio wave is an electromagnetic disturbance consisting of oscillating electric and magnetic fields. If the wavelength is long enough, a single oscillation will take a long time to pass through the lab. For argument’s sake let’s say the wavelength is two light-years. The fields in the lab will take one full year to go from zero to a maximum and back to zero.7 If in our laboratory the field was zero in December, it will be maximum in June.
The slowly changing fields will mean that the behavior of electrons will slowly change with time. For the winter months, when the fields are smallest, the electrons, atoms, and molecules will behave normally. In the summer, when the fields are at their maximum, the electrons will move in strange orbits, and atoms will be squashed in directions perpendicular to the magnetic field. The electric field will also distort the shapes of atoms by pulling the electrons and nuclei in opposite directions. The Laws of Physics will appear to change with the seasons!
What about the Higgs field? Can it change with time? Remember that normal empty space is full of Higgs field. Imagine that an evil physicist invented a machine—a “vacuum cleaner”—that could sweep away the Higgs field. The machine would be so powerful that it could push the universe, or part of it, up the hill to the top of the mountain in the middle of the Higgs-scape. Bad things would happen; atoms would disintegrate, and all life would terminate. What happens next is surprisingly simple. Pretend the Higgs-scape really is a Landscape with a high hill separating two valleys. The universe would act like a small, round BB ball, balancing precariously on the knife-edge between falling to the left and falling to the right. Obviously the situation is unstable. Just a tiny tap one way or the other would send the ball plummeting toward a valley.
If the surface of the Landscape were perfectly smooth, without any friction, the ball would overshoot the valley, climb up the other side, and then roll back past the valley, up the hill, over and over. But if there is the smallest amount of friction, the ball will eventually come to rest at the lowest point of one of the valleys.8
That is how the Higgs field behaves. The universe “rolls” around on the Landscape and eventually comes to rest in a valley representing the usual vacuum.
The bottoms of valleys are the only places where an imaginary ball can stand still. Placed on a slope, it will roll down. Placed at the top of a hill, it will be unstable. In the same way, the only possible vacuum with stable, unchanging Laws of Physics is at the bottom of a valley in the Landscape.
A valley does not necessarily have to be the absolute lowest point on the Landscape. In a mountain range with many valleys, each surrounded by peaks, some of the valleys may be quite high, higher in fact than some of the summits. But as long as the rolling universe arrives at the bottom of a valley, it will remain there. The mathematical term for the lowest point of a valley is a local minimum. At a local minimum, any direction will be uphill. Thus, we arrive at a fundamental fact: the possible stable vacuums—or equivalently, the possible stable Laws of Physics—correspond to the local minima of the Landscape.
No mad scientist is ever going to sweep the Higgs field away. As I mentioned earlier, just to sweep out one cubic centimeter of space would require all of the energy radiated by the sun in a million years. But there was a time roughly fourteen billion years ago when the temperature of the world was so high that there was more than enough energy to sweep away the Higgs field from the entire known universe. I am referring to the very early universe, just after the Big Bang, when the temperature and pressure were tremendously large. Physicists believe that the universe began with the Higgs field equal to zero, i.e., up at the top of the hill. As the universe cooled, it rolled down the slope to the valley that we now “inhabit.” Rolling on the Landscape plays a central role in all modern theories of cosmology.
The Higgs-scape has a small number of local minima. That one of the minima should have vacuum energy as small as 10-120 is incredibly improbable. But as we will see in chapter 10, the real Landscape of String Theory is far more complex, diverse, and interesting. Try to imagine a space of five hundred dimensions with a topography that includes 10500 local minima, each with its own Laws of Physics and constants of nature. Never mind. Unless your brain is very different from mine, 10500 is far beyond imagining. But one thing seems certain. With that many possibilities to choose from, it is overwhelmingly likely that the energy of many vacuums will cancel to the accuracy required by Weinberg’s anthropic argument, namely 119 decimal places.
In the next chapter I want to take a break from technical aspects of physics and discuss an issue having to do with the hopes and aspirations of physicists. We will come back to “hard science” in chapter 5, but paradigm shifts involve more than facts and figures. They involve esthetic and emotional issues and fixations on paradigms that
may have to be abandoned. That the Laws of Physics may be contingent on the local environment, somewhat like the weather, represents a devastating disappointment to many physicists, who have an almost spiritual feeling that nature must be “beautiful” in a certain special mathematical sense.
CHAPTER FOUR
The Myth of Uniqueness and Elegance
“God used beautiful mathematics in creating the world.”
— PAUL DIRAC
“If you are out to describe the truth, leave elegance to the tailor.”
— ALBERT EINSTEIN
“Beauty is worse than wine, it intoxicates both the holder and beholder.”
— ALDOUS HUXLEY
What Physicists Mean by Beautiful
The anthropic controversy is about more than scientific facts and philosophical principles. It is about what constitutes good taste in science. And like all arguments about taste, it involves people’s esthetic sensibilities. The resistance to anthropic explanations of natural facts derives, in part, from the special esthetic criteria that have influenced all great theoretical physicists—Newton, Einstein, Dirac, Feynman—right down to the current generation. To understand the strong feelings involved, we must first comprehend the esthetic paradigm that is being challenged and threatened by dangerous new ideas.
Having spent a good part of a lifetime doing theoretical physics, I am personally convinced that it is the most beautiful and elegant of all the sciences. I’m pretty sure that my physicist friends all think the same. But most of us have no clear idea of what we mean by beauty in physics. When I have raised the question in the past, the answers varied. The most common was that the equations are elegant. A few answered that the actual physical phenomena are beautiful.
Physicists no doubt have esthetic criteria by which they judge theories. Conversations are peppered with words like elegant, beautiful, simple, powerful, unique, and so on. Probably no two people mean exactly the same thing by these words, but I think we can give broad definitions on which physicists will more or less agree.
If there is a difference between elegance and simplicity, it is too subtle for me. Mathematicians and engineers also use these terms more or less interchangeably, and they mean roughly the same thing as they do to physicists. An elegant solution to an engineering problem means one that uses the minimal amount of technology to accomplish the task at hand. Making one component serve two purposes is elegant. The minimal solution is the most elegant.
In the 1940s the cartoonist Rube Goldberg specialized in designing “Rube Goldberg machines,” which were fanciful, silly solutions to engineering problems. A Rube Goldberg alarm clock would have balls rolling down roller-coaster tracks, hammers tapping birds who pulled strings, and the whole thing ending with a bucket of water being poured on the sleeper. A Rube Goldberg machine was a decidedly inelegant solution to a problem.
Solutions to mathematical problems can similarly be evaluated in terms of elegance. A proof of a theorem should be as lean as possible, meaning that the number of assumptions, as well as the number of steps, should be kept to the minimum. A mathematical system such as Euclidean geometry should be based on the minimal number of axioms. Mathematicians love to streamline their arguments, sometimes to the point of incomprehensibility.
The theoretical physicist’s idea of elegance is fundamentally the same as the engineer’s or mathematician’s. The General Theory of Relativity is elegant because so much flows out of so little. Physicists also like their axioms simple and few in number. Any more than is absolutely essential is inelegant. An elegant theory should be expressible in terms of a small number of equations, each of which is simple to write. Long equations with too many symbols crowded together are a sign of an inelegant theory or perhaps a theory that is expressed in a clumsy way.
Where did this esthetic taste for simplicity come from?1 It’s not just engineers, mathematicians, and physicists who derive a sense of satisfaction from a neat solution to a problem. My father was a plumber with a fifth-grade education. But he relished the symmetry and geometry of well-placed piping. He took deep professional pride in finding clever ways to minimize the pipe needed to run a water line from one point to another—without violating the esthetic rules of parallelism, rectangularity, and symmetry. It wasn’t because of the money he could save by cutting down on materials. That was trivial. His pleasure at an ingenious simplification and an elegant geometry was not so different from my own when I find a neat way to write an equation.
Uniqueness is another property that is especially highly valued by theoretical physicists. The best theories are ones that are unique in two senses. First of all, there should be no uncertainty about their consequences. The theory should predict all that is possible to predict and no more. But there is also a second kind of uniqueness that would be especially treasured in what Steven Weinberg calls a final theory. It is a kind of inevitability—a sense that the theory could not be any other way. The best theory would be not only a theory of everything, but it would be the only possible theory of everything.
The combination of elegance, uniqueness, and the power to answer all answerable questions is what makes a theory beautiful. But I think physicists would generally agree that no theory yet devised has fully lived up to these criteria. Indeed, there is no reason why any but the final theory of nature should be perfect in its beauty.
If you asked theoretical physicists to rank all theories esthetically, the clear winner would be the General Theory of Relativity. Einstein’s ideas were motivated by an elementary fact about gravity that every child can understand: the force of gravity feels the same as the force due to acceleration. Einstein performed a thought experiment in an imaginary elevator. His point of departure was the fact that in an elevator it is impossible to distinguish between the effects of a gravitational field and the effects of upward acceleration. Anyone who has been on a high-speed elevator knows that for the brief period of upward acceleration, you feel heavier: the pressure on the bottoms of your feet, the pull on your arms and shoulders feel exactly the same whether caused by gravity or the elevator’s increasing velocity. And during the deceleration you feel lighter. Einstein turned this trivial observation into one of the most far-reaching principles of physics: the principle of equivalence between gravity and acceleration, or more simply, the equivalence principle. From it he derived the rules that govern all phenomena in a gravitational field as well as the equations for the non-Euclidean geometry of space-time. It is all summarized in a few equations, the Einstein equations, with universal validity. I find that beautiful.
This brings up another facet of what beauty means to some physicists. It’s not just the final product of Einstein’s work on gravity that I find pleasing. For me a great deal of the beauty lies in the way he made the discovery: how it evolved from a thought experiment that even a child can understand. And yet I have heard physicists claim that if Einstein had not discovered the General Theory of Relativity, they or someone else would have soon discovered it in a more modern, more technical, but in my opinion, much less beautiful way. It’s interesting to compare the two routes to Einstein’s equations. According to these alternate-world historians, they would have attempted to build a theory along the lines of Maxwell’s electrodynamics. Maxwell’s theory consists of a set of eight equations whose solutions describe wavelike motions of the electromagnetic field. Those same equations also imply the ordinary forces between magnets and between electric charges. It is not the phenomena but the form of the equations that would have been the inspiration for modern-day theorists. The starting point would have been an equation for gravitational waves, similar in form to the equations describing light or sound waves.2
Just as light is emitted from a vibrating charge, or sound from an oscillating tuning fork, waves of gravity are emitted by rapidly moving masses. While the equations describing the waves are mathematically consistent, trouble would arise when the waves were allowed to interact with massive objects. Inconsistencies would arise that do no
t occur in Maxwell’s theory. Undaunted, the theorists would have searched for extra terms to add to the equations to make them consistent. By trial and error they would find a series of successive approximations, each being better than the last. But at any given stage, the equations would still be inconsistent.
Consistency would be achieved only when an infinite number of terms were summed together. Moreover, when all the terms were added, the result would be exactly equivalent to Einstein’s equations! By a series of successive approximations, a route would have been found to a unique theory that would be equivalent to general relativity. There would be no need to ever think about accelerating elevators. The mathematical requirement of consistency, together with the method of successive approximations, would suffice. For some this is beautiful. It could hardly be called simple.
As for the elegance of the equations, I will display them in the wonderfully simple form that Einstein derived.
This small box with a few simply placed symbols contains the entire theory of gravitational phenomena: the falling of stones, the motion of the moon and earth, the formation of galaxies, the expansion of the universe, and much more.
The approach advocated by the modernists, although yielding the same content, would lead to an open-ended infinity of successive approximations. In that form the equations are distinctly inelegant.
Nevertheless, I have to admit that while the “modern derivation” may have missed the elegance of the Einstein equations, it did one thing rather well. It demonstrated the uniqueness of the theory. At each level of approximation, the extra terms needed to restore consistency are uniquely determined; the theory is unambiguous. Not only does it describe how gravity works, but it also shows that it could not have been otherwise.
The Cosmic Landscape Page 12