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PRACTICING SCIENCE WITHOUT mathematics is inconceivable to me. Mathematics is the unifying language of the universe; it is the language in which the laws of nature are written—or, in Galileo’s words, “Mathematics is the language in which God has written the universe.”
I’m going to spare you the full working through of my math, but I am going to briefly retrace the journey I took as my idea germinated. In the process, I hope to share some of the beauty hidden in that math.
As we saw in the context of the string-theory landscape, in physics, a collection of many possible beginnings is generally known as the “space of initial states.” If we are speaking about universes, each of these initial states might produce a universe, although almost certainly not all of them do. In my case, the collection of energies presented in the string-theory landscape offered me the space of initial states on which I could perform my calculations.
I was aware that starting with a pool of multiple beginnings opened the door to the possibility of multiple universes—a multiverse. However, rather than assuming that outcome at the start, I wanted to calculate and compare the individual chances of success for each of these initial states. Perhaps the answer would turn out to be that all of them except one had a zero chance to bring a universe into being. Or perhaps many of them were likely to produce their own universes. But I wouldn’t know until I sat down to do the calculations.
If the calculations showed that only one universe came into existence, then I would be happy to concede that we had to find a different way to address the serious problem raised by Penrose’s claim that our universe had almost no chance to exist. But if my calculations showed that many of these initial states were each capable of producing a universe—if it led to a picture of the cosmos where our universe is almost as unremarkable as a grain of sand on a beach—then I and the rest of the physics community had to start to consider our universe as part of a multiverse, with all of the incredible, Everett-style implications that came along with it.
From there, my mind went to the type of mathematics required for a full implementation of my idea and a plan of the next steps I needed to take. The equations I had to solve for the wave function of the universe propagating through the vast string-theory landscape of energies were very complicated. But if somehow I could do this calculation, then estimating the chance for our universe to exist was straightforward: the probabilities for each possible universe starting out on the landscape were simply given by the square of the respective wave-packet solution. Once I knew these solutions, I could compare which infant universes, out of all our choices, were the most likely to have survived creation and “inflated.” And perhaps, finally, I could resolve the question of whether the probability that a universe like ours had to emerge out of this structure was higher or not compared to other possible universes.
The next day, I began the calculations. It was challenging; the mathematics involved were highly complex and required significant work. I had to follow the evolution of all the branches of the wave function of the universe as they spread throughout the vast complicated structure of the landscape energies. But the difficulty did not matter; it was too exciting. If it worked, this approach would reveal how the interplay of quantum and gravitational forces could explain (instead of postulate) the origin of our universe—and, quite possibly, its emergence from the multiverse.
Far from reaching the end of the cosmic story, I felt that, in many ways, the cosmological expedition into the workings of the universe had just begun. Science was poised to make another leap, this time beyond the borders of our universe and back in time to the instant of its creation. Waiting there, hiding in the cracks of the foundations of our beliefs, was the ghost of Everett—and, waiting with him, the theory of the multiverse itself.
This scientific leap, as I envisioned it, was captured in that one phrase that came to me at the coffee shop: quantum mechanics on the landscape of string theory. And the theory I was building from it would come to be called the theory of the origin of the universe from the quantum landscape multiverse—or, for short, the quantum landscape multiverse. To this day, it is one of my proudest creations.
To show how I ended up with a theory of the quantum landscape multiverse while trying to understand the unlikely origin of our universe, I need to go into more of the science involved in the calculation and the derivation of the answer. For starters, my theory of the origin of the universe from the quantum landscape multiverse relied on treating the entire universe as a quantum wave packet.
We know for a fact that at its earliest moment, our universe was about a few Planck lengths—smaller than the tiniest quantum particle we know. I reasoned that this fact justified my thinking of applying quantum theory to the whole universe in considering our infant universe to be a branch of the wave function of the universe.
If the wave function of the universe is a bundle of waves, meaning it contains individual branches of quantum wave packets, each of which can potentially seed a universe, then it will likely give rise to a number of worlds rather than just one. Based on the wave-particle duality of quantum mechanics, we are allowed to think of these branches of the wave function of the universe as either a bundle of waves or a beam of quantum particles.
Next, let’s allow the wave function of the universe (with all its branches) to run loose on the landscape of string theory, and see what happens.
So far, the emerging picture of a multiverse from the wave function of the universe seemed close to the spirit of Everett’s many-worlds interpretation of quantum mechanics. But not for long. My first attempt at solving the quantum equations that would allow me to find out which energy site our universe chose from the landscape produced a surprising result.
To grasp this situation intuitively, think of the marbles that our gaggle of physicists accidentally sent rolling down the ragged terrain of mountains and valleys in the Rockies (or in the Lake District, depending on your preference). We already know that the marbles rolled until they settled into the lowest possible points in the valleys.
Like the contours of Earth’s gravitational potential energy on a mountain range, the string-theory landscape has its own energy valleys—a whole landscape of vacua. The string landscape has billions of energy valleys, vacua whose depths are spread randomly, ranging from low to high energies. You may recognize vacua as the plural of vacuum. But in the quantum-size string-theory world, especially the compactified eleven-to-four-dimensional string-theory landscape world, a vacuum is not empty. Rather, it is the default or stable state—it is the point where our theoretical marble can stop rolling.
These near-endless chains of vacua are what supply a landscape of potential energies on which the branches of the wave function can settle. So if we continue with our marble analogy, the landscape of energy valleys is like the gravitational potential of a physicist’s classical mountainous landscape. The marbles are the branches of wave-universes or, equivalently (using wave-particle duality), a beam of quantum particles in the wave function of the universe trying to travel through this terrain. As these wave-universes move through the landscape, they sample the energies of the various landscape valleys on which they may eventually settle.
Instead of the example of marbles rolling down mountains to settle in a variety of valleys, perhaps a more useful example to illustrate what was missing in my previous thinking—and the idea that came to me in the coffee shop—is the analogy between the wave function of the universe traveling through the string-theory landscape and a quantum system we are more familiar with: a beam of electrons going through a long piece of wire capable of conducting electricity.
Understood from a quantum perspective, a piece of wire is actually a long chain of billions of atoms and fluxes of electrons trying to travel through the chain without being stuck in any atomic site. In this case, the branches of the wave function of the universe are like the beam of electrons trying to travel along the atoms of that wire, with the energy valleys corresponding to the potential electrical energies contained in that chain of individual atoms of the wire. If we have a perfectly clean piece of wire (a perfect conductor) made of identical atoms with identical spacing and with no impurities or air bubbles, then electricity is conducted evenly throughout the wire, which means that none of the electrons in the beam get trapped inside the wire. All the electrons travel successfully from one end to the other without any losses, thus giving us a perfect electric current. (In real life, wires are not perfect. They may contain bits of impurity or air bubbles. Some of the electrons in the beam may get trapped inside the impure chain and never travel all the way through. In this case, the electric current will suffer a loss, since more electrons are going in than are coming out of the wire.)
In our analogy, the chain of atoms in a perfect wire corresponds to a regular or periodic chain of landscape energy vacua, where all of the vacua have the same energy. But such a perfectly ordered landscape of energies is really bad news if we are trying to harvest universes out of them. Here is why: If you compare the electrons going through the wire to the wave function of the universe going through the landscape, then having a perfect conduction of quantum wave packets means they travel all the way through the landscape without ever getting stuck in any of its energy valleys. If none of the wave packets are confined and settle on top of any of the landscape energy valleys from which they can draw sufficient Big Bang initial energy to fire up a universe, then no actual universes will grow. An orderly landscape with a chain of periodic identical vacua is barren—it looks like a flat, uniform desert.
And so, my final step in putting QM on the landscape—just as we did previously when imagining marbles rolling down a mountain and just as physicists do when they study the quantum behavior of electrons going through a piece of wire—was to solve the quantum equation, a Schrödinger-type equation that describes how a wave or a quantum particle moves under the influence of an external force (potential energy) and the probability the particle has of taking a particular path of motion. I could use these equations to find out what happens to the wave function of the universe when it travels along an “impure wire”—the randomly scattered potential energy valleys of the landscape.
Quantum formalism, when applied to the wave function of the universe, is referred to as quantum cosmology. It is an advanced version of regular quantum theory, but it addresses the motion of waves in abstract spaces, like the space of energies of the landscape, instead of the motion of quantum particles through the real space-time composed of length, width, height, and time.
Quantum cosmology offers a set of equations and rules that describes what happens to the wave function of the universe propagating on abstract spaces, such as the space of energies of the landscape, and how a real universe in a physical space-time is spit out from these initial waves and energies. One of its founding fathers is Bryce DeWitt, the same scientist who defended Everett (and who rebranded his clunky-sounding “universal wave function of the universe” as “the many-worlds interpretation of quantum mechanics”). The other is John Wheeler, who mentored Everett. The quantum equation that gives the probabilities for the wave function of the universe is known as the Wheeler-DeWitt equation.
The Wheeler-DeWitt equation in quantum cosmology is the equivalent of the Schrödinger equation in regular quantum mechanics. And by using a wave function of the universe on the landscape, it means that the beam of branches gives rise to a collection of universes in the same manner as in Everett’s many-worlds multiverse. In other words, Everett’s many-worlds theory is embedded within this theory.
By applying the Wheeler-DeWitt equation and the quantum probability rules to my idea of a wave function of the universe propagating through the string-theory landscape, I could derive how our universe was selected and which energy site on the landscape it chose for its Big Bang. Or so I thought back then. As it turned out, the math was harder than I anticipated.
Despite how alluring the concept that had come to me in the coffee shop was, the mechanics of the solution were exceedingly complicated. As you can imagine, a landscape with 10^600 valleys where our universe could potentially settle meant the required math was horrendous. I had a choice: I could make some crude assumptions to simplify the landscape—say, reducing the size of the landscape to contain two energy valleys instead of trillions of them, in which case the equation became manageable and could be solved by hand—or I could attempt to solve the real equation, without simplifications and with all 10^600 vacua in it, no matter how long that might take.
I suspected that similar equations might have been solved before by condensed-matter physicists, who study materials like the piece of wire with the electrons. So I consulted with my condensed-matter colleagues at UNC and spent the next six months taking a crash course in condensed-matter physics. It helped me understand and identify mathematically similar structures in the string-theory landscape, which exist in condensed matter in the form of exotic materials known as “quantum dots” and “spin glass.” Luckily, complicated mathematical methods (like random matrix theory, which might sound familiar to the advanced reader) for solving these types of problems had been developed in great detail by the condensed-matter scientists.
My hunch turned out to be correct. By the end of my detour into condensed-matter physics, I could finally solve the equation for the wave function of the universe on the true, unabridged landscape of string theory. If I had chosen the easier way and simplified the landscape to only two energy valleys, my answer would have been wrong.*
One of the keys to my solution is the fact that energies in the landscape vacua come in all different sizes and distances. The landscape, in other words, looks nothing like a flat, uniform desert; it’s full of peaks, valleys, and hills. Moreover, the distribution of all these asymmetrical features is irregular; it is disordered, nothing like our “perfect wire.” This disorder turned out to be crucial.
To visualize what happens in a disordered landscape, let’s get back to our example of electrons passing through a wire. If instead of the perfect wire, we have a wire made of an insulating material, a material full of impurities and disorder (for example, glass), then the energies in its chain of atoms are also irregular and disordered. Electrons cannot go all the way through the material; they become trapped. That’s why such materials are called insulators. In the large, visible world, we know this to be true: If I shoot an electric current into your kitchen window, the beam of electrons gets confined inside the glass and stays there. Once the electrons are trapped—localized is a better term for physics—inside the glass, they are these tiny wave packets confined to energy sites located along the individual atoms in the chain.
So what happens to our electrons-and-glass example in the microscopic world? If we were to use an atomic microscope to observe the electrons inside the glass, we would see that the electrons’ localized behavior is a textbook example of quantum interference. Think back to our double-slit experiment with light and atoms. Like an ocean wave crashing against a rocky shoreline, the electron wave packet “smashes” along the chain of atoms and tries to push through the wire. But instead, it keeps getting scattered and broken into two parts, a reflected and a transmitted wave, at each of the atom’s sites. The more scattering sites the electron wave goes through, the more of these reflected and transmitted waves we have. The result is a bunch of quantum waves inside the material, which we know (from chapter 2) will add up and interfere at every single point.
In the case of a disordered chain of atoms, these waves are out of phase with each other due to this disorder. They are “disoriented” from being scattered from irregular atomic sites. Therefore, their interference pattern is destructive (a phenomenon you saw visualized back in figure 4). For our electrons in a particle form traveling along the wire, this means that an electron will be found trapped around some atomic site (where most of its wave is concentrated) but will likely not be found anywhere else in the wire. (If I had oversimplified the landscape to two vacua instead of zillions, then there would have been only two scattered pieces of the wave to add up. The two-vacua kind of landscape would completely lose the complex interference pattern and localization. For this reason, any simplification of the string landscape into a two-vacua object would have produced the wrong physical result.)
In short, like the electron going through an insulating material, the beam of tiny quantum universes trying to travel through the disordered string landscape gets trapped in the various energy valleys inside the landscape. Rather than passing quickly through the energy field, these tiny wave packets resembling quantum universes get stuck. The disordered distribution of vacua energies on the landscape was the key factor that triggered the localization of the branches of the wave function on its vacua. Once these tiny quantum universes “localize” on a particular landscape energy site, they take that vacua’s energy, which drives them through the explosion of Big Bang inflation and makes them grow large.
And this was the picture that emerged out of the equations: As the quantum waves try to make it through the landscape, they get trapped in some vacua, take the vacua’s energy to go through an inflationary expansion, and create real, macroscopic universes out of these infant quantum universes. But since different quantum wave-universes settle in different vacua, their Big Bang energies are also different, because it all depends on what landscape energy vacua they find themselves in.
There is an easy way this energy differential can be visualized. If you recall, when I talked about destructive and constructive interference of waves, I used the example of a concert hall where the waves from all the instruments in the orchestra add up to amplify in certain seats in the hall (the expensive seats) and cancel each other out in other seats (the cheap seats). Let’s think for a moment of the individual seats in the concert hall as being specific energy vacua on the landscape. The interference among the branches of the wave function of the universe looks similar to the interference of the sound waves from the orchestra in the concert hall. Therefore, a quantum wave packet would concentrate mostly around one site and fall off to zero everywhere else, like a concert hall where there is only one expensive seat—one good seat where the music is amplified—and the rest of the seats in the hall are cheap seats, where the music can barely be heard. This single good seat is the vacua on which an infant quantum universe sits. But which “seat” on the landscape would our infant universe have chosen?
The first time I got to the end of my equations and looked at my solution, I felt dumb, because the answer I got was nonsense. Even worse, it took me back full circle to the original problem of our universe’s unlikely origins. The solution I found asserted that the most probable universe was the one that had started at the lowest energies on the landscape, which meant that, according to this calculation, a low-energy Big Bang had the highest chance to produce a universe. Translation of this result: our high-energy Big Bang universe once again had the smallest chance of coming into existence!
With hindsight, I know that this nonsense answer should not have been a surprise. I should have expected it before I sat down to calculate. I should have anticipated that quantum particles, like marbles rolling down mountains, will search for the lowest energy valleys to settle in because they are more stable there. I should have had foreseen this before attempting to solve the Wheeler-DeWitt equations. But I didn’t.
After the first time I worked through the equations, I spent any free moment going back through a photographic copy of each step and calculation, trying to find what I assumed was a mistake. What had I missed? Where had I gotten off track? After a lot more thinking and many more walks in the solitude of the hot and humid North Carolina trails, I finally realized what I was missing. It turns out I was missing a lot!
In my first try, I understood how the branches of the wave function of the universe localize on the various landscape vacua. But I had missed a crucial piece of the puzzle: the separation or decoupling of the different entangled branches of the wave function of the universe from one another as they produce infant universes. Including entanglement among the branches of the wave function in my equations was not the final chapter in calculating the probability of our origin. To complete the project, I had to identify a way to decouple the entangled branches in the wave function of the universe as they were about to inflate and create their own universes.
Unlike entangled quantum particles, big classical universes cannot add up, interfere, or become entangled under the same sky. Entanglement, as we will see in more detail in the next chapter, is a purely quantum effect that doesn’t exist among classical objects. Thus, it needs to be wiped out before our quantum universe, and the others entangled with us, can grow and make the transition from microscopic quantum wave to macroscopic classical universe. In physics, this process of decoupling, the mechanism that destroys entanglement, is known as decoherence.
Overlooking decoherence had taken me full circle back to the mystery of our origin—the solutions I found incorrectly indicated that the most likely universe is one that starts from the lowest Big Bang energy.
Decoherence would turn out to be the key to our final result—that, in fact, the most probable universes produced out of the quantum landscape were universes that started out at very high energies, just like our own universe had! Our result demonstrated, first, that in contrast to Penrose’s estimate, our origin is a very likely one—there is absolutely nothing special or exclusive about our beginning; and second, that the story of the origin of the universe could now be calculated and derived using the laws of nature, and not just postulated.
It was a thrilling moment—but I wouldn’t be able to dwell on it for long. I was about to discover that a quantum selection process was in fact at work, one that gave those infant universes unequal chances of existence—and that helped explain how our universe turned out the way it did.