Our Mathematical Universe: My Quest for the Ultimate Nature of Reality

by Max Tegmark

In an “island universe” model where space is infinite but all the matter is confined to a finite region, almost all members of the Level I multiverse would be dead, consisting of nothing but empty space.

The larger the scale we observe, the more uniformly filled with matter our Universe looks (Figure 4.6). Barring conspiracy theories where our Universe is designed to fool us, the observations thus speak loud and clear: space as we know it appears to continue far beyond the edge of our Universe, teeming with galaxies, stars and planets.

although these other parts of the Level II multiverse are in the same space as we are, they’re more than infinitely far away in the sense that we’d never reach them even if we traveled at the speed of light forever.

Level II universe.

The eternally inflating regions in the figure, which I’ve drawn as thin vertical bars separating the U-shaped Level I multiverses, will in fact expand rapidly, and eventually, parts within them will stop inflating, giving rise to additional U-shaped regions.

Level II

This example may sound silly, and if a real fish were to think this, we might be tempted to laugh at it. But could it be that what we humans think of as empty space is also some form of medium? Then the last laugh would be on us!

medium

This means that we observers are easily tricked into making the same mistake as the fish: because we observe space to have the same properties everywhere in our Universe, we’re tempted to mistakenly conclude that space is like that everywhere else as well.

our cosmic history has been a gravitational tug-of-war between dark matter trying to pull things together and dark energy trying to push them apart. Because galaxy formation is all about pulling things together, I think of dark matter as our friend and dark energy as our enemy.

Cars are created by car factories, rabbits are created by rabbit parents and solar systems are created from gravitational collapse in giant molecular clouds. So it’s quite reasonable to assume that our Universe was created by some sort of universe-creation mechanism (perhaps inflation, perhaps something totally different).

all the other mechanisms we mentioned naturally produce many copies of whatever they create; a cosmos containing only one car, one rabbit, and one solar system would seem quite contrived. In the same vein, it’s arguably more natural for the correct universe-creation mechanism, whatever it is, to create many universes rather than just the one we inhabit.

Eternal inflation predicts that our Universe (the spherical region of space from which light has had time to reach us during the 14 billion years since our Big Bang) is just one of infinitely many universes in a Level I multiverse where everything that can happen does happen somewhere.

many numbers we’ve measured in physics a new meaning: they’re not telling us something fundamental about physical reality, but merely something about our location in it, forming part of our cosmic postal code.

Everything we call real is made of things that cannot be regarded as real. —Niels Bohr

he argued that the sharp corners of the tetrahedron explained why fire was painful, that the ball-like shape of the icosahedron explained water’s ability to flow, and that the unique ability of cubes to be compactly stacked explained Earth’s solidity.

An atom is simply named according to the number of protons it contains (1 = hydrogen, 79 = gold, etc.), so what the alchemists failed to do was clearly to play Legos with the protons and move them from one atom to another. Why couldn’t they do it? We now know that they failed not because they tried something impossible, but merely because they didn’t use enough energy!

Since the electric force causes equal charges to repel each other, the protons in atomic nuclei would fly apart unless some more powerful force held them together.

stable ghostlike particles known as electron neutrinos, muon neutrinos and tau neutrinos have been discovered—we encountered them in the last chapter, and they are so shy that they barely interact with other particles at all: if a neutrino crashes into the ground, it typically passes right through Earth and emerges unscathed on the other side, and continues into space.

string theory suggests precisely this: that if we could slam them together with vastly (perhaps ten trillion times) more energy than today, we’d discover that everything is made of tiny vibrating strings,

One of the most important discoveries in physics has been that there are additional quantities which, just as energy and momentum, appear to always be conserved: electric charge is the most familiar example, but there are also other kinds of conserved quantities known, with names such as isospin and color.

If you could carefully measure the voltage between two points in a beam of light, you’d find that it oscillates over time; the frequency f of this oscillation (how many times per second it oscillates) determines the color of the light, and the strength of the oscillation (the maximum number of volts you measure) determines the intensity of the light.

Light. oscillation. superposition.

depending on their frequency (by increasing frequency, we call them radio waves, microwaves, infrared, red, orange, yellow, green, blue, violet, ultraviolet, x-rays, gamma rays), but they’re all forms of light and they’re all made of photons.

Light

People knew that the temperature T of something is a measure of how rapidly its particles are moving around, and that the typical motion energy E of a particle was given by the formula E = kT, where k is a number known as Boltzmann’s constant.

E=kT. Boltzmann's constant

I discovered that his freethinking didn’t go down well in his home village either: you’ll see in the photo I took (Figure 7.4) that the small town of Alpbach has buried their most famous citizen ever in a quite modest grave right at the edge of the cemetery. …

Shrodinger fact

Schrödinger altered the classical description of the world in two ways: The state is described not by positions and velocities of the particles, but by a wavefunction. The change of this state over time is described not by Newton’s or Einstein’s laws, but by the Schrödinger equation.

A key part of it is to add a loophole to the second item mentioned above, postulating that change is only governed by the Schrödinger equation part of the time, depending on whether an observation is taking place. Specifically, if something is not being observed, then its wavefunction changes according to the Schrödinger equation, but if it is being observed, then its wavefunction collapses so that you find the object only in one place.

the concept of observation or not

Schrödinger equation Equation that lets us predict how the wavefunction will change in the future

Schrödinger equation

Whereas traditional physical processes would be described by mathematical equations, the Copenhagen interpretation had no equation specifying what constituted an observation, that is, exactly when the wavefunction would collapse.

Copenhagen interpretation. Observation

An even simpler example is a pencil balanced on its tip, where a microscopic nudge of the initial tilt can determine the direction in which it will ultimately come crashing down. Whenever such chaotic dynamics are at play, the initial position of a single atom can make all the difference, so if that atom is in two places at once, you’ll end up with macroscopic things in two places at once.

superposition. Be at two places at once.

This gives them properties both of traditional particles (they’re either here or there) and of waves (they can be in several places at once in a so-called superposition).

But the physics of smaller particles are different than the object of their congregation

This was when my head really started to spin, because the Quantum Cards experiment is just one particular example of how microscopic quantum weirdness gets amplified into macroscopic quantum weirdness.

Just answered your own question there, physics is different on different scale

Whereas the Level I and Level II kinds are far away in our good old three-dimensional space, the Level III ones can be right here as far as these three dimensions are concerned, but separated from us in what mathematicians call Hilbert space, an abstract space with infinitely many dimensions where the wavefunction lives.

Hilbert space

careful experiments show that large objects never act like they’re in two places at once, even if you don’t look at them. In particular, they never display wavelike properties that make so-called quantum interference patterns.

Feynman had emphasized that quantum mechanics splits our Universe into two parts: the object under consideration and everything else (referred to as the environment). However, I felt that an important piece of the quantum puzzle was missing here: your mind. As Everett’s work had shown, understanding the process of observation requires us to include a third part of our Universe as well: your mental state as an observer,

the concept of observation. your own mind?

the entropy of an object decreases while you look at it and increases while you don’t. Decoherence is simply a measurement that you don’t know the outcome of.

The Everett interpretation had Level III (quantum) parallel universes as its hallmark, but you can see in the figure that you can safely ignore them, because they’re all indistinguishable. In this sense, the Level I and Level III multiverses are unified: as long as you have an infinite space with a Level I multiverse, you can ignore all its Level III parallel universes, since they’re in practice all just identical copies.

as long as you have an infinite space with a Level I multiverse, you can ignore all its Level III parallel universes, since they’re in practice all just identical copies.

Back when quantum mechanics was born, there were indeed many physicists who believed that quantum mechanics would prove to work only on the atomic scale. Well, no longer! The simple double-slit interference experiment (Figure 7.7), hailed by Feynman as the mother of all quantum effects, has been successfully repeated for objects larger than individual elementary particles: atoms, small molecules and even the soccer ball–shaped carbon-60 “Bucky Ball” molecule.

larger object obey the quantum mechanics as well?

This mathematically simplest quantum theory also predicts a censorship effect called decoherence, which hides most such weirdness from us, mimicking wavefunction collapse.

Decoherence happens constantly in your brain, debunking popular suggestions about “quantum consciousness.”

The wavefunction and Hilbert space, which constitute arguably the most fundamental physical reality, are purely mathematical objects.

For fun, I sometimes compare how the same news story is reported online by MSNBC, FOX News, the BBC, Al Jazeera, Pravda and elsewhere. I find that when it comes to telling the truth, the whole truth, and nothing but the truth, it’s the second part that accounts for most of the differences in how they portray reality: what they omit.

External reality The physical world, which I believe would exist even if we humans didn’t Consensus reality The shared description of the physical world that self-aware observers agree on Internal reality The way you subjectively perceive the external reality

The difficulty of linking external reality to consensus reality reached a new record high with the discovery of quantum mechanics, manifested in the fact that we physicists still argue about how to interpret the theory today, about a century after its inception.

Today, the grand challenge of theoretical physics is unifying quantum mechanics with gravitation. Based on this historical progression of examples, I predict that the correct mathematical theory of quantum gravity will break all previous records in being difficult to interpret.

The mathematical description of the external reality that theoretical physics has uncovered appears very different from the way we perceive this external reality.

Einstein’s revolutionary relativity theory never won the Nobel Prize,* Einstein himself dismissed Friedmann’s expanding-universe discovery, and Hugh Everett never even got a job in physics.

fact

Sirius A, the brightest star in our night sky (center), and a star orbiting the monster black hole at the center of our Galaxy (right), which is four million times more massive than our Sun.

Sirius A

we saw that there’s something that’s arguably even more fundamental than our three-dimensional space and the particles within it: the wavefunction and the infinite-dimensional place called Hilbert space where it lives. Whereas particles can be created and destroyed, and can be in several places at once, there is, was and always will be only one wavefunction, moving through Hilbert space as determined by the Schrödinger equation—and the wavefunction and Hilbert space are purely mathematical objects.

External Reality Hypothesis (ERH): There exists an external physical reality completely independent of us humans.

Mathematical Universe Hypothesis (MUH): Our external physical reality is a mathematical structure.

If we assume that reality exists independently of humans, then for a description to be complete, it must also be well defined according to nonhuman entities—aliens or supercomputers, say—that lack any understanding of human concepts. Put differently, such a description must be expressible in a form that’s devoid of any human baggage like “particle,” observation or other English words.

all physics theories that I’ve been taught have two components: mathematical equations and “baggage” —words that explain how the equations are connected to what we observe and intuitively understand.