Quantum theory is strange and counterintuitive, but it’s very precise. Lots of analogies and broad concepts are presented in popular science trying to give an accurate description of quantum behavior, but if you really want to understand how quantum theory (or any other theory) works, you need to look at the mathematical details. It’s only the mathematics that shows us what’s truly going on. 

Mathematically, a quantum object is described by a function of complex numbers governed by the Schrödinger equation. This function is known as the wavefunction, and it allows you to determine quantum behavior. The wavefunction represents the state of the system, which tells you the probability of various outcomes to a particular experiment (observation). To find the probability, you simply multiply the wavefunction by its complex conjugate. This is how quantum objects can have wavelike properties (the wavefunction) and particle properties (the probable outcome).

No, wait. Actually a quantum object is described by a mathematical quantity known as a matrix. As Werner Heisenberg showed, each type of quantity you could observe (position, momentum, energy) is represented by a matrix as well (known as an operator). By multiplying the operator and the quantum state matrix in a particular way, you get the probability of a particular outcome. The wavelike behavior is a result of the multiple connections between states within the matrix.

Okay, maybe not. Actually, as Richard Feynman showed, the behavior quantum objects are governed by a summation of possible histories known as a path integral. In order for a quantum particle to get from point A to point B, it has to take a certain path. Since quantum objects could take lots of different paths (unlike classical particles), you have to calculate each possible path and sum them in a particular way to determine the probability that the quantum object will arrive at point B.

It turns out that there are multiple ways mathematically to describe quantum behavior. These three are just the most popular, but there are at least nine versions of non-relativistic quantum theory, each with a different mathematical formulation. But despite these mathematical differences, each of them will make the same physical predictions. When you apply these different mathematical models to a real situation, they give the same answer every time. So which one is right? Experimentally, all of them. So it really comes down to using the version you want for a particular situation. It seems the reality of quantum theory is as indefinite as quantum theory itself.

You might think this overlapping magisteria is simply due to quantum theory’s counterintuitive properties, but that’s not the case. Take, for example, the most intuitive and clear theory of Newtonian physics. In the Newtonian model, objects are solid and real, and their motion is governed by vector forces. Push on an object, and it accelerates in that particular direction. Easy peasy. Of course this view (what we might call vector mechanics) isn’t the only mathematical formulation we have.

Another approach is analytical mechanics, where instead of looking at forces we look at constraining quantities such as the object’s energy. If I toss a baseball it isn’t free to move just anywhere. Its motion is constrained by the kinetic energy of its motion and potential energy it could gain or lose from gravity. Those constraints determine the baseball’s equation of motion. Two versions using this approach are Lagrangian and Hamiltonian mechanics. There are other mathematical formulations of classical mechanics as well, such as Gauss’ principle of least constraint, and Appell’s formulation of generalized work. Each of these uses different mathematical methods, starts with different governing principles, and still gives the same physical prediction. So which one is right? Is motion all about forces, or limiting energy, or something else?

One of the fundamental ideas of science is that scientific models must be testable. There must be a way to distinguish an accurate model from an inaccurate one, and the ultimate arbiter is experiment and observation. But nothing in this idea requires that there must be one unique valid theory. Since multiple mathematical formulations each agree with experiment, they are all equally valid.

That’s perhaps the most wondrous thing about all of this. Mathematics does contain truths about our physical universe. It allows us to develop an understanding and precision we can’t get any other way. But this mathematical truth isn’t a single absolute view. Instead, mathematics is so powerful and so subtly interconnected that there are multiple roads to understanding. Two areas of mathematics that seem widely separate often reveal elegant connections, and those connections are often central to our understanding of physical reality. It’s a wondrous and amazing thing.

So to really understand the subtle beauty of physics, you do need to understand mathematics. It isn’t always easy to understand, but it’s well worth the effort.

Miss the beginning of the series? It all starts here.

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Suppose I were to pick two numbers. I won’t tell you what those numbers are, other than to call them A and B. If I asked you what there sum is, A + B, you couldn’t tell me the answer without knowing A and B, but you could tell me one important thing: the answer will be a number. That might seem trivially obvious, but it actually says something important about numbers. Any two numbers when added together will give you a number. That’s because numbers under addition form what is known as a group. 

Vectors can be added to make another vector.

In the physical sciences, lots of things are groups, and this has real physical implications. For example, vectors are groups under addition, so if you add up a number of vectors, the result will be a vector. Since forces are vectors, if you have a number of forces acting on an object you can add them all up to find the total or net force which determines how an object will move. Newton’s laws of motion rely upon the fact that vectors are a group.

Basically, a group is any collection of things (numbers, vectors, etc.) that can be connected by some mathematical operation (addition, multiplication, etc.). There are some specific rules that have to be obeyed, but the key is how members of the group are related under their operations. While it seems trivial to say that numbers or vectors are groups, the idea of groups is much more abstract. Take, for example, rotations.

A Rubik’s cube demonstrates how rotations are a group.

Rotations form a group, where a combination of rotations is equivalent to other combinations of rotations. A Rubik’s cube is a good example of this. If someone scrambles a Rubik’s cube by rotating different parts of it, you don’t need to know what specific rotations they used. Instead you can use a process where you use different methods to return the cube to its unscrambled position. The fact that a Rubik’s cube can be solved is due to the fact the the rotations of the cube form a group. The combinations of rotations you use to solve it are equivalent to the original rotations used to scramble it.

Where things get interesting is when you look at the types of things that are always the same within a group. For example, on Earth, locations are defined by their latitude and longitude. Latitude is defined by the angle north or south of the equator, while longitude is defined as the angle east or west of the prime meridian. But suppose we used a different reference frame. Instead of the equator, suppose we started with a different circle around the Earth, for example. The original latitude and longitude are used because they are convenient, but we could use all sorts of coordinate systems if we wanted. In fact, since we could always shift from coordinate system to another, the set of possible coordinate systems is a group. If we shifted from the original coordinate frame to some new one, then our “latitude” and “longitude” would change, but the distance between any two points on Earth wouldn’t change. Mathematically, we could say that distance is invariant under a coordinate transformation.

These invariant quantities are related to the symmetry of a group. In terms of the physical universe, they describe the symmetries that exist in the cosmos. For example, imagine a star in the middle of empty space. With no other stars in the area to determine the star’s position, there is a symmetry of motion. If you could take that star and shift it some distance in a particular direction, everything would look exactly the same. This is known as translational symmetry. Mathematically, this means the linear motion of an object is invariant under a change of position. In physics we call this conservation of momentum. It turns out that the mathematical symmetry of space is connected to the physical quantity of momentum. Which brings us to the most brilliant mathematical physicist of all time, Emmy Noether.

Emmy Noether used the mathematics of group theory to show that every symmetry within a group that describes a physical phenomenon is connected to a conserved physical quantity. So, translational symmetry means there is a conservation of linear momentum. Rotational symmetry means there is a conservation of angular momentum. Conservation of charge, energy, the connection between electric and magnetic fields, are all the result of group symmetry. This relation is now known as Noether’s theorem, and its power is hard to overstate. What Noether showed was that all the conserved quantities that exist in the Universe exist because of a symmetry that exists within the abstract mathematical concept of a group. Some of the most beautiful and powerful mathematics is physically connected to the way our Universe works. Mathematics doesn’t just describe the world around us. The most basic connections within mathematics describe the bedrock of physical processes.

Everything we study within physics depends upon Noether’s theorem, from dark energy to the Higgs boson. It has transformed the way we view the cosmos, and it demonstrates the real power of mathematics when it comes to understanding the Universe.

Next time: Given that mathematics is so powerfully central to our understanding of physics, does math tell us what is true about reality? The last part of the series starts tomorrow.

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In science, physical quantities are represented by numbers. A block has a mass of 42 kilograms, a car moves at 25 meters per second, or a star is 142 light years away. In mathematics these are known as real numbers. One of the basic properties of real numbers is that any real number multiplied by itself is a positive number. For example, 3 x 3 is 9, and (-4) x (-4) is 16. But imagine a number where its product is actually negative. For example, what if some number i works so that i x i is -1. Such a number is known as an imaginary number. The term imaginary number often gives the impression that these numbers are merely mathematical abstractions, but when it comes to the physical universe, imaginary numbers are quite real. 

Complex numbers on a plane.

While any two imaginary numbers will always yield a negative number, when you add imaginary numbers to real ones, things start to get interesting. For example, 2 + 4 = 6 and 3i + 2i = 5i, but 2 + 3i can’t be simplified to a real or imaginary number. Such combinations are known as complex numbers, and they have a real number part and an imaginary number part. Because of this, complex numbers share a similarity with the vectors we talked about last time. Just as real numbers can be imagined as points on a number line, complex numbers can be imagined as points on a number plane. Any point on this complex plane can be represented by a distance (magnitude) and direction from your origin.

Circularly polarized light.

This makes complex numbers extremely useful when describing certain physical phenomena. For example, light is an electromagnetic wave that has properties such as amplitude and wavelength, but it also has an orientation known as its polarization. Complex numbers can describe light wave as it travels through space can be represented as a complex number, where the real and imaginary parts tell us the orientation of the wave. While we could describe the polarization with real numbers, complex numbers can do it much more simply and elegantly. The same is true with many other physical processes, such as oscillating currents in a circuit, or the vibration of a bridge under stress.

Given the oddness of imaginary numbers, it’s still tempting to think of them as merely a useful mathematical tool. But it turns out there are some physical processes where imaginary numbers are not just useful, they are downright required. Take, for example, the strange world of quantum theory.

The real (blue) and imaginary (red) parts of a wavefunction tell us where an object is likely to be.

The physics of quantum objects is difficult to wrap our heads around, and there are lots of interpretations to help us understand. One of the most popular versions is known as the Copenhagen interpretation. In this view, a quantum object is described by a wavefunction. Just as the function of motion for a baseball can be determined by Newton’s laws of motion, the wavefunction of an object can be determined by the Schrödinger equation. But there’s a fundamental difference. The wavefunction by itself doesn’t tell you what a quantum object is doing. Instead, if you multiply the wavefunction by itself (technically by its complex conjugate) then you get a real function that tells you what the object is likely to do. Mathematically, a solution to Schrödinger’s equation contains imaginary numbers. It has to in order to accurately describe the way a quantum object behaves.

So rather than simply being abstract mathematics, or merely a useful tool, the very fabric of matter depends upon imaginary numbers. At its heart, our world is (mathematically) an imaginary world.

Next time: Complex numbers are useful because of the relationship between real and imaginary numbers. It turns out that in mathematics the relationships between numbers is what really matters. As we’ve come to understand this, we’ve also come to realized that mathematical relationships are also at the very heart of the cosmos.

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Newton’s laws are a story of arrows. Not physical arrows (although that’s sometimes the case) but a kind of mathematical arrow known as a vector. 

The simple concept of a vector is something that has both a quantity (magnitude) and a direction. For example, if you’re traveling in a car, the velocity of your car is a vector that determines both your speed and the direction in which you are moving. Newtonian physics is largely concerned with forces, which are vectors representing the push or pull on an object in a particular direction. One of the key aspects of Newton’s laws is that forces are exerted between objects. If you push on an object to get it to moving, you can feel the object pushing back against your hand. That makes sense if two objects are physically touching, but what about if they aren’t touching. What about, say, the Earth and the Moon?

The forces on an object represented as arrows.

The Moon doesn’t drift through space, but rather orbits the Earth. Newton reasoned the Earth must be pulling on the Moon to keep it in orbit. A force of gravity, if you will. Newton’s breakthrough was to recognize that the gravitational force that pulls apples to the Earth is the same gravitational force that holds the Moon in orbit around the Earth, or the planets in orbit around the Sun. Every body in the universe must pull upon every other body with a mutual attraction. This law of universal gravity worked so well Newton was able to derive Kepler’s laws of planetary motion as a consequence of gravity. It connected earthly motion with heavenly motion and laid the foundation for modern astrophysics.

A magnetic field lines revealed by iron filings near a magnet.

But there was still the issue of how distant objects can feel such a force. How does the Moon “know” the Earth is nearby and “know” to be pulled towards it? For a time you could brush aside the question and focus on solving Newtonian problems, but the question raised its head again in the study of electricity. By the 1700s we understood that electricity came in two forms of charge, positive and negative. Like gravity, charges exerted forces on other charges, even separated by some distance. Sure, the force got weaker with distance, but it was always there.

One proposed answer is that charges must reach out to each other with some kind of energy. That is, a charge is surrounded by a field of electricity, a field that other charges can detect. Charges possess electric fields, and charges interact with the electric fields of other charges. For this to work, the electric field must be a vector field. At each point in space the electric field must have a magnitude and a direction. When a charge detects an electric field, it therefore knows the magnitude and direction of the force acting on it.

For any vector field you can trace paths through the field known as field lines. If you start at a particular point in the vector field, you can take a small step in the direction the vector is pointing. At your new position take a step in the direction of the vector. Keep doing that, and you trace a line following the vectors of the field. If you trace field lines for an electric field, it turns out that they always draw a path from some positive charge and to some negative charge. This is because positive charges always push other positive charges away, while negative charges always pull positive charges towards them. As a result, electric field lines always spread out of positive charges and collapse into negative ones. In mathematics this property is known as divergence.

Divergence (left) and curl (right).

Field lines that have a beginning and an end will always be divergent. But what if you traced a path through space and eventually found yourself back where you started? Instead of a finite line, it would form a loop without end. In mathematics this is known as curl. It turns out that magnetic fields have exactly this property. While charges are always either negative or positive, magnets always come in pairs of poles (north and south). Magnetic field lines spread out from north poles, but they always loop back on themselves, coming together at a south pole. Because of magnetic curl, you can never have just a north pole or just a south pole, and in fact if you break a magnetic in half you will get two magnets.

So the electric field is a vector field with divergence, and the magnetic field is a vector field with curl. But mathematically a general vector field can have a combination of divergence and curl. The mathematics would therefore seem to imply that electric fields and magnetic fields are simply two parts of the same vector field. This is actually true. Electric and magnetic fields are each part of a more general electromagnetic field. Not only that, electric fields that change over time (say from moving charges) can induce magnetic fields, and changing magnetic field can induce electric fields. Together these changing fields can waves of electromagnetism we call light. Light isn’t a separate physical phenomenon, but is due to the fact that electricity and magnetism are connected both physically and mathematically.

Now you might be wondering about gravitational fields. Since the forces of gravity are similar to the forces between charges, doesn’t that imply that there should be a partner to gravity? Some sort of magnetogravity field? It would if gravity was a vector field, but it turns out gravity is a different kind of field altogether.

A metric tensor is often visualized as a rubber sheet, though it’s not a very accurate picture.

Although the simple representation of vectors is as an arrow, the mathematics is a bit more subtle. Not everything that has a magnitude and direction is a vector. For example, the Sun radiates light outward (direction) at a specific rate (magnitude), but that isn’t a vector. Mathematically, vectors are defined by things like how they add together, and how they transform in different frames of reference. While vectors can be represented by arrows in a physical space, that’s not the only thing they can be. The concept of a vector can also be generalized. You might remember from last time that gravity is actually a geometric property of bendable space and time. For this you need a mathematical quantity that can not only describe changes in quantity and orientation, but also volume and twisting and shear. This is done through a generalization of vectors known as tensors. Gravity isn’t a vector field, but the properties space and time that describe gravity are a tensor field known as the metric tensor. Since vector fields are a special case of tensor fields, gravity can be approximated as a vector field of gravitational force. Tensors are also useful in electromagnetism, since an electromagnetic field can also be described as a single tensor field.

So the arrows of simple forces pointed us to the general concepts of vectors and tensors, and it led us to a deeper understanding of both electromagnetism and gravity.

Next time: Just as geometry and vectors can be generalized into more abstract forms, so can numbers. By generalizing numbers beyond the familiar we’ve been able to understand the imaginary aspects of quantum theory.

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Stretch a string between two points, and you get a straight line. Walk one end of the string around the other and you get a circle. These two shapes, circles and lines, form some of the basic shapes of geometry. 

While early astronomy focused on the marking of time, such as when the rising Sun would be most northerly, or the number of days until the crescent Moon returned, the simple geometry of curves gave us a connection to the heavens. The Sun, Moon, and even the stars seemed to trace circles around the Earth. Likewise, a dropped ball traced a linear path to the ground, and fire seemed to rise linearly upward. The geometry of circles and lines was nothing less than the sacred geometry of Heaven and Earth.

Around 300 BC, Euclid of Alexandria formalized this geometry in a 13 book opus known as The Elements. He started with five basic assumptions (axioms) about lines and circles:

1. You can always draw a straight line segment between two points.
2. You can always extend a straight line segment indefinitely
3. For any line segment you and draw a circle where the line segment is the radius and one end is the center.
4. All right angles are the same.
5. If two lines aren’t parallel, they will eventually cross if you extend them far enough.

From these axioms he developed a formal method of proofs and theorems (elements), showing that if these initial axioms are true, then other aspects of geometry must also be true. Euclid gave us the language of geometry, and by extension the language of much of modern mathematics. This geometric language could then be used to describe the motion of heavenly bodies. If you wanted to know when Mars and Jupiter would appear close in the sky, or when Venus would appear as the morning star, you could calculate it with geometry.

That same geometry also gave us tools to measure heavenly motions more accurately. We could triangulate the positions of planets against the stars, and calculate their true paths about the Earth. It soon became clear that planets did not move in circles. Lines and circles were so incredibly useful that many proposed solutions still focused on them. Perhaps a planet’s motion about the Earth was circular, but not centered on Earth. Perhaps it was a combination of circles (epicycles) on top of other circles to trace out a planet’s path. Perhaps the planets move in circles about the Sun rather than the Earth. All of these improved upon the simple model of circular motion about the Earth, but they were always just a bit off from the true motion of the planets. The solution came from Johannes Kepler, who proposed not circular orbits about the Sun, but elliptical ones.

Conic sections as slices of a cone.

An ellipse is part of a family of curves known as conic sections. If you take a line segment and trace one end about a circle, the line traces out a cone. If you then slice the cone with a plane, you can form four different types of curves. Straight through, and you get a circle. At an angle, and you get an ellipse. Parallel to the edge of the cone and you get a parabola. Steeper than the edge of the cone and you get a hyperbola. In this way a circle is just a special case of a larger geometric family. By generalizing circles to ellipses, Kepler devised a set of three basic rules for planetary motion that came to be known as Kepler’s laws. They were extremely accurate, and were much simpler than offset circles and epicycles.

Each point in space has a unique coordinate.

Around the same time as Kepler, René Descartes was developing a new approach to geometry. Although our understanding of geometric forms had improved over the centuries, geometry still followed the methods of Euclid. Imagine a line bisecting a circle, envision a sphere enclosed by a cube. Geometry was about lines, curves and shapes in relation to each other, and this could be complicated. Kepler’s second law for example, imagined a line connecting a planet to the Sun to sweep out an equal amount of area for equal amounts of time. Descartes imagined that space could be filled with a grid as a reference frame. In this way, each point in space can be represented by a unique set of numbers (coordinates), and a curve can be represented as a function related one coordinate to others. With this analytic geometry, Descartes connected geometry to algebra, giving us even more tools to describe curves and forms.

Newton’s geometry of space and time is still used in introductory physics classes.

Analytic geometry also allowed us to look at motion not just as a path through space, but also as a path through time. Each position in space has three coordinate numbers marking its location, and by adding a fourth coordinate representing time we can create a geometry of where and when. When Isaac Newton developed his laws of motion, he described motion in terms of speed and acceleration. Using analytical geometry he could connect these functions of time to curves in space, tracing an object’s path through space and time. This same approach also allowed Newton to prove that Kepler’s laws of motion were the result of a universal force of gravitational attraction, ushering in the age of astrophysics.

The Euclidian geometry of space and time was so powerful that it’s validity seemed unquestionable. What else could the cosmos be if not an extent of space existing in time? Combined with the accuracy of Newtonian physics, it felt as if we’d reached the pinnacle of understanding. But in the 1800s Bernhard Riemann began to explore alternatives to Euclidian geometry. The coordinates of Descartes were a way of mapping out Euclid’s geometric space, but what if the relations between these coordinates could be distorted. We might imagine a Euclidian surface as a sheet of paper marked with a grid. If the sheet were made of rubber, stretching or bending the sheet would distort the shape of the grid. Some rules of geometry would still apply on the sheet, but not necessarily the five axioms of Euclid. Just as circles are just one example of a conic section, Euclid’s geometry is just one member of a much larger geometric family.

Riemannian manifolds come in a variety of shapes.

This gave rise to a more general form of geometry known as Riemannian geometry, where space could be a malleable manifold rather than a rigid background. The connections between points in space are determined by the structure of the manifold, and the old rules of Euclid can be stretched or even broken. Two circles of the same circumference might have different lengths radii. Parallel lines might eventually cross. Two right angles might not be the same when compared to each other. Just as Descartes connected geometry to algebra, Riemann connected geometry to topology. Geometry was no longer limited to a fixed background grid.

But surely none of this applied to the Universe at large. Sheets of paper and rubber balls can be distorted into different shapes, but space is not a physical material. Surely it must be rigid and absolute. Surely space and time must be Euclidian.

But it’s important to note that Euclid’s axioms were assumptions. They seem intuitively true for space and time, but assumptions can be wrong. One of the big assumptions about time in the Universe, was that it’s the same everywhere. If we sync two clocks, they should always read the same time even if they are speeding on a starship or light years away. But if space time were the absolute grid against which everything is measured, then the speed of an object must always be relative to that grid, even the speed of light. If you were speeding along relative to the cosmic spacetime grid, you would measure a different speed for light than if you were standing still. But it turns out space and time aren’t absolute frames, light is. Light forms a geometric connection between space and time, and the geometric rule that connects space and time is that its speed will always be a universal constant.

This is the insight Albert Einstein brought to physics. Riemann was right. The key to geometry is how a manifold is topologically connected. For our Universe, light is the connection, and space and time distort in whatever way necessary to preserve that connection. It is the general relativity of space and time.

The geometry of space is not Euclidian.

Perhaps the most amazing aspect of Einstein’s theory is that gravity — the force that causes the planets to trace their elliptical geometry around the Sun — is itself simply a consequence of geometry. The distortions of space and time mean that objects don’t always move in a straight line. Their path can be distorted, making it look like they are being pulled by a gravitational force. Newton’s law of gravity was an intellectual triumph, but it also represented an incomplete understanding of geometry. Our Universe does have a sacred geometry after all. It’s not the fixed geometry of a rigid and invisible grid, but the luminous geometry of light.

Next Time: Newton’s laws of motion brought vectors to physics. These mathematical arrows pointed us to a subtle property of nature leading to revolutionary new understanding of light.

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Mathematics is the language of science. From arithmetic to group theory, mathematics builds the very foundation of scientific models. We might be inspired by an idea or analogy, but the precision of science requires a mathematical structure. Perhaps the most fundamental thing we’ve learned about the cosmos is that it has a deep connection to mathematics.

This connection has raised the question about just why mathematics is so effective. Perhaps it’s simply due to the fact that we follow mathematical models where they are useful, and discard them where they are not, making applied mathematics self-selecting. Perhaps it’s because as evolved primates within this physical universe the mathematics we think is “pure” is simply a reflection of how our universe works. Regardless of the cause, mathematics seems unreasonably effective as physicist Eugene Wigner once argued. It’s so powerfully useful that some folks such as Max Tegmark have proposed that the structure of the Universe could simply be the structure of mathematics itself. Gallons of ink have been spilled on all sides.

But the very power of mathematical models within science also raises a wall separating those with the mathematical training to understand these models from those without. This is particularly seen in the popularization of science where (with some exceptions) equations never appear, and the focus is on broad analogies rather than the underlying maths. This reinforces the misconception that established science can be overturned simply be a new idea, and that the mathematics is merely a minor detail. The reality is that an idea can be a spark, but fire of knowledge is only captured with specific mathematics, and it’s in the mathematics where much of the beauty an elegance of a scientific model lies. The mathematics of science can be deep and subtle, and its nature is not often discussed.

So for the next several posts I’ll try to present some of the mathematical beauty behind several scientific theories:

1. Geometry – From Aristotle’s conception of earthly lines and heavenly circles, to Kepler’s elliptical forms, geometry has played a central role in astronomy. As we followed geometry into more abstract concepts, it opened the doors to the beginning of time.
2. Vectors and Fields – Forces have both a quantity (magnitude) and direction. In mathematics we call them vectors. From that simple mathematical concept arose the first unified field theory.
3. Complex Numbers – It was long thought that any number multiplied by itself is a positive number. But what if a number multiplied by itself was negative? This idea was so odd it came to be known as imaginary. It turns out that imaginary numbers open the door to very real physics.
4. Group Theory – We often think of mathematics as numbers, or at least equations. But it can also be about relationships and connections. Often how different parts of a model connect is the key to understanding the model on a deeper level.
5. Formalism – Is mathematics simply a set of connected rules, or is it something more? Does mathematics limit what we can know about the physical universe?

We’ll start with geometry, and how a simple curve tracing the motion of a planet led us to an exploration of space and time itself. It all begins tomorrow.

Paper: Wigner, E. P. The unreasonable effectiveness of mathematics in the natural sciences. Communications on Pure and Applied Mathematics. 13: 1–14 (1960). DOI:10.1002/cpa.3160130102

Paper: Max Tegmark. Is “the theory of everything” merely the ultimate ensemble theory? arXiv:gr-qc/9704009 (1998).

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Today is pi day, which means an obligatory post on that most famous of irrational numbers. Most know the number from geometry, as it was originally defined as the ratio of a circle’s circumference to its diameter. But it turns out there are some deep and subtle connections between pi and physics we still don’t fully understand. Take, for example, the Riemann Zeta Function. 

The function is usually denoted as ζ(s), and it looks like a simple sum. Start with the number 1, then add 1/2 to the power of s, then add 1/3 to the power of s, 1/4 to the power of s, and so on. Add that sum forever and you get ζ(s). What does this have to do with pi? Well it turns out that if s is an even number, then our sum will be related to a power of pi. For example, ζ(2) = pi²/6. This is true for any even number, so that ζ(2n) is equal to pi to the 2n power, multiplied by some rational fraction. So it turns out that pi is buried within this infinite series.

But that’s just for the positive even integers. For the negative even integers the sum is simply zero. Thus ζ(-2n) = 0, which are known as the trivial zeros of the Riemann Zeta Function. For ordinary real numbers, the negative even numbers are the only values for which the Riemann Zeta Function is zero. Things get really interesting, however, if we allow for complex numbers.

For any real number multiplied by itself, the answer will always be positive, but there are imaginary numbers that yield a negative number when multiplied by themselves. This are usually denoted by iy, where y is a real number and i represents the square root of -1. Thus the square of i3, for example, is -9. By combining real and imaginary numbers you can get complex numbers, which can be represented by z = x + iy. Complex numbers have a real part x, and an imaginary part iy.

So, if we consider complex numbers, are there any complex zeros where ζ(z) = 0? The answer is yes. In fact we’ve found more than 10 trillion complex zeros, and we know that there are an infinite number of them. What’s interesting about all the known complex zeros is that they can be written as z = 1/2 + iy. In other words, the real part of the number is always 1/2. With trillions of these zeros all having the same form, it would seem clear that all the complex zeros must have their real parts be 1/2. This is known as the Riemann hypothesis, and while it certainly seems to be true, no one has yet been able to prove it absolutely. The Riemann hypothesis is one of the great unsolved mysteries in mathematics.

Complex numbers and circles are related.

It turns out that the Riemann Zeta Function (and thus the Riemann hypothesis) shows up in a wide range of physics theories, from describing the energy levels of certain quantum systems to subtle aspects of string theory. The connection between the Riemann hypothesis and physics is strong enough that some have proposed using experiments with quantum systems as a way to prove the hypothesis (though I doubt that would satisfy mathematicians). There is both deep mathematics and deep physics buried in the Riemann Zeta Function, and we still don’t fully understand it.

Of course all this trouble began when we introduced complex numbers into the function. When I first mentioned complex numbers, I said they could be written as z = x + iy. It turns out they can also be written as a number to an imaginary power. When written that way, complex numbers are multi-valued. That’s because there is a connection between complex numbers and circles. For a circle, if you walk around it you eventually return to your starting point. For complex numbers, increasing the imaginary power is like walking around a circle, so if you add a particular number to the imaginary power you get the same complex number you started with.

That particular number happens to be 2 times (you guessed it) pi.

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Suppose you measure the time a planet takes to orbit the Sun (its period), and compare it with a measure of its distance from the Sun (the semi-major axis). If you plot the logarithm of periods for all the planets (and even Pluto) vs the logarithm of their semi-major axes, you find they all lie along a line with a slope of 3/2. If you do the same for the moons of Jupiter, or Saturn, you get the same slope. The same is true for asteroids, comets, exoplanets and binary stars. They all follow the same 3/2 relation, and fall along the same line so long as they’re orbiting the same body.

The Kepler relation for planets and moons. Credit: Professor Kenneth R. Lang, Tufts University

This relation is true regardless of which scientific model we use to describe planetary motion. It is simply a description of the relationship between a planet’s period and distance. What makes the relation useful is its mathematical precision. Any theory of planetary motion should be able to derive this 3/2 relation as a consequence of the model. Not 4/3, not 8/5, but 3/2. It just so happens that Newton’s law of gravity makes just such a prediction. If we assume that masses are attracted to each other with a force inversely proportional to the square of their distance, then through a bit of calculus and geometry we can show the 3/2 relation must hold to an outstanding degree of precision.

There’s a reason why astronomical models are mathematical in nature. It’s only through mathematics that we can give precision to our description of the heavens. It’s one thing to say that planets close to the Sun orbit faster than more distant ones. It’s quite another to say, “how much faster?” To do that, we need to describe the relation mathematically.

The extra term in general relativity, which is the first-order correction to Newtonian gravity, makes all the difference.

Without mathematical precision we have no way to distinguish between similar models. Take, for example, Newtonian gravity vs. general relativity. In Newton’s model, gravity is a simple force between masses. In general relativity gravity is a consequence of the warping of spacetime. While the description of gravity in these two models is radically different, both models derive the 3/2 relation for planetary motion. There’s actually very little to distinguish the models as far as planets are concerned.

An example of orbital precession. Credit: Wikipedia

There are some small differences, such as their descriptions of the motion of Mercury. Both models predict that Mercury’s orbit will precess over time due to the gravitational interaction with other planets, but general relativity predicts Mercury will precess just a bit more. The difference in the two predictions is only about one-tenth of an arc second (which is 1/3600 of a degree) per orbit, but our measurements are precise enough to show that Einstein was right, and Newton was wrong.

With mathematics, we can even test models we don’t entirely understand. While we don’t know the underlying cause for dark matter, we do understand the affect it must have on the way galaxies cluster together. We also know what affect alternative “modified gravity” models would have. Even with an incomplete understanding of either approach, we can look at what the mathematics of these models tell us.

Red indicates the observed clustering of galaxies at different scales. Black is the dark matter prediction, while blue shows modified gravity predictions. Which model do you think works? Credit: Scott Dodelson

What we find is that dark matter agrees with the clustering observations, while modified gravity does not. What’s more, the observational data best agrees with a specific form of dark matter known as cold dark matter. With mathematics, we don’t need to know everything about dark matter. We can simply derive the consequences of different assumptions and see what works. If we let the mathematics lead the way, we can often learn something about the universe.

Perhaps the most extreme case of letting the math lead us is in quantum theory. Despite its success as a scientific model for nearly a century, there is still a great deal of debate over the interpretation of quantum theory. Are there wavefunctions that collapse when observed? Does it imply the existence of parallel universes? The behavior of quantum systems are so deeply counterintuitive that we struggle to find a meaning behind the mechanism.

Various interpretations for quantum theory. Credit: Wikipedia

However there is complete agreement on the mathematical formulation of quantum theory. The various interpretations of quantum theory rely upon the same mathematical results, which is why there’s no clear resolution to the argument. It’s also why a popular approach to teaching quantum theory is “shut up and calculate.”

While that might seem an unsatisfactory approach to scientific understanding, it’s produced the most precise scientific model ever devised. One property of the electron (and its heavier counterparts, the muon and tau) is its magnetic moment, which is often expressed in terms of a dimensionless g-factor. Experimentally this g-factor has a value of 2.0023318416, accurate to 0.26 parts per trillion. The predicted value from quantum electrodynamics (QED) is 2.0023318361. The difference is only 5 parts in a billion, but even that may be enough to hint at deeper physics yet to come.

I found a Higgs boson! Credit: Particle Zoo.

When we talk about astrophysics to the general public, we often rely on appealing images and rough analogies to convey the meaning behind the mathematics. It’s a good way to convey our understanding of the universe, but it also gives the impression that the analogies are what science is all about. This is why so many “alternative” science promoters focus on broad concepts. “Clearly dark matter is due to plasma tubes!” they’ll say. “Black holes are just concentrated unobtainium!” The idea is the key, and the mathematics is left as an exercise for the reader.

For real scientific models, however, the mathematics is central, and the ideas are a useful way to lead you toward the mathematical formalism. Its the unity of physical models and mathematical formalism that produces the most powerful and accurate scientific models. Precision is key, and without it, science is merely a numbers game.

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In my introductory physics courses, I occasionally implement what I call “high stakes labs.” Take a set of measurements for a devices such as a ball launcher, use those results and your knowledge of physics to predict an outcome (such as where the ball will land) and then test your prediction. The catch is that the grade for the lab depends upon the accuracy of your prediction. It is stressful and challenging, but it demonstrates a common aspect of science and engineering. You can’t simply look up the answer in the book. You have to test your ideas in the real world.

In terms of high stakes labs, the highest of stakes was likely the early manned space program. Not only does everything from the propulsion system to life support need to function correctly, the rocket needs to land in the correct location.  Calculating trajectories is not easy, as anyone who has played Kerbal Space knows.  It involves complex mathematics such as analytic geometry, and it absolutely needs to be correct.

When NASA prepared to launch Alan Shepard as the first American in space, they used a computer to calculate his capsule’s trajectory. Her name was Katherine Johnson. Johnson’s forte was analytic geometry, and she was very, very good at it.  She began her work as part of a pool of “computers” that worked through the complex mathematics necessary for orbital predictions. Her skill as a mathematician was so impressive that when NASA first began using electronic computers to calculate trajectories, Johnson was asked to verify the results.

Johnson worked for NASA’s Langley Research Center for more than three decades, co-authored 26 papers, and ensured that American astronauts reached their destination. She also happened to be a black woman in a field dominated by white men, which was a whole other kind of high stakes lab.

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My research area is computational astrophysics. This means I use computers to analyze astronomical data or model astrophysical systems. Most of my work is done through an application known as Mathematica, which is a powerful computational program. Like any application, Mathematica has advantages and disadvantages, but it has one property that is absolutely essential: it is Turing complete.

A program that is Turing complete is capable (at least in principle) of computing anything that can be computed. So the only real limitation to an astrophysical model is computing power and memory. As a result, I can run a model in Mathematica on my laptop. If it takes too long, or needs too much memory, I can run the same model on a cluster of computers. With enough grant money I could run it on a large supercomputing cluster. My only limitation is that of the current hardware.

Turing completeness is central to computational astrophysics, though it is not a particularly rare property for applications. For example, Excel and Minecraft are Turing complete. One consequence of this is that any Turing complete application can model any other Turing complete application, so in principle you could program Excel to run Minecraft, and then program Minecraft to run Mathematica. It would be horribly impractical to do that, but it’s possible in principle.

The idea of computational completeness was developed by Alan Turing. He was a mathematician and computer scientist who lived in the mid-1900s, and was central to the cracking of the German enigma code in WWII. It was his work on computability that lead to his understanding of enigma.

Knowing an application is Turing complete, you can then choose your application based on other needs, such as ease of coding, or scalability. These are two of the reasons I use Mathematica. It is geared toward scientific computing, and the same program can be run on a laptop or computer cluster without recoding.

Completeness gives us the confidence that computational models can provide an accurate description of astrophysical processes. We are not simply limited to models that can be done by hand. We can use computers to extend our understanding of the universe.

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Today is March 14, which many celebrate as Pi Day since the month and day mark 3.14, which is approximately pi.It is also Albert Einstein’s birthday, so it seems fitting to ask whether pi can exist in a universe as Einstein described it.  Just for fun, I’m going to outline why the answer is no, and then explain why that answer is wrong.

The value of pi (3.14159…) is defined as the ratio of the circumference of a circle to its diameter.  But in a physical universe where (as Einstein demonstrated) space is curved, the ratio of circumference to diameter isn’t pi.  For example, if you drew a circle around the Earth, the ratio of circumference to diameter would actually be a little less than pi.  This is because the mass of the Earth curves space around it, making the diameter of your circle a bit longer than it should be.

Geometry of curved space. Credit: The Airspace

This is actually a way you could define a region of space as being curved.  Draw a circle around a region of space, find the ratio of circumference to diameter, and if the value is less than pi then that region of space is curved.  The smaller the ratio, the more strongly that region of space is curved.  If you drew a circle around a black hole, its diameter would be infinite, so the ratio would be zero.

So if we define pi as the ratio of circle’s circumference to diameter in physical space, then pi would actually have lots of values depending the curvature of space around the circle, and none of them would be 3.14159…

Of course that isn’t how pi is defined.  The circumference/diameter definition applies for a mathematically ideal circle, where space isn’t curved.  It can also be defined in other ways, such as an infinite series pi = 4 – 4/3 + 4/5 – 4/7 + …  The geometry definition is just a simple (and perhaps the oldest) one. The key point is that pi is a mathematical concept, not a physical one. 

So even though physical space is actually curved, pi still exists and has the value we all know and love.  So celebrate the day, because it is a perfect excuse to have a slice of pie. 

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There’s a popular video from Numberphile that shows how the sum of all the integers, adding them up forever is -1/12.  If that ties your brain in a knot, it should.  How is it possible for 1 + 2 + 3 + … to be a negative number?  The answer is that it’s not…quite.  It’s not that the video is wrong, but rather than it’s misleading.  There is a way to assign a value of -1/12 to the sum, and it does have important applications in physics, but your intuition that adding all the integers should be infinity is correct.  The catch is that dealing with infinities (unlike a finite sum of numbers) your answer can depend on just how you do the sum.

The key to understanding this mathematical weirdness is a concept known as convergence.  If we take the sum 1 + 2 + 3 + … , with each number we add the sum gets larger and larger.  Because of this it seems obvious that the sum must be infinity, but technically it means the sum doesn’t converge.   In other words, as you keep adding numbers, there is no finite value that the sum approaches.  In contrast, the sum 1/2 + 1/4 + 1/8 + … does converge.  You might recognize it from Zeno’s famous paradox, where to run a marathon you first have to run half the distance, then half the remaining distance, then half of that, and so on.  With each number we add, the sum gets closer and closer to 1.  So the infinite series converges at 1.

In the video, they look at three infinite series:

S1 = 1 – 1 + 1 – 1 + …

S2 = 1 – 2 + 3 – 4 + …

S3 = 1 + 2 + 3 + 4 + …

They then claim that S1 = 1/2, from which they show 2 S2 = S1, and 3 S3 = – S2, thus S3 = -1/12.  Of course if you look at each of these sums, you can see that none of them converge.  The S1 sum keeps toggling between 1 and 0, and the S2 sum keeps oscillating between positive and negative numbers.  So how can you say that the sums of these are finite numbers?

You can’t.  Not in the way most people add numbers.  But there is a way to sum divergent series so that they produce a finite number.  It known as a Ramanujan summation, and basically it is a way of reordering things so that you get a finite number.  It is a way of connecting convergent and non-convergent series, and it is a way of assigning these non-converging sums a unique number.  Because of this, it is very useful in high level mathematics.  So you are right in thinking that if you add up all the integers you will get infinity, but there is a way to generalize the summation to get a finite result.

This seems like a bunch of mathematical trickery, but it isn’t.  It is a very useful tool for an aspect of physics known as renormalization.

To get an idea of how renormalization works, imagine a particle in a box.  The more energy a particle has, the faster it moves.  So a particle in a box will bounce around with a speed dependent on its energy.  If we take all the energy out of the particle, it will just be stationary somewhere in the box.  The key point is that for a simple particle it is possible to take all its energy way.  So if you then imagine an endless sea of particles boxes, the lowest possible energy the sea can have is zero.

Energy levels for a quantum particle in a box.

But things are different for a quantum particle in a box.  A quantum particle can have particular energy levels, but because of the fuzzy nature of quantum particles you can’t take all the energy away from the particle.  At its lowest possible energy it still has just a bit of energy.  So if you imagine an endless sea of quantum particles, the lowest possible energy the sea can have is a bit + a bit + a bit + … = infinity.

In other words, the sum of a quantum sea is not convergent.  But this doesn’t make any sense.  If there is a sea of particles all in their lowest possible energy the sum shouldn’t be infinity.  But because there is more general way to do infinite sums, you can use that to get a finite result.  In other words, you can renormalize things to get an answer that makes physical sense.

The example I gave is overly simplistic, but in quantum physics these apparently infinite sums show up all the time, and we use renormalization to get meaningful results.  In particular, the meaningful results we get agree with experiments.

The important thing to keep in mind is that while things like the Ramanujan sum are not simple sums like we’re used to, but they are a mathematically rigorous way to deal with infinite series, and this mathematical formalism is used in physical models that really work.

But the simple sum of integers is not -1/12.

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