Saturday, February 16, 2013

Electrons, Strings and Spooky Action

After I wrote the Atoms series in this blog, I felt for just a moment I knew the atom's tricks, but the truth, surprisingly, is that no one does. Atoms are weird, especially the electrons within them. I warn you this story doesn't end neatly.

Modeling the Electron

Looking at Bohr models, these fundamental building blocks of matter seem straight forward enough. Below right is a Bohr diagram of a phosphorus atom. Electrons are shown in grey and the nucleus is shown in green.

Each electron whirrs around the atomic nucleus, confined to a shell (each circle) that corresponds to its energy. Its electron configuration can be written as 1s22s22p63s23p3. You can learn how to write these configurations in Atoms Part 4A. Even the quantum mechanical model doesn't seem too far out if you think of electrons as clouds shaped by where one might be at any given time. Phosphorus has six electrons in its 2p orbital, for example. Below left is a diagram of how that orbital is organized into three suborbitals, each one filled with two electrons, one per lobe.

Despite all of this understanding, there is a heart of weirdness in every electron. If you are like me and you only feel comfortable when you can get a mental image of what you're thinking about, the electron is destined to continuously throw stones at its own reflection. The days of thinking of them as little billiard balls or tiny planets orbiting a tiny Sun are long over.

Measuring the Electron

Let's start by going over what we do know: As far as anyone knows, electrons aren't made of any smaller parts. They are simply full of "electron stuff." Each electron is a magnetic dipole (a tiny magnet) and an electric monopole (a tiny charge). It has an intrinsic quantum spin, which we'll explore in detail, and a rest mass. As an electron approaches the speed of light, its mass is better described as relativistic mass as Einstein's theory of special relativity comes into play. Its mass, from an observer's point of view, increases and ultimately approaches an infinite value as it approaches the speed of light.

The size or diameter of an electron is very difficult to say. The classical radius of an electron is about 2.8 x 10-15 m, based on its mass, its charge and the permittivity (resistance against an electric field) of empty space. It's an estimate of how big the electron would need to be if all of its electrostatic potential energy is converted to mass. This number is used in modern classical theories like Thomson scattering but it is no longer thought of as the actual size of an electron because quantum mechanics, needed to understand it, regards the electron as a cloud of probability with an indefinite outer edge. Mathematically, it treats the electron as a size-zero point. In quantum field theory, it is an excitation in a field so size doesn't have any relevance. In string theory, an electron would be a string with a length that is a compromise between its tension (wants to make it smaller) and Heisenberg's uncertainty principle (resists being a single point). You can't measure an electron by the size of hole you can shoot it through because it doesn't need a hole, as in the case of quantum tunnelling. Its size does not appear to be strictly the size of its wave function because these functions are probabilistic, meaning that they ultimately extend to infinity.

An electron, however, can have linear momentum and kinetic energy as it flies through space-time. Inside an atom, its energy is confined to specific energy shells.

Each of these electron qualities (except size) has a value you can measure very precisely. Some values, like velocity and charge, can be measured using classical mechanics. Values, like linear momentum and mass, must sometimes be relativistic - you need to use the theories of relativity in order to measure them when an electron is traveling close to light speed. Other electron qualities are described by quantum numbers. Its magnetic dipole is a quantum number. So is its intrinsic spin and the energy and shape of its orbital inside an atom. What makes these numbers "quantum" is that when you measure any of them you will get only discrete values. For example, in an atom, an electron's energy is confined to a limited set of values. It's not allowed to possess any energy in between those values. This concept is explored in detail in the article, Atoms Part 2. Below, electrons in an oxygen atom are allowed in two (ground state) energy shells (black) and at least two excited energy shells (grey) but not in between them.

Quantum mechanics gives scientists a great deal of information about electrons but it also means that electrons are slippery in their hands. You can't know everything about an electron at the same time. This is the essence of Heisenberg's uncertainty principle. You can't pin down both its momentum (or velocity) and its location, for example. Young's famous double slit experiment means that electrons also show both wave and particle behaviour at the same time. Whether an electron acts like a wave or a particle depends on how you decide to measure it. Maybe it's more accurate to say how you see the behaviour depends on how you measure it, to get way from the idea that the electron physically changes thanks to your eyeballs happening to be on it. The Copenhagen interpretation of quantum mechanics says that the wave function of an electron collapses when you observe it, and it implies that quantum mechanics does not describe the electron's reality - it deals only with probabilities - that what you measure will be this value or that. An electron really does have a particle/wave nature to it and it really can be completely described by a wave function. However, quantum mechanics, while offering a mathematical description, does not let us probe and examine an electron as if it were an ordinary (classical or Newtonian) object.

What Electrons Do

Despite the complex quantum nature of their electrons, atoms tend to follow rules, which make it possible to predict how they will act with one another (this is chemistry, explored in Atoms Part 4A, B, C D and E) and how they react to changes in energy by vibrating (incandesecence, described in Atoms Part 3) or emitting light (luminescence, described in Atoms Part 2), all of which intimately involves the atom's electrons. Scientists also have a good grasp on how atoms stack together in various materials - interactions between countless electrons in the outermost (valence) orbitals of the atoms in these materials determine how they stack themselves. Electrons can even smear quantum aspects of themselves by sharing interconnected valence orbitals (Atoms Part 4C). The metallic bond (Atoms Part 4D), for example, is dispersed evenly across the entire metal, allowing it to conduct a current. This electron arrangement is called delocalization. A three dimensional lattice of metal atoms is held together by many shared valence orbitals. The bonding electrons, those in the valence energy shell, do not belong to any particular atom or bond. Instead they exist in orbitals shared across several atoms and/or bonds (they do not exist outside the valence energy shell as many online teaching videos incorrectly show). These molecular (bonding) orbitals are hybrids of the individual atomic valence orbitals, and they are described by a different set of quantum numbers than atomic orbitals are.

The "smearability" of electron valence orbitals is thanks to the wave nature of the electrons themselves. I came face to face with this weird ability while researching the article "Quasiparticles." In that article I explored how the concept of the atom evolved from a billiard ball into the modern quantum mechanical model, how quantum field theory works and how phenomena like quasiparticles emerge from complex three-dimensional systems of atoms inside solid materials. The phonon, a quantum vibration, is an example of a quasiparticle that we'll encounter later on in this article.

Electrons can also pair up their spins in order to align their magnetic moments, showing off their inherent magnetism within magnetic objects. Magnetism can be quite complex. Each electron has an intrinsic spin and a charge. Magnetism arises from this moving charge, turning the electron into a tiny bar magnet. Electrons moving inside orbitals also contribute to the magnetism of materials. I explored these interesting phenomena in five Magnetism Explained articles.

Electrons are fascinating multifaceted entities. But there is one thing electrons do that seems to defy explanation altogether: spooky action at a distance. Albert Einstein famously came up with the phrase itself and quantum entanglement, as it is scientifically called, has bugged scientists ever since.

Spooky Action

In the early twentieth century, Einstein and his contemporaries were busy figuring out the quantum nature of the electron. They realized a consequence of their developing theory was that no two particles of matter, including electrons, could share all the same quantum numbers at the same time. This is the essence of the Pauli exclusion principle. It means that only two electrons can share a single energy orbital in an atom.

Quantum Spin

Two electrons are allowed, rather than just one, because electrons come in one of two possible quantum spins, usually called up and down for convenience. This spin quantum number is a measure of the electron's intrinsic angular momentum. Sometimes this number gets confused with the angular momentum number, which describes the orbital angular momentum, the shape in other words, of an electron orbital. That number is very important in chemistry because it influences bonds and bond angles between atoms. The electron's intrinsic angular momentum, on the other hand, is sometimes interpreted to mean that the electron is a tiny spinning sphere. This quality, however, is a purely quantum mechanical concept that doesn't have a counterpart in classical mechanics. In classical mechanics, any object can have angular momentum. It is a product of two measurements: rotational inertial and rotational velocity. It's what keeps the gyroscope spinning upright, below.

Kiko2000; Wikipedi
An electron is a little different from a gyroscope. The gyroscope experiences something called external torque - friction robs it of energy and it eventually falls over. In a system where there is no external torque - inside the vacuum of space - angular momentum is conserved. It doesn't get lost anywhere. That's what keeps planets, stars and galaxies spinning. All electrons come with, and can never lose, exactly the same angular momentum value. It is an intrinsic part of their "electron stuff," even though an electron, according to quantum mechanics, has no diameter about which to spin. If an electron had classical angular momentum, it would be altered when it collides with other particles.

Electrons confined to atoms have set values not only for angular momentum but also everything else, except energy. Here, they have a few choices, as I mentioned earlier, and these choices place an electron in one of a set of possible energy shells. A single energy shell can and usually does contain more than one orbital. Because of quantum mechanics, two electrons in an atom can only share the same orbital (same energy and same angular momentum number) if they do not share the same intrinsic spin direction - one can spin up and one can spin down. You can think of the spin in classical terms as one with a right-hand spin and the other with a left-hand spin.

This facet of electron behaviour is part of what gives various atoms their size. New orbitals must be layered on top to accommodate increasing numbers of electrons in larger atoms. Innermost electrons tend to get squished a bit closer to the (positively charged and repellent) nucleus as you go up in atomic number. If electrons could all share the same energy, they would all squeeze into the lowest energy orbital possible and everyday matter, as we know it (as well as chemistry, magnetism, electricity and even light), would be vastly different, if possible at all. As you will see later on, electrons will occasionally "break the rules" and do just that.

The weirdest thing about the quantum spin of an electron is that it exists as a superposition of the two (up/down) states. This phenomenon, first described by Paul Dirac in the1930's, arises from solutions to Schrodinger's equation, which is a core part of quantum mechanics. Most of the time, scientists don't need to think of the spin this way, but it's what makes spooky action spooky, as we'll see. This may seem to contradict the Pauli exclusion rationale I just talked about and you might be asking, how do you even know there are two spins then? You can very quickly get into some dicey territory as you consider the ramifications of the Pauli exclusion principle and superposition. What happens if one electron of an entangled pair is in a collision that changes its energy and momentum? Does the other electron's energy and momentum change? The answer as I understand it is no. For example, in an experiment done last year in Austria, researchers got a group of 14 calcium ions caught in an electromagnetic trap to become an entangled coherent unit. Once entanglement is achieved, outside noise destroys it at a rate proportional to the square of the number of entangled units. Using this logic, I suspect that the collision of one electron would be considered enough noise to instantly turn the pair into an incoherent state. Try this Discover blog article, where physicist Sean Carroll starts out questioning some of celebrity physicist's Brian Cox's "A Night With The Stars" lecture about quantum entanglement and ends up having "fun" attempting to reconcile the Pauli exclusion principle and quantum entanglement. Not only entertaining in its own right, it shows how an expert grapples with these concepts.

Spin can indeed be measured, but it is neither up nor down until you measure it. This is reminiscent of the electron cloud we encountered in Atoms Part 4A. The electron is neither here nor there and everywhere all the same time until you measure its location. When you measure the electron's spin, the quantum uncertainty, expressed as a wave function, collapses and at that point the spin of that electron has a 50/50 chance of being up or down. It will be one or the other. This concept, in and of its self, may be difficult to buy but is not too difficult to digest. The implications of it are.

Spooky Action Breaks The Speed Limit

If you take a pair of free electrons and make sure they are both in exactly the same quantum state, they are now entangled in the quantum sense. You can then separate them and measure the spin of one of the electrons. You will discover this one is spin up for example. If you observe the other electron you will find it's wave function has collapsed into down spin. Physicists have been able to take measurements over such precise intervals that the information "received by" the second electron must happen faster than the speed of light!

Einstein thought this scenario through several decades ago and the idea of faster-then-light information transfer went against everything he knew about special relativity - one of his great theoretical legacies. He called it spooky action at a distance. Scientists around this time had thought up ways to accurately measure the speed of light and they always found it to be the same value, no matter whether the light source was moving toward or away from the measurer or the measurer was stationery. If the speed of light never changes, it means time and space have to give instead, based on Einstein's theory called special relativity. Realizing that space and time are flexible, Einstein went on to build a new theory of gravity based on space-time curvature. Relativity also means that the momentum (you can also call this relativistic mass of any particle with mass will approach infinity as it is accelerated to light speed. A particle without mass, such as a photon - travelling at light speed - also has momentum, but it is not infinite and it is calculated slightly differently. Relativity also means that an object travelling faster than light speed would, according to an inertial frame of reference, be travelling backward through time. This would violate causality, which in scientific terms can be described as a violation of the second law of thermodynamics. For example, a smashed glass of wine will never reassemble itself - every action has a one-way time arrow, an idea I explored in Time.

The speed of light is the speed limit of the universe. The violation of the second law of thermodynamics extends the speed limit from objects with mass to the speed limit for information travel as well, and no such violation has ever been measured. Almost every physicist considers light speed to be the speed limit of the universe, for objects and for information. Fields of force such as magnetic, electrical and gravitational fields, all propagate at the speed of light.

So how do electrons get away with it? One answer is that they do not break the second law of thermodynamics because no information is actually transmitted - you can't know whether the initial spin is up or down before you measure it and collapse the second electron, so you can't send information this way. Many theorists find this argument too easy. Even though you can't dictate the information, something somehow is telling the second electron what spin to collapse into. Not only do electrons get entangled, but photons, the calcium ions I mentioned, and even macroscopic objects like the diamonds Dr. Cox referred to in his lecture can form quantum-entangled pairs as well. It seems that these particles and objects, at least from a quantum mechanical point of view, are not limited to the dimensions of space-time we experience.

If we accept that all the physical laws must remain intact, entanglement means that two electrons, separated by distance, somehow can act like one single entity. If you do a one eighty, the question becomes how does the universe make one electron look like two? Like the black hole problem, this question seems to be about the nature of space-time.

How To Explain Spooky Action > Remove A Spatial Dimension

There have been a few theoretical attempts to explain the universe with quantum entanglement and all the known theoretical laws intact. One that stands out is the Holographic principle, a possibility I explored in the article, Holographic Universe. In a nutshell, this theory suggests that everything in the universe is a four-dimensional hologram of an underlying two-dimensional reality. The edge of the universe is like the inside of a balloon and according to this principle the entire universe is, in reality, confined there. The four dimensions we experience are a hologram projected from a two-dimensional information structure painted on the inside of the balloon. The idea was born as a way to explain how a black hole can seemingly erase all the quantum information encoded in the material that it gobbles up, and still not violate the second law of thermodynamics. Gerard 't Hooft and Leonard Susskind used the mathematics of string theory to take a four-dimensional black hole (includes time) and turn it into a two-dimensional information structure called a worldsheet. 't Hooft reasoned that incoming and outgoing particles (some are emitted from the black hole through Hawking radiation) deform the sheet, leaving imprints of their quantum information. Susskind was able to describe these imprints mathematically as holograms.

Theorists have attempted to connect this principle to the quantum entanglement problem and some suggest that the bridge between the two might be the Bohm interpretation, also called the Broglie-Bohm theory and the pilot-wave theory, of quantum mechanics. As I understand this theory, which came together in the 1960's, the position and velocity of any particle is defined by its wave function through something called a guiding equation, which allows the wave function to evolve over time. The guiding equation takes into account all the wave functions of all other particles in the universe by encapsulating them into one giant wave function. It attempts to describe an electron's environment as one that is dependent on a statistical average of what all other particles in the universe are doing, and in this way the electron is able to exchange information instantaneously with all other particles, including the entangled partner. In fact, in the eyes of the Bohm interpretation, the electron's spin is not an intrinsic property of the electron at all - it is its wave function in relation to the wave function of the device used to measure it.

If you connect this interpretation to the Holographic principle, you can imagine the wave functions of all particles in the universe mathematically encoded (and interconnected) on the two-dimensional information sheet.

In quantum field theory, all particles of force and matter can be described by a series of mathematical values called operators, which are continuously varying wave functions, where some or maybe all wave functions are entangled with others.

One wave function in two dimensions can look like two separate particles in four dimensions. Going backwards, you can take two separate electrons in three spatial dimensions and condense the dimensions into two. The two electrons are a hologram of one wave function located on a two dimensional information sheet, shown below.

Scientists see one electron that is able to correlate with another distant electron in a way that classical mechanics can't account for. They call this phenomenon nonlocality. Locality in quantum physics is not a straight cut concept. Instead it is subtle and open to interpretation, when you consider the uncertainty underlying where a particle is and what it is doing as well as the effects of various fields on it. In addition, the particle itself may create fields - each electron creates an electric and magnetic field that weakens with distance but, according to quantum mechanics, has infinite range. This is part of what I think Dr. Carroll was getting at in his blog article. In contrast, all the physical laws in classical mechanics and relativity depend on locality: a particle is a point-like object and it is influenced directly only by its immediate surroundings, whether they are force fields or other particles.

The holographic principle, largely built from string theory, allows entangled electrons to interact outside the bounds of four-dimensional space-time. If we could un-glue ourselves from space-time and see this underlying two-dimensional reality, we would see the electrons not as two distinct particles moving separately through space and time but as some kind of single entity. If we took a second look at Young's double slit experiment, we would not be so shocked to see electrons, shot one at a time, somehow build up a perfect wave interference pattern, as shown right.

The electrons in the experiment are, after all, according to this principle, interconnected within a two-dimensional structure and they are not confined to our space-time. Keep this general thought in mind as you read on. It is, by accident, the same conclusion another researcher, through a very different line of reasoning, recently comes to, except that instead of condensing the four dimensions we experience, he expands them.

Besides offering a coherent way to look at a black hole, the Holographic principle is attractive for two other reasons as well. The entropy of the universe increases as the information structure "balloon" expands. Entropy, according to the second law of thermodynamics, can be looked at as a measure of disorder or degrees of freedom. The universe is currently in a state of low entropy. It has intricate structure or a high degree of order in other words. It is in a state of thermodynamic disequilibrium. The universe is evolving toward equilibrium as any closed system does. All stars will eventually die out and black holes will evaporate. It is moving toward a state of maximum entropy or thermal equilibrium, a fate called heat death. An expanding information structure offers a mechanism for this increasing entropy - new "bits" of information are continuously added to the world sheet as it grows.

The second attractive feature of this principle has to do with gravity, a force so familiar to us that still a mystery because, unlike all the other forces in nature, it cannot be described from a quantum mechanical point of view as a force particle. Using the same string theory mathematics used in the Holographic principle, gravity becomes an emergent phenomenon of a two-dimensional reality, rather than a fundamental interaction like the other forces. This idea is closely linked with the entropy mechanism. In 2009, physicist Erik Verlinde modeled gravity as an entropic force by adding up all the tiny degrees of freedom (microscopic entropy) encoded on the two-dimensional information structure of the Holographic principle. In this theory, derived from both classical Newtonian gravity and relativity, gravity emerges as a large-scale phenomenon, one that you don't experience on the microscopic scale, reflecting what appears to happen in nature.

Despite the promise of the Holographic principle, I feel uneasy about an underlying flat reality. Does this mean everything, including us, breaks down into quantum information written on a two-dimensional sheet, a bit like the Matrix movies except that in this case, the "wool pulled over our eyes" is the reality? Whatever I feel, this is one of the few theories I know of that can explain a universe with all of its locality-based physical laws and quantum entanglement intact. It implies that electrons really exist on a two-dimensional "sheet," and it is only because of our three-dimensional viewpoint that the connections between them seem spooky.

How To Explain Spooky Action > Add a Spatial Dimension

I recently read a fascinating article called "Strange and Stringy" by physicist Subir Sachdev, in the January 2013 issue of Scientific American. This article forced me to revisit the notion that electrons are tapped into a reality that is essentially hidden from us. Spooky action at a distance has a distasteful magical air about it when you can use only the tools of quantum mechanics and relativity to look at it. As we move beyond Einstein's four-dimensional space-time once again, string theory seems to set the course . . .

Subir Sachdev is a condensed matter researcher. This is a field of physics that studies the interactions of atoms and molecules in close proximity to each other and the properties these interactions manifest in materials. In his article, he describes how he (heroically, I think) approached string theorists in an attempt to understand how the atoms in superconductors coordinate themselves in ways that look a lot like quantum entanglement.

A superconductor is a material that, when cooled below a specific critical temperature, shows some remarkable qualities. Its electrical resistance drops to zero and it ejects all magnetic field lines from its interior, a phenomenon called the Meissner effect. Subir Sachdev's work focuses on what electrons are doing inside the material as it changes from an ordinary conductor or resistor into a superconducting state. Although superconducting materials may be pure metals and metal alloys, they can also be made of other materials, including ceramics and carbon nanotubes.

Electrical conduction is fairly well understood, especially in metals, which tend to be good conductors. Metals have relatively few electrons in their outermost (valence) orbital, and these electrons can adjust their energy very easily. The electrons holding the lattice of atoms together share all the bonds equally. This means they are delocalized and the valence orbital (part of the electron's wave function) is smeared across the entire metal. It doesn't exactly mean that millions of electrons squeeze into one single orbital. That violates the Pauli exclusion principle. Instead, countless individual two-electron orbitals overlap with each other, creating hybrid molecular orbitals. You can't distinguish one molecule from another one inside a metal because of this bond sharing.

When an electric potential is applied to the metal, the electrons adjust their state slightly and move in one direction as a single wave, creating an electric current. If you remember that an atomic orbital is described using quantum numbers, you can describe the metal's "electron sea" as a quantum effect. When electrons drift in the direction of an electric field, the metal experiences some internal resistance as they maneuver through and collide with the lattice arrangement of atoms. Atoms hold to the lattice arrangement, thanks to chemical bonding, but when these collisions occur, the atoms vibrate within the lattice. At any temperature above absolute zero, there is always some vibration in the lattice and the collisions add to it. The atoms have increased kinetic energy, which we can feel when a current-filled wire gets warm, for example.

As you lower the temperature of a conductor, electrical resistance decreases as the frequency of collisions decreases. The atoms in the lattice vibrate less vigorously. But even at absolute zero, where atoms reach their lowest possible kinetic energy and the material no longer contains any thermal energy, drifting electrons will still experience some resistance. In this case, the average free path for an electron is determined more by impurities and defects in the metal lattice. No metal is a perfect lattice and these defects impede the current just a tiny bit, even at absolute zero.

When you cool down a superconductor, you get a different result. At first, the electrical resistance drops as it does in an ordinary conductor, but when it reaches a specific temperature, before absolute zero, called the critical temperature, electrical resistance suddenly drops to zero. There is absolutely nothing impeding the electrons as they drift through the material. How is that possible?

In a regular conductor, the chemical bond (the valence electron orbital) is shared among electrons. The electrons share one quantum aspect in other words, but they still, for the most part, act like a sea of individual electrons. In a superconductor, the drifting electrons don't act like individual particles. Instead they act like a fluid made up of pairs of electrons called Cooper pairs. Each Cooper pair acts like a single particle.

The electrons in these pairs exchange phonons, which act like an attractive force that overcomes their natural repulsion for each other. A phonon is a quantum mechanic description of a special kind of vibration. The whole lattice vibrates at exactly the same frequency throughout. The vibration is perfectly uniform, unlike the vibrations that are set up in the lattice of an ordinary conductor. Those vibrations can be described using the rules of classical mechanics. Just like the energy spectrum of an electron in an excited atom, the phonon vibration energy has gaps or forbidden energies. These gaps owe themselves to quantum mechanics once again, where energies only take on discrete values, not a continuous spectrum. This means you need a specific minimum amount of energy in order to vibrate (excite) the lattice. If the lattice is cold enough, it can't vibrate at all and that means drifting electrons cannot be scattered. They slip right through, as if the material was invisible to them. The critical temperature is where the thermal energy of the lattice drops below the minimum energy required to excite the lattice. In a slightly different way of looking at it, it is where the thermal vibration of the lattice is gentle enough that it doesn't break apart the relatively weak phonon bond between the electron pairs. Critical temperature is specific to each superconducting material, usually between 20K and 1K. Solid mercury becomes a superconductor at 4.2K, for example. The Cooper pair fluid is a superfluid. That means it can flow without losing any energy whatsoever. Current will flow around and around a perfect superconductor loop forever. You can make superconducting electromagnets out of a coil of superconducting wire. These are used in MRI machines and in the Large Hadron Collider.

Researchers recently discovered new kinds of superconductors, ones with much higher critical temperatures, usually around -76K. These are the materials that Dr. Sachdev is trying to understand. And understanding them may lead to a deeper understanding of quantum entanglement, string theory, a new way to look at spooky action at a distance, and ultimately a new way to look at the entire universe.

He is taking clues from the more conventional low-temperature superconductors. Within these materials, Cooper pairs do not act like paired electrons in ordinary ionic and covalent chemical bonds or even those shared across the "electron sea" in metals. Instead, thanks to their phonon bond, they act like one particle instead of two. This means that the electrons no longer obey the Pauli exclusion principle because each one of the pair condenses into the same quantum state. You can use the word condense here because this process actually leaves a band gap (a now-empty energy level) above the electron pair inside each metal atom. The phonon interaction between electrons in a Cooper pair is surprisingly long-distance, on the order of hundreds of nanometers. This distance is greater than the average electron-electron distance inside the material. It implies these electrons form an entangled pair.

How can these two separate particles act like one particle? The electrons themselves are particles called fermions. They have a spin of 1/2. Electrons, like all fermions, have half integer spins and they must all follow the Pauli exclusion principle. No two fermions can share all the same quantum numbers at the same time, except . . .

When you exchange two identical particles, it is mathematically equivalent to rotating each particle by 180 degrees. Whole integer spin particles (force particles or bosons) don't change the sign of their wave function when one is swapped for the other. They have a symmetric wave function. But when two fermions are exchanged, their wave function changes because of their 1/2 spins. Somehow the phonon vibration gets around this by allowing the wave functions of two electrons to be symmetric to each other so they can act like one single wave function. A Cooper pair therefore acts like one particle with a total combined spin of one (1/2 + 1/2). This means it follows a different set of rules, called Bose-Einstein statistics. These particles are bosons and they all have whole integer spins. An infinite number of bosons ( can condense into the same quantum state; they can occupy the same space at the same time. Inside a superconducting material, Cooper pairs condense into the same lowest possible energy quantum state. Dr. Sachdev describes this as pouring water into a glass but instead of filling up the glass like you expect, the water you pour in forms a thin layer of ice on the bottom that never gets thicker.

Superconductor condensation is one way to get two electrons to act like one particle. A phonon vibration turns two fermions into one boson. This phenomenon itself says something interesting about how electrons can change their guise, but this doesn't translate into the behaviour of entangled electrons, where there should be no phonons present in air or a vacuum and no new force particle (boson) shows its presence. The types of superconductors that Dr. Sachdev is working on, however, reveal true electron entanglement within a material.

In high-temperature superconductors, such as barium iron arsenide, the electrons seem to follow rules similar to Bose-Einstein statistics when they are very cold. But the attractive force between the electrons is not a phonon. It doesn't come from the vibration of the material's lattice. Instead it comes from the electron spin itself.

In this material, you can replace any fraction of arsenic sites in the lattice with phosphorus atoms. If you swap in a small amount of phosphorus, the material forms a spin-density wave, an example of a new kind of quantum physical state. A spin density wave is a modulation in the density of up versus down electron spins in the material. Electrons arrange their spins in a lowest possible energy ground state, reminiscent of the electron spin arrangements you find in magnetic materials. In this case, on half the iron sites the electron spins are more likely to be up and on the other half they are more likely to be down. As you increase the amount of phosphorus, you gradually lose the spin density wave. It disappears altogether when you reach 30% replacement. Now, inside the material, the electron spin at any site in the lattice is equally likely to be up or down.

If you raise the temperature of the material at this point, something very interesting happens. You don't get a superconductor or a spin-density wave. You get a new state altogether, called a strange metal. In this quantum state, all the electrons have a 50% chance of being up or down and, like the quantum entangled pair, they don't choose between them but instead form a giant entanglement across the whole material, where the spin of every electron is in a superimposed up/down state. A strange metal is neither an ordinary conductor nor a superconductor. The electrical resistance increases linearly instead of with the square of the temperature as you heat it.

Describing all the possible electron interactions across this material is an almost impossible task. Surprisingly, string theory offers a solution . . .

When you look at string theory, you won't find anything about quantum entanglement. What you will find is an elegant set of complex mathematic formulas. Using these equations you can visualize particles of matter (fermions) and force (bosons) as very tiny vibrating one-dimensional strings, rather than points. Each particle is a unique vibration. Strings avoid the problem of infinite values for forces such as gravity inside black holes. General relativity describes the gravity of a black hole as an infinite space-time curvature. Infinities make it impossible to know anything about the physical processes going on inside the black hole. String theory makes corrections to Einstein's equations for gravity that grow more significant as the distance scale gets smaller. They can be made to work where general relativity fails. The catch is that space-time geometry itself must give way as relativity gives way. At normal distances and with normal gravity, the corrections disappear and string theory looks just like general relativity, but in black holes, where distance approaches zero and gravity approaches infinity, string theory takes over and it can describe the physics going on with great (but not total) success.

String theory uses the geometry of something called branes instead of the geometry of space-time. Branes describe spatial dimensions, anywhere from zero (a point) to up to 11, depending on which one of several string theories you choose from. Strings may vibrate freely across these spatial dimensions as well as one dimension of time, or they may be stuck to branes at one or both of their ends.

What Dr. Sachdev found is that you can take the almost-impossible job of mathematically describing every possible entangled electron interaction in a strange metal and transform it using the formulas in string theory into a relatively simple task. These formulas treat the entanglement process like spatial distance. In this way the depth of entanglement, from just a few pairs of entangled electrons to all the electrons in a material being entangled, acts as a fourth spatial dimension. This extra spatial dimension only shows itself when you are looking in the right place, for example at a quantum phase transition, "like [looking at] a figure in a pop-up book" in his words. I've drawn my interpretation of this idea below.

Like the holographic principle, two entangled but separate electrons look like one single particle, if the four dimensions of space-time are altered.

If you are now asking if it's the whole electron or just part of it, the spin, that's tangled up in the extra dimension, I don't know. I'm not sure if anyone does. In fact, the author is not saying that he believes there is any physical reality beyond the mathematics, some kind of five-dimensional universe like the one I attempt above, buzzing with strings. He claims only that the mathematics translate well into condensed matter problems.

There is a downside to mathematically simplifying a system. You don't get a real view of an actual material in all its complexity. You certainly don't get a sudden revelation of matter's stringy insides. However, Sachdev and his colleagues are discovering that string theory formulae not only describe strange metals better than anything else, they can also describe superfluids, another quantum phase of matter.

While Dr. Sachdev's article does not indulge in much speculation, he emphasizes the astounding point that they are using mathematics originally designed to describe black holes and the universe just as it exploded into being to predict how electrons behave in matter. This involves a huge step back, a rethink of the problem and then recognizing the potential and getting up to speed with another's unfamiliar work, but he has shown the payoff can be immense.

What Is An Electron?

I would like to have been able to place the electron in some kind of real object type of framework for us, where we can sit down and visualize it and go, "Yes. That's an electron." Quantum mechanics and string theory, both complex and elegant mathematical frameworks, don't allow us this satisfaction. Instead, they seem to leave almost everything open for interpretation. It's up to physicists to either try to make a real object out of math, or to convince us that the math is the object (shiver).

Does an electron ultimately live in two or five dimensions rather than "our" four? In this sense, condensed matter physics might be the first rigorous testing ground for string theory. It seems possible that the successes of these approaches imply an as yet unknown physics underlying the universe. The Holographic principle can be interpreted to mean that the universe condenses down into a kind of two-dimensional mathematical framework. What does that say about the ultimate reality of an atom, or anything? Is five-dimensional space-time any better? What's lurking in there that we don't know about?

It's a scary and hugely exciting time to be watching physics evolve! Elusive little electrons are trying to tell us something about the universe, more we can fathom.