Beyond the God Particle
Page 28
Note that if a gluon collides with a quark, the gluon will be absorbed and the quark will be accelerated. It is, perhaps, one of the most astounding aspects of modern science that the simple idea of a symmetry yields up the photon, and quantum electrodynamics likewise yields the correct theory of the strong interactions when it is adapted to quark color. This theory is called quantum chromodynamics (QCD), and it is a stunning success.
Quarks thus interact with one another by the exchange of gluons. We can draw the appropriate Feynman diagrams and learn how to compute them. The force is strong because the “color charge,” g, the analog of the electric charge, is large.
Figure A.37. Quark Exchanging Gluons. A red quark scatters off of a blue quark. The quarks exchange color through the (red, anti-blue) gluon that hops between them, giving rise to the strong force that confines the quarks. The force becomes weaker at higher energies, a prediction of the theory that is well confirmed by experiment. The proton is held together by gluon exchange among the quarks. Every 10–24 seconds or so, a gluon hops between quarks in the proton.
One of the most remarkable discoveries about QCD is that the coupling strength of the interaction of the quarks to the gluons, denoted by g, becomes weaker as we push the quarks together to extremely short distances. This is known as “asymptotic freedom.” Conversely, at large distances, the quark-gluon coupling becomes very large. This leads to a strong pull on the quarks, preventing them from being separated and isolated in the lab. It also turns out that, because of this strong coupling, only quantum-bound states that are composed of quarks that have an exact color neutrality—a perfect balance of all three quark colors at any instant in time—can exist. This means we can have only combinations of (rby), which are the baryons, or (), which are the anti-baryons; by () we mean the anti-color of (q), or the color-neutral quantum combination of π+ ↔ u, which are the mesons. The color gauge theory, QCD, therefore neatly explains the pattern of strongly interacting particles found in nature, and why the quarks can never be really freed from their prisons.
While it is very hard to compute the properties of the theory when g is large, the fact that g becomes small at short distances means that fairly precise calculations using Feynman diagrams can be performed revealing the collisions and scattering of individual quarks at very high energies. It also means that at very high energies, for example, in the collisions produced at the LHC, the individual quarks and gluons collide and leave traces of their collisions. This leads to a bizarre phenomenon, nature's own version of a prison break, known as a quark jet (gluon jets can also occur).
At the LHC, a proton with seven trillion electron volts of energy (7 TeV) collides head-on with a proton of the same energy. At the highest energies, or shortest instants of time, the individual quarks are resolved and behave as though they were almost free particles. Therefore, collisions occur in which a pair of quarks, perhaps a u and a d collide head-on. This quark and antiquark are scattered through very large angles, ripped out of the proton and antiproton, while the remaining debris, the other quarks and gluons of the original proton and antiproton, continue to move forward in their original directions of motion. For a brief moment the quarks are free, moving at very high energies, and they can travel perhaps a hundred times the distance within which they are normally confined, away from the debris of quarks and gluons of the shattered proton and antiproton. The quarks, for a brief moment in time, have broken out of their confining prison cells.
But then the interaction becomes strong and the vacuum itself begins to rip apart in the vicinity of the collision. Pairs of quarks and antiquarks and gluons are ripped out of the vacuum, a turbulent plasma of matter from the point of collision, like the long arm of the law apprehending the escapees. The liberated quarks become shackled by this flurry of new matter and antimatter. Soon all quarks and gluons are recaptured, reassigned to new pions and protons and neutrons. The liberation of the quarks is over.
Nonetheless, the indelible footprint of the escaped quarks remains. Two very well-defined blasts of particles, called jets, mostly composed of pions, stream off into space in the directions of the original u and d escapees. These jets of particles clearly mark the original quark paths and carry the full energy of the temporarily liberated quarks. These jets are the conspicuous tracers of high-energy quark, and gluon, collisions.
At the LHC a pair of gluons can collide to produce a Higgs boson. The decay signature of the Higgs boson is reconstructed in the detector. There are many decay modes of the Higgs boson, but the one first glimpsed at the LHC was a pair of gammas, two high-energy photons, in the process g + g → higgs → γ + γ. In this way, nature's most enigmatic particle, the Higgs boson, is pulled from the vacuum's Higgs field that surrounds us and gives mass to all the other particles.
SPIN
Any rotating body has spin—a top, a CD player, the earth, the washing machine basin on the rinse cycle, a star, a black hole, a galaxy—all have spin. So, too, quantum particles, molecules, atoms, nuclei of atoms, the protons and neutrons in the nucleus, the particle of light (photons), electrons, the particles inside of protons and neutrons (quarks, gluons), etc. But while large classical objects can have any amount of spin, and can stop spinning altogether, quantum objects have “intrinsic spin” and are always spinning with the same total intrinsic spin.
An elementary particle's spin is one of its defining properties. We can never halt an electron from spinning, else it would no longer be an electron. However, we can rotate a particle in space, and the value of its spin, as projected along any given axis in space, will change, just as it does for a classical spinning top. The difference in quantum physics is that we can only ask what value does the spin have when projected along a given axis, because that is what we can measure—asking about things we cannot measure is meaningless in quantum physics.6
Let's discuss the rotational motion of a classical object. Linear physical motion is measured by something called momentum. In Newtonian physics this is simply mass times velocity. Note that this combines the concept of matter (mass) and concept of motion (velocity), so it represents a kind of measure of “physical motion.” This is a vector quantity, since the velocity is a vector, having both a magnitude (speed) and a direction (of motion) in space. In general a vector can be visualized as an arrow in space with both magnitude and direction.
Likewise, physical rotational motion is measured by a (pseudo-vector) quantity called “angular momentum.” Classically, angular momentum involves the way in which mass is distributed throughout the object, which is the “moment of inertia.” If a body is large, with a large radius, when it spins there is a lot more matter spinning than if the same amount of mass were distributed within a smaller radius. So, not surprisingly, the moment of inertia, I, increases with the size of the body. In fact, it's mass times “the (approximate) radius of the body squared,” or roughly I = MR2 with M the mass and R the “radius” of the body. This can be made very precise using calculus.
Spin also involves the “angular velocity,” how fast the object is actually rotating. Angular velocity is usually denoted by ω (omega) and is “so and so many radians of rotation per second.” (360 degrees equals 2π radians. So, for example, 90 degrees corresponds to π/2 radians; radians are a more mathematically natural way to measure angles than degrees because a circle with a radius of one has a circumference of 2π. Therefore, spin is just the product of the moment of inertia times the angular velocity, or S = Iω. (Compare: momentum is mass times velocity and describes motion in a straight line, while spin is moment of inertia times angular velocity—these are very similar constructs.) Spin is also a vector quantity, pointing along the axis of the spin rotation. Here we use the “right-hand rule” to establish the direction of the spin vector: curl the fingers of your right hand in the direction of the spinning motion and your thumb will point in the direction of the spin vector.
Spin is a form of angular momentum, which is a conserved quantity (like energy and momentum) such
that the total angular momentum of an undisturbed isolated system remains forever constant. As a consequence of this, we see that an ice skater, viewed as a physical system, can dramatically increase her spin motion (angular velocity) as she draws her arms inward. The spin angular momentum is S = Iω = MR2ω, which must stay the same as she pulls her arms in. Pulling her arms in decreases R, while M stays the same. So, the angular velocity ω must increase to compensate the decrease as the rotational velocity increases. In fact, R2 becomes four times smaller if the skater simply decreases her arms’ outward distance, R, by a half, so her angular velocity must increase approximately fourfold, which is why this is such a dramatic stunt.
Angular momentum, which was a continuously varying quantity in Newtonian physics, changes its character drastically in quantum mechanics—it becomes quantized. Angular momentum is always quantized in quantum mechanics. All observed angular momenta as measured along any spin axis are discrete multiples of = h/2π, where h is Planck's constant. All the particle spin and orbital states of motion we find in nature have angular momenta that can have only the exact values
Angular momentum is always either an integer or a half-integer multiple of in nature. We don't see this quantization effect for very large classical objects because they can have such enormous angular momenta, many times greater than . Only at the level of exceedingly tiny systems, atoms, or the elementary particles themselves, do we observe the quantization of angular momentum.
Angular momentum is therefore an intrinsic property of an elementary particle or an atom. All elementary particles have spin angular momentum. We can never slow down an electron's rotation and make it stop spinning. An electron always has a definite value of its spin angular momentum, and that turns out to be, in magnitude, exactly /2. We can flip an electron and then find its angular momentum is pointing in the opposite direction, or –/2. These are the only two observable values of the electron's spin when measured along any chosen direction in space. We say that “the electron is a spin-1/2 particle,” because its angular momentum is the particular quantity, /2.
Particles that have half-integer multiples of for their angular momentum, that is,
are called fermions, after Enrico Fermi, who helped pioneer these concepts (with Pauli and Dirac). The main fermions we encounter in most of our discussions are the electron, the proton, or the neutron (and quarks, which make up the proton and the neutron, etc.), and each has angular momentum /2. We refer to all of these as “spin-1/2 fermions.”
Particles, on the other hand, that have angular momenta that are integer multiples of , such as 0, , 2, 3,,…and so on, are called bosons, after the famous Indian physicist Satyendra Nath Bose, who was a friend of Einstein and who developed some of these ideas. There is a profound difference between fermions and bosons that we'll encounter momentarily. Typically, the only particles that are bosons and that will concern us presently are particles like the photon, which has “spin 1,” or one unit of angular momentum; the quantum particle of gravity, the graviton, which has yet to be detected in the lab, and has “spin 2,” or 2 units of angular momentum; and other particles that are made of quarks and antiquarks, called mesons, that have “spin 0,” or 0 units of angular momentum. Orbital motion also has angular momentum. All orbital motion, in quantum theory, has integer units of for angular momentum, hence, 0, , 2, 3,,…and so on.
EXCHANGE SYMMETRY
Elementary particles are so fundamental that they have no identifying labels. For example, any two electrons cannot in principle be distinguished from each other. There is no difference between any two electrons in the universe. The same is true of photons, muons, neutrinos, quarks, etc. The quantum effect of this identity symmetry depends strongly upon spin.
Now, in everyday life, the category of “things” that we encounter called “dogs” is very large, and no two dogs are identical. However, all electrons are precisely identical to each other. Electrons carry only a very limited amount of information. Any given electron is exactly identical to any other electron. The same is true of the other elementary particles. Therefore, any physical system must be symmetrical, or invariant, under the swapping of one such particle with another. In a sense, nature is very simple-minded in the way it treats electrons in that it doesn't know the difference between any two (or more) electrons in the whole universe.
“Exchange symmetry” implies that swapping two identical particles must leave the laws of physics invariant because the particles are identical. At the quantum level this implies that our swapped particle waves must give the same observable probability as the original. But probability involves taking the “square” of the waves, or more properly, the square of their “wave functions.”7 This condition, however, implies two possible solutions for the effect of the exchange on the wave function, that is, the exchanged wave can either be symmetrical, +1 times the original one, or else it can be anti-symmetrical, –1 times the original one. Either case is allowed, in principle, because we can measure only the probabilities (the squares of wave functions). Quantum mechanics allows both possibilities, so nature finds a way to offer both possibilities, and the result is astonishing.
BOSONS
For bosons, upon swapping two particles in the wave function, we would get the + sign.8 With this result, we find an important effect—two identical bosons can be located in the same quantum state. In fact, by considering lots of bosons localized in the same region of space, described by one big wave function, we can actually prove that the most probable place for all the bosons in a system is piled on top of one another. So, it is possible to coax a lot of identical bosons to share the same little region in space, almost an exact pinpoint in space. Or, the identical bosons can be coaxed readily into a quantum state with the exact same value of momentum. Thus, we say that bosons condense into compact, or “coherent,” states. This is called Bose–Einstein condensation.
There are many variations on Bose–Einstein condensation and all kinds of phenomena that have in common many bosons in one quantum state of motion. Lasers produce coherent states of many, many photons all piled into the same state of momentum, moving together in exactly the same state of momentum at the same time. Superconductors involve pairs of electrons bound by crystal vibrations (quantum sound) into spin = 0 bosonic particles (called “Cooper pairs”). In a superconductor the electric current involves a coherent motion of many of these bound pairs of electrons sharing exactly the same state of momentum. Superfluids are quantum states of extremely low-temperature bosons (as in liquid 4He), in which the entire liquid condenses into a common state of motion that becomes completely frictionless. It has to be the isotope 4He in order to get a superfluid (2 protons + 2 neutrons in the nucleus), because the isotope 4He is a boson, while the other common isotope 3He is not (with 2 protons + 1 neutron in the nucleus, it is a fermion; see below). Bose–Einstein condensates can occur in which many bosonic atoms condense into ultra-compact droplets of very large density, with the particles piling on top of one another in space.
FERMIONS
For fermions the rule is that we get the (–) sign in front of the wave function. This holds for any particle with fractional spin, such as the electron with spin 1/2. From this we can prove that no two identical fermions can occupy the same quantum state at the same time. This is known as the Pauli exclusion principle, after the brilliant Austrian-Swiss theorist Wolfgang Pauli. Pauli proved that his exclusion principle for spin 1/2 comes from the basic rotational symmetries of the laws of physics. It involves the mathematical details of what spin-1/2 particles do when they are rotated. Swapping two identical particles in a quantum state is identical to rotating the system by 180º in certain configurations, and the behavior of the spin-1/2 wave function then gives the minus sign (see note 8).
The exclusion property of fermions largely accounts for the stability of matter. For spin-1/2 particles there are two allowed states of spin, which we call “up” and “down” (“up” and “down” refer to any arbitrary direction in s
pace). Thus, in an atom of helium, we can get two electrons into the same lowest-energy orbital state of motion. To get the two electrons in one orbital requires that one electron has its spin pointing “up,” and the other has spin pointing “down.” However, we cannot then insert a third electron into that same orbital state because its spin would be the same, either up or down, as one of the two electrons already present. The exchange symmetry minus sign would force the wave function to be zero.
In other words, if we try to exchange the two electrons whose spins are the same, the wave function would have to equal minus itself and must therefore be zero! Hence, for the next atom, lithium, the third electron must go into a new state of motion, that is, a new orbital. Thus, lithium has a closed inner orbital, or “closed shell” (i.e., a helium state inside of it), and a sole outer electron. This outer electron behaves much like the sole electron in hydrogen. Therefore, lithium and hydrogen have similar chemical properties. We thus see the emergence of the Periodic Table of Elements. If electrons were not fermions and did not behave this way, every electron in the atom would rapidly collapse into the ground state. All atoms would behave like hydrogen gas. The delicate chemistry of organic (carbon-containing) molecules would never happen.
Yet another extreme example of fermionic behavior is that of the neutron star. A neutron star is formed as the core of a giant supernova implodes while the rest of the star is blown out into space. The neutron star is made entirely of gravitationally bound neutrons. Neutrons are fermions, with spin 1/2, and again the exclusion principle applies. The state of the star is supported against gravitational collapse by the fact that it is impossible to get more than two neutrons (each with spins counter-aligned) into the same state of motion. If we try to compress the star, the neutrons begin to increase their energies because they cannot condense into a common lower-energy state. Hence, there is a kind of pressure, or resistance, to collapse, driven by the fact that fermions are not allowed into the same quantum state.