GAUGE SYMMETRY
It has been known for several hundred years that electric charge is conserved in any physical process. This conservation law is fundamental to the classical theory of electric and magnetic fields, or electromagnetism.
We see an example of electric charge conservation when we consider the decay of the neutron, . The neutron is electrically neutral, having zero electric charge. When it decays, we are left with a positively charged proton, a negatively charged electron, and a neutral (anti) neutrino. The positive charge of the proton identically equals the opposite of the negative electron charge, and the neutrino has zero electric charge, so the final products of the neutron decay have a zero total electric charge. Electric charge conservation is an exact conservation law in all physical processes—we have never seen a net gain or loss of electric charge in any physical process. The existence of this conservation law implies some hidden symmetry in nature.
Electromagnetism, or “electrodynamics,” is the physical description of electric and magnetic fields, and electric charges and currents, and it was formulated in a classical (non-quantum) framework over the entirety of the nineteenth century. The pinnacle achievement is usually considered to be the formulation Maxwell's equations, discovered in 1861 by James Clerk Maxwell, a succinct and complete set of equations that summarized all known aspects of electrodynamics, which allow us to compute the electric and magnetic fields anywhere in space and time, given any choice of electric charge and electric current distributions.9
Maxwell's classical theory of electrodynamics makes no sense without the conservation law of electric charge. The underlying continuous symmetry that leads to this, however, appeared, at first, to be somewhat obscure. Electric charges are the sources of electric fields, much like mass is the source of a gravitational field in Newton's theory of gravity. An electric field is just the electric force exerted on an electric charge at any point in space. When electric charges move, they become electric currents and produce magnetic fields. Magnetic fields, in turn, produce forces on moving electrons (electric currents). In fact, a pure electric field in space becomes a combined electric and magnetic field if we simply move through it.
The Maxwell theory does not allow solutions to its equations in which a source or a sink, an electric charge, simply disappears into nothingness. Even if an electric charge falls into a black hole, the black hole itself will have the same value of the electric charge that it swallowed.
If we probe deeper into the structure of Maxwell's theory, however, we find that there is something even more fundamental than the electric and magnetic fields called a gauge field. The gauge field is related to the electric and magnetic fields in a peculiar way: If we are given the gauge field in any region of space and time, we can always calculate the values of the electric and magnetic fields in that region. However, we cannot reverse this process. That is, given electric and magnetic fields in the same region of space and time, we cannot determine exactly what gauge field produces them. In fact, we can always find an infinite number of gauge fields that would produce the same observed electric and magnetic fields.
Moreover, while electric and magnetic fields are easily measured in the lab, we cannot directly measure the gauge field by theory or experiment. Even a zero value everywhere for the electric and magnetic fields, that is, a vacuum, does not determine the value of the gauge field—infinitely many different gauge fields exist that produce zero values of the electric and magnetic fields. The gauge field is therefore a hidden field, not amenable to any measurement that would determine its exact form.
The concept of a gauge field was first considered as a tool for conveniently expressing electric and magnetic forces by various scientists in the early to mid-1800s. Often different people would write down different gauge fields, in different forms, and it was always unclear whether or not they were describing different phenomena. In 1870, Hermann Ludwig Ferdinand von Helmholtz, a famous contributor to the theory of electromagnetism, showed that different forms of gauge fields can lead to the same physical consequences, that is, to the same electric and magnetic fields. One can continuously transform one gauge field into another, and the physics stays the same. This is essentially the first example of a new symmetry transformation of electrodynamics—a “gauge transformation”—though its implication as a fundamental symmetry of nature was not appreciated at the time.10
In fact, if we turn this around and insist that, as a symmetry principle, the gauge field must always be a hidden field and can never be determined unambiguously, then we do find something remarkable: it is this gauge symmetry that implies that electric charge must be conserved! We can continuously transform our chosen gauge field into another one, without changing the values of the electric and magnetic fields, and this is the symmetry that leads to the conservation of electric charge. This hidden symmetry is called “local gauge invariance.”11
It was in the twentieth century, with the development of quantum mechanics, and the effort to include both the electron and electromagnetism into one completely consistent theory, that the symmetry of gauge invariance emerged as the overarching theme. In fact, this has been the dominant theme in all of twentieth-century physics—all forces are now known to be governed by “gauge symmetries” and are called “gauge theories.”
All particles are described in quantum theory by waves, through their wave functions. The wave function is denoted by ψ(x, t) and is a complex number–valued function of space and time. The probability of finding the particle at space location x and at time t is determined by the mathematical square of the wave function |ψ(x, t)|2. The information about the particle's momentum is determined by the wavelength of the wave, and the energy by the frequency through the formulae E = hf, “energy equals Planck's constant times the frequency,” and p = h/λ, “momentum equals Planck's constant divided by the wavelength.” Despite the fact that this energy and momentum information is always present in the wave function and can readily be extracted from it by differentiating it with respect to t or x, we can never measure the wave function directly because the wave function involves complex numbers that don't make sense as physical observables. Only the (absolute squared) magnitude of the wave function, which is the probability, can actually be measured.
We can ask: “What would happen if we somehow changed the phase of the wave function without changing the observable probability at any point in space and time?” We keep the probability of finding the electron at any point in space the same. We call this a “gauge transformation.” But, in making this change, there is apparently nothing invariant here. This would affect the derivatives of the wave function with respect to t and x, and those determine the energy and momentum. This is evidently not a symmetry of the original quantum state, but rather it seems to produce a new quantum state with different observable energy and momentum.
Let us now suppose that there is some other quantum particle wave that modifies the derivatives with respect to t and x. And let us further suppose that when we change the electron's wavelength or frequency, we are simultaneously required to modify the new field in such a way as to keep the derivatives with respect to t and x the same. The net effect is that we have kept the probability, the energy and the momentum invariant under our transformation. Together with the gauge particle, we can maintain both the original incoming total energy and the momentum, even though we scramble the unobservable phase of our electron's wave function. Thus, the term “gauge” means that the actual determination of the physical momentum of the electron requires the presence of the calibrating “gauge” field. Only the electron wave function, together with the “gauge” field, yields a physically meaningful description of the electron. The presence of the new gauge field in the derivatives causes the interaction of the photon with the electron (see note 11).
The gauge theory asserts that, if the electron is given a physical kick, if an electron is accelerated, then the gauge field is actually shaken off—it is emitted as an independent p
article wave with a physical momentum of its own, and the electron recoils to conserve energy and momentum. The gauge field becomes a true physical entity and is radiated out into space. From the point of view of a distant observer, an accelerated electron has radiated a new particle, the photon.
Light is emitted from accelerated charges. This occurs in countless physical processes, such as the scattering of an electron off of an atomic nucleus, or an atom, or another electron. It can be observed readily in the laboratory. At very low energies, it is the way in which electrons emit the photons from a campfire. Accelerated electrons radiate the microwaves that heat our coffee in a microwave oven, or transmit the evening news into our living rooms, or cause the sun to shine.
We can graphically represent a physical process by a set of Feynman diagrams that represents the quantum computation. These diagrams tell us precisely how to compute the quantum outcome, the probability of a given process, provided that the strengths of the interactions are known and are not too large. We can often visualize a process through Feynman diagrams even when we cannot compute the result. A graduate student, writing from Cornell University where Feynman developed this technique, commented, “At Cornell, even the janitors use Feynman diagrams.”12 With the full machinery of Feynman diagrams we can compute the scattering rate for two beams of electrons to arbitrary precision, including many diagrams that represent detailed quantum corrections to the basic result. The experimentalist can compare the calculations with the results measured in the lab, and these are found to agree to extremely high precision.
YANG–MILLS GAUGE THEORY
The modern era of gauge theories began with a remarkable paper of Chen Ning Yang and Robert Mills in 1954.13 These authors asked a straightforward question: “What happens if we extend the gauge symmetry of the electron to larger symmetries?” The symmetry of electrodynamics involves, as we have seen, the phase of the electron wave function. This is called “U(1) symmetry.”
Yang and Mills turned to the next more complicated symmetry, “SU(2),” the symmetry of the rotations sphere in three real dimensions (or the symmetry of the rotations of two particles, such as (u, d) quarks or (ve , e−), that is, rotations in 2 complex dimensions). It turns out that this symmetry leads to a more general form of a quantum gauge theory called a “Yang–Mills theory.” SU(2) has three gauge fields, hence three photon-like objects, and now the gauge fields themselves carry charges, unlike the case of electrodynamics in which the photon carries no electric charge. Moreover, the Yang–Mills construction works for any symmetry. Symmetry thus becomes partially fundamental to the basic structure of a quantum theory of forces.
In the Standard Model electroweak theory, the symmetry is the “product group” of SU(2) × U(1), with 4 gauge fields, W+, W–, Z0, and γ fully and accurately described by the Yang–Mills theory. The Higgs boson, as we have seen, causes the W+, W–, and Z0 to become heavy, while the photon γ remains massless. Likewise, as we've seen, the quarks carry 3 “colors,” and the resulting SU(3) gauge theory has 8 gauge bosons, known as the gluons.
Indeed, all known forces are based upon gauge theories. Yet, there are four completely different structures, or styles, of gauge invariance. Einstein's theory of gravity contains a coordinate system invariance, that is, it doesn't matter what coordinate system you use, or how you choose to move, inertially or non-inertially through space and time, to describe nature. This leads to gravity as a bending and reshaping of geometry, governed by the presence of energy (equivalent to mass) and matter. Particles must then emit and absorb gravitons, which are the gauge fields, or the “quanta,” of gravity. The Newtonian gravitational theory is recovered only as an approximation at low energies (slow systems, without too much mass). The description of the remaining nongravitational forces in nature is based upon the Yang–Mills theory of SU(3) × SU(2) × U(1) as codified by the Standard Model.
THE WEAK FORCE AS A GAUGE THEORY
Let's now consider in a little more detail how the weak interactions are described by a gauge symmetry that unifies them together with the electromagnetic force. Taken together, the quarks, leptons, and the gauge symmetries (including Einstein's general relativity) provide a complete accounting of all observed laboratory physics to date and define what is called the “Standard Model.”14
Recall that, subsequent to Becquerel et al., yet more than 65 years ago, Enrico Fermi wrote down the first descriptive quantum theory of the “weak interactions.” Fermi had to introduce a new fundamental constant into physics to specify the overall strength of the weak interactions, called GF, and it represents a fundamental unit of mass, which sets the scale of the weak forces, about 175 GeV.
In the 1960s the weak forces were found to involve a gauge symmetry based upon the symmetries SU(2) × U(1) (by Sheldon Glashow, Abdus Salam, and Steven Weinberg, and this was perfected as a quantum theory by Gerhard ‘t Hooft and Martinus Veltman). Let us now describe the gauge symmetry of the weak interactions.
We see that, within each generation, the quarks and leptons are paired. That is, the red up quark is paired with the red down quark, the electron neutrino with the electron, the charm quark with the strange, the top with the bottom, and so forth. We thus imagine that the electron and its neutrino are a single entity that lives in a two-dimensional space, with one axis meaning “electron” and the other “electron neutrino.” The quantum state is an arrow in this space that can point in any direction. When the arrow points along the electron axis, we have an electron. Rotating the arrow, we have a neutrino. The rotations we can do on the arrow form the symmetry group, called SU(2).
So we now imagine an electron neutrino particle wave with a given momentum and energy. Then we perform a gauge transformation that rotates this into an electron, which has negative charge, and also scrambles the electron momentum and energy. To make this into a symmetry, we need to introduce a gauge field, the W+ that can restore the total energy and momentum, and rotate the quantum arrow back to its original electrically neutral “electron neutrino” direction. In a sense, the gauge field rotates the coordinate axes, so the arrow is now pointing back in the original direction, relative to the coordinate system, and we get back the original neutrino we started with. This is completely analogous to what we do with quark color, where the gauge rotation from one color to another is compensated by the gluon field. This requires a total of three new gauge fields, W+, W–, Z0, in addition to the photon γ. In fact, electrodynamics and the weak interactions now become blended together into one combined entity called the “electroweak interactions.”
There is, however, an enormous difference between the photon and these three new gauge fields. The photon is a massless particle, while the W+, W–, Z0 are very heavy particles. The forces that are produced by the quantum exchange of W particles between quarks and leptons give rise to the weak force that Fermi was describing 65 years ago. As we've seen, the Higgs field in the vacuum causes the W+, W–, Z0 to become massive.
The strength of the Higgs field in the vacuum is already determined by Fermi's theory to be 175 GeV. The field implies the existence of a new particle, the Higgs boson, the necessary quantum of the Higgs field. All the matter particles, and the W+, W–, Z0, get their masses by interacting with the vacuum-filling Higgs field (unlike a superconductor, however, the photon does not interact with this particular field and remains massless). The Higgs field is “felt” by the various particles through their “coupling strengths.” For example, the electron has a coupling strength with the Higgs field, ge. Therefore, the electron mass is determined to be me = ge × (175 GeV). Since we know me = 0.0005 GeV, we see that ge = 0.0005/175 = 0.0000029. This is an extremely feeble coupling strength, so the electron is a very-low mass particle. Other particles, like the top quark, which has a mass mtop = 175 GeV, has a coupling strength almost identically equal to one (suggesting that the top quark is playing a special role in the dynamics of the Higgs field). Still other particles, like neutrinos, have nearly zero masses and therefore nearly zero
coupling strengths.
All of this sounds like a spectacular success, and it is, but there is a slight shortcoming—there is, at present, no theory for the origin of the coupling constants, such as that of the electron ge. These appear only as input parameters in the Standard Model. We learn almost nothing about the electron mass, swapping the known experimental value, 0.511 MeV, for the new number, ge = 0.0000029. Furthermore, we are clueless as to what generates the mass of the Higgs boson itself.
The Standard Model did successfully predict the coupling strength of the W+, W–, Z0 particles to the Higgs field. These coupling strengths are determined from the known value of the electric charge and another quantity, called the weak mixing angle, measured in neutrino scattering experiments. So the masses of W and Z, MW and MZ (note that the W+ and W– are particle and antiparticle of each other and must have the same identical masses; the Z0 is its own antiparticle), are predicted (correctly) by the theory. The W+ and W– have a mass of about 80 GeV, and the Z0 has a mass of about 90 GeV. These have been measured to very high precision in experiments at CERN, SLAC, and Fermilab.
Symmetry and its spontaneous breaking through of the Higgs particle, therefore, completely controls the mass generation of all the particles in the universe. And it appears that it is the Higgs boson, the quantum of the Higgs field, that was discovered in the two experiments, ATLAS and CMS, at CERN on July 4, 2012, with a mass of mh = 126 GeV. “What generates the Higgs boson mass?” is now the most important scientific question of our time.
Beyond the God Particle Page 29