In contrast to O(3), we can also visualize the group O(2) × O(2) × O(2), consisting of three copies of the group O(2). This group also has three generators (angles) that can produce the infinite number of elements of the group. However, in contrast to O(3), the order of rotation of a pair of angles commutes, and therefore this three-dimensional group forms an abelian group, which was not the case for O(3).1
We now move to a two-dimensional group in the complex plane, SU(2). Similar to our construction of U(1) in one complex dimension, we can now form the group SU(2) in two complex dimensions. SU(2) stands for special unitary Lie group in two complex dimensions. In contrast to the case of O(2) and U(1), SU(2) is not identical to O(3). SU(2) is the group of 2 × 2 complex matrices. If you multiply two of these matrices in one order, you get a third matrix, but if you multiply them in the opposite order, you get a different matrix. As in the case of O(3), SU(2) is a nonabelian group. Similar to O(3), the Lie group SU(2) has a finite number of generators—namely, three—that can generate the infinite number of elements in SU(2). Although SU(2) and O(3) are different Lie groups, they do share the same Lie group algebra, which describes the order or sequence of pairs of rotations. The identity operations for O(3) and SU(2) are different (the identity operation is the one that takes the group back to itself under rotations). For O(3), it only takes 360 degrees of rotation to get back to the original state, whereas it takes 720 degrees of rotation in O(3) to get back to the corresponding original SU(2) state.2 Because of this difference, SU(2) is said to have two degrees of rotation such that, in physical terms, SU(2) can describe a particle’s quantum orbital angular motion as well as spin, the spin being understood to be the quantum mechanical spin or degree of freedom.
The nonabelian Lie group SU(2) plays a special role in particle physics and in the standard model in particular. It was this group that Yang and Mills used to describe the mirror symmetry of protons and neutrons; under the influence of strong interactions they look exactly alike, and both are designated as nucleons. Yang and Mills generalized Maxwell’s electromagnetic equations to reach this result. They generalized the U(1) group invariance of Maxwell’s equations for electromagnetism to an SU(2) group invariance of the equations. Yang and Mills called the SU(2) description of nucleons the “isotopic spin group.”
We can, in fact, increase the number of dimensions of the complex space being acted on by the special unitary groups from 2 to N, thereby forming the special unitary group SU(N), where N equals any number greater than one. In particular, the nonabelian group SU(3) is the group of 3 × 3 complex matrices. This group forms the basis of the theory of strong interactions called quantum chromodynamics. Before the development of QCD, Murray Gell-Mann and Yuval Ne’eman used SU(3) to describe the many new particles discovered during the 1960s. The number of generators of the Lie group SU(3) is eight, corresponding to the eight independent rotation angles in three complex dimensions. (For the group U(3), there would be nine generators.) This is why Gell-Mann called his theory of particles the Eightfold Way, alluding whimsically to the Buddhist path to enlightenment.
INVARIANCE: ABSOLUTE OR RELATIVE?
We recall that symmetry in nature means that under a set of transformations, the laws of nature remain the same, or invariant. A fundamental invariance is the one discovered by Galileo. It says that the laws of physics and the results of experiments are the same when expressed in an inertial frame of reference, defined to be one that moves at a uniform speed, when compared with another inertial frame of reference also moving at a different uniform speed. We have experienced this when sitting in a stationary train, looking out the window, and watching another train moving past us slowly. We get the strange feeling that the train opposite is not moving but we are moving, even though we are not. In Galileo’s day, he envisioned ships when thinking about this invariance.
Newton was well aware of this symmetry, called Galilean invariance, and used it to construct his laws of motion. He postulated the existence of an absolute space and an absolute time, and incorporated Galilean invariance by saying that objects move relative to the absolute space and that the simultaneity of events in that space was absolute. Nineteenth-century physicists such as Maxwell believed that there had to be a medium in space that carried electromagnetic waves. This medium, called the “ether,” in effect would represent Newton’s absolute space, because bodies in space were moving relative to the absolute frame associated with the ether. This was an analogy with Newton’s notion that bodies move with respect to the frame of his absolute space. At that time, scientists believed that the speed of light was not independent of the motion of the source of the light. That is, light would move at different speeds relative to the absolute reference frame of the ether.
The null experiment of Michelson and Morley, performed in 1879 to detect (finally) the “ether,” showed, surprisingly, that the motion of the earth did not affect the speed of light. This indicated that the speed of light was a constant, independent of its source, and therefore there was no absolute frame of reference associated with an ether. The idea of an all-pervasive ether in space was so entrenched in the physics of the 19th century that physicists such as Hendrik Lorentz attempted to explain the Michelson–Morley experiment and its implications for light speed by inventing an electrodynamics of electron interactions. In this theory, Lorentz claimed that the matter making up the interferometry rods of the experiment contracted along the direction of the motion of the earth around the sun, canceling out the putative effect of the ether on the light signal, and thereby saving the idea of the ether.
In 1905, Einstein postulated that the speed of light is constant with respect to the motion of every inertial frame and independent of the motion of the source of light. He discovered a new symmetry of nature in the form of his special relativity theory, which overthrew the older Galilean invariance based on Newton’s postulate of an absolute space and time. Now clocks and measuring rods appeared to observers to be affected by their motion. Clocks slowed down as they were accelerated to the speed of light, and measuring rods shrank in size as they were propelled close to the speed of light. (Einstein ignored the idea of the ether.) Moreover, Einstein’s postulates in special relativity, including the postulate that the speed of light is constant with respect to all inertial frames, led him to discover that energy could be converted into mass times the square of the speed of light.
German mathematician Herman Minkowski expressed brilliantly the invariance symmetry of special relativity—contained in the group transformation laws discovered by Hendrik Lorentz in 1904—by picturing spacetime as four-dimensional with three space dimensions and one time dimension. The invariance of the laws of nature with respect to the Lorentz transformations is described by the group SO(3,1). This Lie group is a special orthogonal group of transformations in three spatial dimensions and one time dimension. This group replaced the more restrictive symmetry or invariance group associated with what is called the group of Galilean transformations in which the speed of light was treated as an infinite quantity. SO(3,1) describes the transformation of two coordinate reference frames moving with constant relative speed to one another in special relativity. The Galilean group of transformations can be considered approximately true for slowly moving objects such as the horses and carriages or ships of that era.
In classical physics as well as particle physics, the continuous symmetry invariance of equations implies the existence of a conservation law in a theory. German mathematician Amalie (Emmy) Noether derived the basic conservation laws of energy and angular momentum from the translational and rotational symmetry of the field equations of any physics theory. From her work we can derive the conservation of electric charge from the gauge invariance of Maxwell’s field equations. Gauge invariance is one of the most fundamental invariance symmetries of nature. Maxwell’s equations can be written in terms of a potential vector field, having direction, in four spacetime dimensions. From this vector field, we can derive the strength of the elect
romagnetic field for Maxwell’s equations. A certain transformation in this field, expressed as a derivative of a scalar field with one degree of freedom, leaves the electromagnetic field strength invariant. The derivative, an arbitrary gradient added to the vector potential, does not change Maxwell’s field or equations. This transformation of the vector potential and the resulting invariance of Maxwell’s equations constitute the gauge invariance of Maxwell’s equations. The vector potential field has no absolute values. Although it cannot be detected physically in classical experiments, it does have a deep physical significance in modern gauge theory.3
GAUGE SYMMETRY AND GAUGE INVARIANCE
The important symmetry called gauge invariance plays a fundamental role in particle physics. When famous mathematical physicist Hermann Weyl published the first unified theory attempting to unify gravity and electromagnetism, he used the idea of gauge symmetry to try to unify these forces, applying it to spacetime in his unified theory. In addition to the spacetime metric of Einstein’s gravity theory of 1916, Weyl introduced a vector field, generalizing the Riemannian geometry used by Einstein to construct his theory of gravity.
In his book, Space, Time, Matter, originally published in 1918,4 Weyl reconsidered Einstein’s original purely metric theory, in which distances between points were measured using Pythagoras’s theorem. Weyl questioned the implicit assumption behind the metric theory of a fixed distance scale or “gauge.” What if the direction as well as the length of measuring rods, and also the unit of seconds in measuring the time of clocks were to vary at different places in space-time? He used the analogy of railway gauges varying from country to country. It was well known to travelers at that time that they sometimes had to change trains at national borders because of incompatible railway track gauges. To create a similar situation in physics, which allowed him to incorporate the electromagnetic vector potential into spacetime geometry, Weyl generalized the coordinate transformation symmetry of Einstein’s gravity theory to a gauge symmetry associated with the electromagnetic vector field. By this means, he was able to create his unified theory.
Einstein dismissed Weyl’s ideas about gauge symmetry of spacetime. He showed that Weyl’s theory would mean that clocks from different parts of space time would tick at different rates when they were brought together to the same place in space. In fact, this would also contradict our understanding of atomic spectral lines, because they would vary from one position in space to another. However, Weyl prevailed much later, in 1929, by publishing an article that applied his ideas of gauge symmetry to the phases of the quantum mechanical wave.5
Prior to Weyl’s paper, in 1927, another theorist, Fritz London, applied Weyl’s early ideas about gauge theory to the quantum mechanical phase of the wave function in Schrödinger’s wave equation.6 In quantum mechanics, we have the duality of wave and particle, as explained by Bohr’s complementarity principle, in which he claimed that the wave and the particle are simply two complementary manifestations of matter. An important physical feature of wave motion is its phase. Two oscillating waves described by x(t) and y(t) have a frequency ν, an amplitude A, and a phase θ (Figure 3.1). When the crest and trough differ by a phase θ of anything other than 180 degrees, then the waves are said to be “in phase” (Figure 3.2). If the phase θ is changed by a specific numerical amount of π or 180 degrees, then the wave motion of y(t) or x(t) is unchanged and the crest of one wave matches the trough of the other wave; they cancel out and the waves are said to be “out of phase” (Figure 3.3).
Figure 3.1 Phase shift in waves. An angle (phase) on the horizontal axis increases with time.
SOURCE: Wikipedia Commons
In quantum mechanics and particle physics, the conservation of electric charge plays a significant role. When waves change phase, they can cause a violation of this conservation of electric charge because the electric charge is the source of electromagnetic fields and waves. However, as shown by London and Weyl, the electromagnetic field arranges itself so that it always cancels out this difference in phase, retaining the conservation of electric charge, which is equivalent to the gauge invariance of Maxwell’s equations.
Figure 3.2 Waves in phase.
SOURCE: Wikipedia Commons.
We have now connected a symmetry of quantum mechanical (wave) equations with a conservation law—namely, the conservation of electric charge. We see that the gauge invariance associated with quantum mechanical wave motion as described by London and Weyl is connected intimately to Emmy Noether’s discovery of the connection between symmetries in nature and conservation laws. We can indeed say that electric charge is the source of the electromagnetic field. This interplay between quantum mechanics and particle physics has been important in the development of QED. It can be applied to Dirac’s relativistic wave equation for the electron or any other spin-½ charged particle such as quarks.
Figure 3.3 Waves out of phase.
SOURCE: Wikipedia Commons.
We must now distinguish between global gauge invariance, or global symmetry, and local gauge invariance. In the case of global symmetry, the changes of phase of the wave equation are the same at every point in spacetime, and this guarantees the global conservation of electric charge. On the other hand, for local gauge symmetry, the phases can change from one point to another in spacetime, but the electromagnetic field always counteracts these changes of phase in such a way that electric charge is conserved. That is, although the waves are out of phase locally from one spacetime point to another, the conservation of charge guarantees that any difference in the phases is canceled, thus preserving gauge invariance.
The concept of local gauge symmetry has played a crucial role in the development of particle physics theories and the standard model. In their nonabelian gauge theory based on SU(2), Yang and Mills extended the idea of phase differences and conservation of charge by replacing electric charge with isotopic spin charge. We recall that the isotopic spin charge is applied to the strong-interaction theory of protons and neutrons. There is a symmetry between the isotopic spin of protons and neutrons in strong-interaction theories that leads to the conservation of isotopic spin charge. Instead of the photon being the carrier of the force between protons and neutrons, as it is in electromagnetism, Yang and Mills invented a vector field B with spin 1 to do that job. In their original article published in Physical Review in 1954, Yang and Mills explained that the B-vector meson had to have a charge and to be massless, to mimic the photon in QED. Yet, there are no known electrically charged particles that are massless, including the photon, and this fact was troubling for Yang and Mills. It turned out to be a serious flaw in their argument for a nonabelian extension of Maxwell’s equations. Despite this, they felt that the idea of this local gauge symmetry and the conservation of isotopic spin through the force field of the B-vector meson was beautiful and elegant, and justified publication in Physical Review. As we will see in later chapters, the B-field was replaced subsequently by the color charged gluon particle in the nonabelian SU(3) strong-interaction theory QCD. The gluon is the carrier of the strong force in the atomic nucleus. It is massless, and the electric charge in the theory is replaced by color charge. Moreover, for the weak interactions, the B-field of Yang and Mills’s original theory became the charged, massive W boson.
Theorists working on gauge theory up to the present have been focusing on how to interpret the vector field in the nonabelian gauge theory published initially by Yang and Mills for SU(2). In 1983, the model of weak interactions being mediated by two massive vector bosons was confirmed by the discovery of the massive W and Z particles. Already during the early 1960s, before the experimental discovery of the W and Z particles, the question had been raised: Where do the masses of the W and Z come from? It seemed unusual that these force carriers should have masses, because the photon and the gluon did not. This search for the origin of the W and Z masses and the masses of the quarks and leptons led to the idea of broken gauge symmetry. Physicists conceived of an initial massless symme
try phase of all the elementary particles, including the W and Z particles, in the early universe, which was then broken spontaneously, thereby producing masses for the elementary particles, except for the photon and the gluon.
The mathematical concepts of group theory, symmetry, and gauge invariance led eventually to the development of the standard model of particle physics based on the group SU(3) × SU(2) × U(1), where the SU(3) sector describes the strong interactions whereas SU(2) × U(1) is the electroweak sector of the model.
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Looking for Something New at the LHC
The main goal of the LHC is to search for the last remaining undetected elementary particle in the standard model of particle physics, which plays a fundamental role in its framework—namely, the Higgs boson. It was important for the experimental runs in 2012 to build up the luminosity or intensity of the proton collisions so that the new data could produce a more significant statistical signal of the new particle resonance that had been discovered in 2011. That new resonance might or might not turn out to be the Higgs boson.
According to the standard model, the Higgs boson is a massive, electrically neutral particle. It has spin 0, which means that it is a scalar boson field; it does not have any directional properties in space, as is the case with the other bosons. Bosons are the carriers of the force fields between matter particles such as quarks and leptons. The bosons that have been discovered so far are the photon, the gluon, and the W and Z bosons. The Higgs boson has possibly been discovered, whereas others—like the graviton and the inflaton—are still hypothetical.
Cracking the Particle Code of the Universe Page 9
