cphil_symmetry
SYMMETRY
by Ray Shelton
OUTLINE
The ancient Greeks apparently believed that the symmetries of nature were directly reflected in the motion of objects: the stars must move in circles because those are the most symmetrical trajectories. The static symmetries like that of the circle are reflected in the dynamical symmetries of motion. Of course the planets do not move in circles as Kepler showed, but Newton recognized that the fundamental symmetries of nature are revealed not in the motion of individual objects, but in the set of all possible motions; that is, the symmetries of the motion are manifested in the equation of motion, rather than in the particular solutions to the equations. Newton’s law of universal gravitation, for example, exhibits spherical symmetry (the force of gravity are always in the same direction, to the center of the attracting body), yet the planetary orbital motions are elliptical. Thus the underlying symmetry of the system is only indirectly revealed to us.
It was not until 1917 that the dynamical implication of symmetry was completely understood. In that year, Emmy Noether (1882-1935) published her famous theorem relating symmetries and conservation laws: every symmetry of nature yields a conservation law, and conversely, every conservation law reveals an underlying symmetry. For example, the laws of physics are symmetrical respect to translations in time; they work the same today as they did yesterday. Noether’s theorem relates this invariance to the law of the conservation of energy. That is, if a system is invariant under translation in time, then the energy is conserved. And if a system is invariant under translation in space, then the momentum is conserved; if it is symetrical under rotation about a point, then angular momentum is conserved. Similarly, the invariance of electrodynamics under gauge transformation leads to the conservation of charge (this is called an internal symmetry, in contrast to the space-time symmetries).
| Symmetry | Conservation Law | |
|---|---|---|
| Translation in time | ↔ | Energy |
| Translation in space | ↔ | Momentum |
| Rotation about a point | ↔ | Angular Momentum |
| Guage Transformation | ↔ | Charge |
What precisely is a symmetry? It is an operation that can be performed (at least conceptually) on a system that leaves it invariant; this is, it carries the system into a configuration that is indistinguishable from the original one. Consider an equilateral triangle ABC; it is carried into itself by a clockwise rotation about its center through 120° (R+) and by a counterclockwise rotation through 120° (R–), and by flipping it about an axis through A perpendicular to the opposite side (RA), or around the corresponding axis through B (RB), or through C (RC). And doing nothing at all (I) is obviously leaves it invariant, so this is also a symmetry operation, albeit a pretty trival one. And so are combination of operations, as for example, clockwise rotations through 240°; but this is same as rotating counterclockwise by 120° (R+R+ = R–). This turns out to be the identification of all the symmetry operations on the equilateral triangle. There are six elements in this set of symmetry operations on the equilateral triangle: I, R+, R–, RA, RB, and RC.
The set of symmetry operations on any system must have the following properties:
1. Closure. If Ri and Rj are elements in the set, then the product, RiRj, which means to perform Ri and then Rj, is also in the set; that is, there exists an element Rk, such that
Rk = RiRj.
That is, the operation (multiplication) is defined on the group.
2. Identity. There is an element I such that IRi = RiI = Ri for all elements in Ri.
3. Inverse. For every element Ri there is an inverse, Ri-1, such that RiRi-1 = Ri-1Ri = I.
4. Associativity. Ri(RjRk) = (RiRj)Rk.
These properties are precisely the defining properties of a group. In fact, the mathematical theory of groups may be regarded as the systematic study of symmetries. The use of multiplicative notation for the group operation is purely a matter of convention, and does not say anything about the nature of the operation. One could just as well used the additive notation. In that case, the unit identity element I is called the zero element and the inverse element is the opposite or negative element. The usual convention is that the additive notation is reserved for Abelian groups. Note that the group elements need not commute: that is, the following communtative relation does not hold:
RiRj = RjRi.
A group in which this relation does hold is called Abelian, being named after the tragically short-lived Norwegian mathematican Niels Henrix Abel (1802-1829). Translations in space and in time form an Abelian group; rotations do not form an Abelian group and are called non-Abelian. Groups may be finite (like the triangle group which has six elements) or may be infinite (like the set of integers, with addition taking the role of group “multiplication” operation.) Groups may also be continuous (such as the the group of all rotations in a plane), in which elements depend on one or more continuous parameters (the angle of rotation, in this case), or discrete, in which the element may be labled by an index that takes on only integer values (all finite groups are discrete).
In physics, the groups of interest are groups of matrices. In elementary particle physics the most common groups are of the type mathematicians call U(n): the collection of all unitary n × n matrices. A unitary matrix is one who whose inverse is equal to its transpose conjugate:
U-1 = U*T.
If these are further restricted to unitary matrices with a determinant 1, the group is called SU(n), where the S stands for “special”, which just means “determinant 1”. And if these are further restricted to real unitary matrices, the group is called O(n), where the O stands for “orthogonal”; an orthogonal matrix is one whose inverse equals to its transpose:
O-1 = OT.
Finally, the group of real, orthogonal, n × n matrices of determinant 1 is SO(n). The SO(n) may be thought of as the group of all rotations in a space of n dimensions. Thus the SO(3) describes the rotational symmetry of our 3-dimensional world, a symmetry that is related by Noether’s theorem to the conservation of angular momentum. Indeed, the entire quantum theory of angular momentum is really closet group theory. It so happens that SO(3) is almost identical in mathematical structure to SU(2), which is the most important internal symmetry in elementary particle physics.
An ordinary scalar belongs to the one-dimensional representation of the rotation group SO(3), and a vector is a three dimensional representation; four-vectors belongs to the four-dimensional representation of the Lorentz group; and the curious geometrical arrangements of Gell-Mann’s Eightfold Way correspond to irreducible representation of the group SU(3).
| Group Name | Matrices in group |
|---|---|
| U(n) | n × n unitary (U*TU = 1) |
| SU(n) | n × n unitary with determinant = 1 |
| O(n) | n × n orthogonal (OTO = 1) |
| SO(n) | n × n orthogonal with determinant = 1 |
In particle physics the important symmetries are:
| U(1): | the symmetry of the electromagnetic field. |
| SU(2): | the symmetry of the weak nuclear interaction. |
| SU(2) × U(1): | the symmetry of the unified electroweak interaction. |
| SU(3): | the symmetry corresponding to the quark theory and the strong nuclear interaction. |
| SU(5): | One of the suggested symmetries of the grand unified theory |
One specific kind of symmetry is at the heart of quantum field theory. This is gauge symmetry, a concept used in quantum field theory to describe a field for which the equations describing the field do not change when some operation is applied to all particles everywhere in space; this is global gauge symmetry. It is also possible to have local gauge symmetry, where the operation is applied in some particular region. The term “gauge” is simply a label that mathematicans use to describe a property of a field. The term was introduced in this context shortly after World War One in 1918 by the German mathematican Hermann Weyl (1885-1955), who was trying to develop a unified theory combining electromagnetism (Maxwell’s equations) and gravity (Einstein’s General Relativity). A gauge transformation is one that changes the value of some physical quantity everywhere at once, and the field has gauge symmetry if it is unchanged after such a transformation. The word “gauge” means simply “measure”, and a gauge, quite simply, is a measuring standard. When measuring most things, the size of the standard may be changed at will, but the results remain the same. For example, the length of a board is the same whether the carpenter’s measuring tape is marked in inches and feet, or centimeters and meters; as long as the proper conversion is made, identical pieces of wood can be cut in both standards. Weyl was apparently thinking of the metal devices used by railroad engineers to measure the distance between the rails of a train track. When the word is applied to fields with gauge symmetry, the fields can be regauged (or remeasured) from different baselines without affecting their properties.
A classic example of a gauge theory is gravity. Imagine a ball sitting upon a step of a staircase. It has a certain amount of gravitational potential energy on that step. If the ball moves down to another step on the staircase, then it loses a specific amount of potential energy, which depends only upon the strength of the gravitational field and the difference in height between the two steps. The gravitational potential energy can be measured from anywhere. It is usually measured from either the surface of the earth or the center of the earth as our baseline, but any step could be chosen as the baseline or any point in the universe as the zero for the measurements. But the difference of potential energy between the two step will always be the same, no matter how the baseline is regauged. So gravity has gauge symmetry and the theory of gravity is a gauge theory.
The theories of gravity and of electromagnetism are both gauge theories, and the requirement of gauge symmetry was one of the key inputs used in the development of the theory of the weak interactions and the theory of quantum chromodynamics in terms of quantum fields. The modern gauge theory was largely created at Brookhaven National Laboratory in early 1954 by Chen Ning Yang and Robert Mills, In a single short paper published in the Physical Review, titled, “Gauge Invariance and Isotopic Spin”, Yang and Mills built the base upon which quantum field theory is constructed. At first their ideas were treated with skepticism and even derided as being pure mathematics, without any physical significance; now the theories of the Yang-Mills gauge symmetries is powerful enough to explain every elementary particle and interaction in existence.
The central idea of the paper occurred to Yang in 1948, when he was a graduate student at the University of Chicago. Yang was born in Hefei, China, in 1922, where his father was a professor of mathematics. The young Yang had studied at the Chinese universities of Kunming and Tsinghua, where he obtained a M.Sc. Yang had left worn-torn China in 1945, to study with Enrico Fermi at the University of Chicago and to work on his Ph.D. which was awarded to him in 1948. In 1949 he joined the staff at the Institute of Advanced Study, in Princeton, where he stayed until 1965. Since he thought that his given name, Chen Ning. might be to difficult for Americans to say, he awarded himself the nickname “Frank”, after Benjamin Franklin, a man that the youthful Yang admired. Yang was impressed with the idea that gauge symmetries could be used as the basis to construct the entire theory of quantum electrodynamics. He said in a telephone interview on 3 April 1983,
“When I was a graduate student, people talked a lot about gauge invariance. People would say that a calculation was not right, for example, because the result was not gauge invariant. It was a kind of check on calculations. But it was not appreciated that gauge invariance is a principle that can generate forces. It had been understood that way for electromagnetism in the 1920s, only somehow people were not paying attention. They also didn’t realize the principle could be moved to new situations.”
Yang began to suspect, not only that the such description might be essential to understanding of electromagnetism, but that there might be other kinds of local gauge fields. He was unable to carry his postulate any further than this. Yang finished his degree in 1948 and went to Princeton, New Jersey. In 1953-54 he spent a year away from Princeton, at the Brookhaven National Laboratory, where he shared an office with the theorist Robert Mills. Yang and Mills together constructed a gauge-invariant quantum field theory for the strong interaction. Their paper began,
“The conservation of isotopic spin is a much discussed concept in recent years. Historically an isotopic spin parameter was first used by Heisenberg in 1932 to describe the charge states (namely neutron and proton) of a nucleon. The idea that the neutron and proton correspond to two states of the same particle was suggested at the time by the fact their masses are nearly equal, and that [many] nuclei contain equal numbers of them.”
They noted that the strong force does not care whether the axis of the isotopic spin is up (proton) or down (neutron).
“Under such an assumption one arrives at the concept of a total isotopic spin which is conserved in nucleon-nucleon interactions. Experiments in recent years on the energy of light nuclei strongly suggest that this assumption is indeed correct.”
The point having been made that isotopic spin is conserved, they busied themselves with the main argument of the paper. They spent three pages working out what a “B field” would be like if it existed, and what is the characteristic of the virtual B particle that transmits it. Today this B particle is called a vector boson, which is jargon for a particle with a spin of one. Yang and Mills hoped that the equations would work out in such a way that this purely speculative B particle would convey a force identical to the strong force. They would thus have an explanation for strong interactions that would be in the same language of quantum electrodynamics, their model theory.
But when Yang and Mills did the mathematics, the calculation seemed to suggest that their conjectured vector boson had electric charge but no mass. This was discouraging: massless particles, such as the photon, are easy to make in experiments, and charged particles, such as electrons, are easy to detect. Yang and Mills could not understand why, if charged, massless vector bosons existed, they would not have already been discovered. They admitted that they had no satisfactory answer. In spite of the unresolved question about the mass of the vector boson, Yang and Mills decided to publish their work. Although the problem would not be fully resolved for fifteen years, Yang and Mills had transformed the role of symmetry in quantum physics. They had given symmetry a new importance by showing how it could create forces and set particles in motion. It was a major step toward unification, toward the day when scientists would be able to trace out the laws of the universe to a single principle, one which would reaches out to cover the whole of nature with a single explanatory web. Yang and Mills had raised the hope that the clue to the diversity of the world to be symmetry.
In the meantime, physicists occupied themselves with the question of how to use the new insight. Yang and Mills, and many of their colleagues, chose to look at the strong interactions and the heavy particles it affected, about which a wealth of data was pouring in from the first particle accelerators. Although they did not realize it, that was a blind alley. The Yang-Mills theory in its original form was of little use. After all, it started from the premise that there exists two elementary spin-1/2 particles of equal mass, and so far as we know there is no such pairs in nature. Yang and Mills themselves had the nucleon system (neutron and proton) in mind, and thought of their model as a way of implementing Heisenberg’s isospin invariance in the strong interactions. The small mass differences between the proton and neutron, 1.29 MeV/c2, would be attributed to electromagnetic symmetry-breaking. For the theory to succeed there had to exist a massless isotriplet of vector (spin -1) particles, which did not. A number of attempts were made to doctor up Yang-Mills theory. Meanwhile, for more than a decade after 1954, the Yang-Mills model languished. Most of the speculation about the Yang-Mills gauge fields — the term was soon applied to any local gauge field — remained hidden in physicist’s notebooks, and gauge symmetries lay dormant until many years later in 1967 when Steven Weinberg, Abdus Salam, and Sheldon Glashow applied them to weak interactions and unified electromagnetic and weak interactions — the electroweak theory which considers the weak and electromagnetic interactions as different manifestations of a single electroweak force — a theory which became the paradigm for all future unified field theories.
The symmetries of space and time give rise to universal conservation laws such as those of energy, of linear momentum, and of angular momentum. These laws require invariances under groups of transformations through space, time, and angular rotation, repectively. But other conservation laws are known to exist, such as the conservation of electrical charge. These also require invariances under appropriate symmetry operations. These invariances give rise physically to predictions such as the existence of new particles and values for their electrical charge, spin, and other quantum numbers such as isospin. Thus these symmetry operations are the fundamental ways of describing the conservation laws that hold in particle interaction. But internal symmetry also help us to categorise particles according to their intrinsic properties. By regarding the neutron and proton as isospin down and isospin up components of a single neuleon, the strong interaction’s indifference to “neutron-ness” and “proton-ness” can be expressed as the invariance of strong interactions to rotation in the abstract isospin space. The group of transformations which achieves these rotations is the Special Unitary group of dimension 2 called SU(2), which acts on the 2-dimensional space defined by the proton and the neutron, redefining the proton and neutron as the mixture of the original particle. Of course, the same must also be true of the pions, which form a 3-dimensional space (π+, π0, π–), and the Δ baryon (Δ++, Δ+, Δ0, Δ–), which form a 4-dimensional space. These are referred to as the 2-, 3-, and 4-dimensional representations of SU(2).
When the conservation of strangeness is added to that of the isospin as a property of the strong interaction, it became clear that the strongly interacting particles are governed by a bigger symmetry group. Although it seems obvious, it took a great deal of work to show that SU(3) is the appropriate group. The transformation of the SU(3) group generated many dimensional representations (multiplets), 1, 3, 6, 8, 10, 27, etc,, each of which is a well-defined quantum-number pattern. It was a triumph for the orginators of the scheme to find that some of these exactly fitted the quantum-number structure of the observed hadrons. The identification of the correct symmetry group for strong interactions, and the assignment of hadrons to the multiplets, led to prediction in 1962 of a new hadron necessary to complete the spin -3/2 baryon decuplet 10. This is the famous Ω– particle with strangeness assignment -3. Its spectacular discovery in 1964 in bubble chamber photographs at Brookhaven convinced a previously world of physics of the validity of SU(3).
The correct symmetry group SU(3) having been found, a major problem remained. It was necessary to explain why the mesons filled some multiplets and the baryons fitted others, but other multiplets have no particles. In particular, it seemed odd that the fundamental 3-dimensional representation should remain unfilled (that is, the most basic representation of SU(3) group). In an unsuccessful symmetry scheme prior to that of Gell-Mann and Me’eman, the proton, neutron, and hyperon were assigned to this triplet, but the logical consequences of such an assignment were incompatible with the experimental evidence.
In 1964, Gell-Mann and George Zweig pointed out that the representation SU(3) which were occupied by particles could be chosen from amongst all the mathematically possible by assuming them to be generated by just two combinations of the fundamental representation. Gell-Mann called the entities in the fundamental representation quarks. Three varieties of quark or flavors, as they are now called, have since then have become known as the up quark, down quark, and strange quark; the up and down labels referring to the orientation of the quark’s isospin. The combinations of quarks which give the occupied representations SU(3) are a quark-antiquark pair for the meson multiplets and three quarks for the baryon multipets.
If nature was perfectly symmetrical, then the work of physicists would be much easier. The grand unified theory would be obvious, because there would be only one force, not four. But nature is full of surprises in the form of broken symmetries. For example, the natural world is not perfectly crystalline or uniform but filled with irregular galaxies, elliptical planetary orbits, etc. The world is full of examples of where symmetry is hidden because it is broken. In fact, the universe would be a rather dull place if symmetry were neer broken. Human life would not exist, because there would be no atoms; life would be impossible. Everything would be perfectly homogeneous and dull. It is symmetry’s breaking, therefore, that makes the universe interesting. The study of broken symmetries explain, for example, the freezing of water. Water in liquid form possesses great symmetry. No matter how we rotate it, it remains water. In fact, even the equations governing water have the same symmetry. However, as we cool water slowly, random crystals form in all directions, creating a chaotic network that eventually becomes solid ice. Although the original equations possess great symmetry, the solution of the equations do not necessarily possess this symmetry.
The reason why these transitions take place is that nature always perfers to be in the lowest energy state. We see examples of this all the time; for example, water flows downhill because it is trying to reach the lowest energy state. Quantum transitions occur because the system started originally in the wrong energy state (sometimes called a “false vacuum”) and would prefer to make a transition to a lower energy state.
According to these unified theories, all interactions we see in the present world are asymmetrical remnant of a once perfectly symmetrical world. This symmetrial world is revealed only at very high energies, energies so high that they can never be accomplished by human beings. The only time that such energies existedwas in the first nanoseconds of the big bang which was the origin of the universe.
If we go back to the beginning of time, to the first moments of creation, the energy of the primordial fire ball was so high that the four interactions were unified as one highly symmetrical interaction. As this fireball of swirling quarks, colored gluons, electrons, and photons expanded, the universe cooled and the perfect symmetry began to break. First gravity was distinguished from the other interactions and then the strong, weak, and electromagnetic interactions became apparent as they froze out of the cooling universe manifesting symmetry breaking. Exotic quanta like charmed particles decayed away, and soon mostly protons, neutrons, electrons, photons, and neutrinos were all that was left. After further cooling, atoms could form — the building blocks for life. The universe from its very beginning to the present may be viewed as a hierarchy of successive broken symmetries — a transition from a simple perfect symmetry at the beginning of time to the complex patterns of broken symmetries we see today.
At the immense energies at the beginning of time, life could not exist. Although interactions were unified and perfectly symmetrical, it was a sterile world. The universe had to cool and the perfect symmeties break down before life could exist. Our world exhibits a broken or imperfect symmetry.