In a previous post, I discussed the importance of symmetries in particle physics, and illustrated with some examples from classical physics. I also discussed Noether's Theorem, which says that every continuous symmetry of the laws of physics gives rise to a conservation law. I want to pick up that discussion here and turn our attention from the classical external symmetries to the internal symmetries of particle physics. Let me start with a review of the external symmetries and of symmetry groups.
The classical1 external symmetries I discussed last time were translational symmetry, which implies conservation of linear momentum; time symmetry, which implies conservation of energy; rotational symmetry, which implies conservation of angular momentum; boost symmetry, which implies conservation of a relativistic analogue of angular momentum; and time reversal and parity. Einstein's Special Theory of Relativity allows us to combine translational and time symmetry in a natural way into a single spacetime displacement symmetry. Relativity likewise allows us to combine rotational and boost symmetry into a single spacetime rotational symmetry. This symmetry is called the special Lorentz symmetry, and its mathematical symbol is SL(2,c). The combination of both displacement symmetry and special Lorentz symmetry, which characterizes the real physical universe, is called Poincare symmetry. Its mathematical symbol is P(1,3). This symbol reflects the fact that spacetime has one timelike dimension and three spacelike dimensions.
When we move from the world of classical physics to the world of particle physics, we find that Poincare symmetry is still an exact symmetry. However, both parity and time reversal symmetry are violated by the weak force. This is one of the most remarkable results in the history of particle physics. In addition to Poincare symmetry, the Lagrangian density describing the dynamics of quantum particles can also have internal symmetries. It is these symmetries that distinguish the various models of particle physics. The standard model of physics2 is based on a combination of three internal symmetries, U(1) x SU(2) x SU(3). An understanding of these symmetries requires that we know something about the mathematical behavior of symmetry groups.
Any two operations allowed by a particular symmetry can be combined to produce another operation that also respects the symmetry. In addition, every operation has an inverse operation that undoes its effect. These properties of symmetry operations imply that the set of operations permitted by a given symmetry form a mathematical group. If the symmetry is a continuous symmetry, like P(1,3), the group is called a Lie group.
I'll illustrate Lie groups with one of the simplest, U(1), also called "the unitary group of rank 1." This is the simplest internal symmetry group of the standard model of particle physics. It is easy to visualize: It is the rotational symmetry of a circle. U(1) represents a continuous symmetry because a circle can be rotated by any angle and remain unchanged. The combination of two rotations is identical to a single rotation by an angle equal to the sum of the angles of the two rotations: A rotation by 30 degrees followed by a rotation by 40 degrees is identical to a rotation by 30+40=70 degrees. Furthermore, every rotation has an inverse. A counterclockwise rotation of 40 degrees followed by a clockwise rotation of 40 degrees is the same as no rotation at all: 40-40=0.
This simple geometrical model is only one way to represent U(1). Another representation is the set of all complex numbers whose absolute value is 1. The product of any two such complex numbers also has absolute value 1, and an argument equal to the sum of the arguments of its factors. If the argument is interpreted as a rotation angle, the correspondence is perfect. In fact, it's better than perfect. A rotation of 270 degrees followed by a rotation of 180 degrees is not 270+180=450 degrees, but 450-360=90 degrees, and it is arbitrary whether a rotation of 120 degrees followed by a second rotation of 120 degrees is written as 240 degrees or -120 degrees. All these ambiguities vanish if you use the complex number representation, because there is exactly one complex number corresponding to a given rotation.
An internal symmetry is so called because it is not a symmetry in physical spacetime. U(1) may represent rotations in a plane, but this plane is outside the space that is visible to our senses. It is tempting to write it off as a purely mathematical abstraction, but it is just as real as physical space. In particular, the U(1) symmetry of this plane is intimately connected to the very real force we call electromagnetism.
The story of how U(1) relates to electromagnetism goes back over a century. Shortly after Maxwell published his equations describing the electromagnetic field, in the closing years of the 19th century, it was realized that the electromagnetic field can be described quite elegantly in terms of the electrostatic potential, already familiar to physicists, plus a previously unrecognized vector potential. The latter has both a magnitude and a direction in space. The vector potential is required for a complete description of changing electric and magnetic fields.
When Einstein applied the Special Theory of Relativity to electromagnetism, he found that the electrostatic and vector potentials were the timelike and spacelike components of a four-dimensional spacetime vector. In other words, the electromagnetic 4-potential is a quantity that has a magnitude and direction in spacetime. Physicists find this description so elegant that the electromagnetic four-potential A is the quantity that appears in the Lagrangian density of particle physics.
There's just one problem: The value of A is not unique. It can be modified by the addition of an arbitrary term of a particular form3 without changing the predicted electric and magnetic fields one iota. The addition of such a term is called a gauge transformation, and the fact that the electromagnetic is unchanged by gauge transformation means that electromagnetism has a gauge symmetry.
There's that word "symmetry" again. If the Lagrangian density of the electromagnetic field is required to have gauge symmetry, a conservation law naturally follows, just as Noether told us it must. But the conservation law in question is already well-known to physicists: It is conservation of electric charge.
To understand how gauge symmetry is connected to U(1) symmetry, we must first explain an important property of charged particles. As a charged particle moves through physical space, it is simultaneously traveling in a circle in the U(1) plane. The rate at which the particle moves around this circle is proportional to the particle's total energy, which includes the particle rest mass. For example, a motionless electron completes its circle in the U(1) plane 124 million million million times each second. Ordinary particles move clockwise while their antiparticles move counterclockwise.
The Lagrangian density for a charged particle in an electromagnetic field has contributions from the electromagnetic field itself, which depend on A in a gauge-symmetric way; contributions from the mass and kinetic energy of the charged particle; and contributions from the interaction of the particle with the electromagnetic field. None of these contributions depend on the position of the particle on its circular path in the U(1) plane. Only the kinetic term is even aware of the existence of a U(1) plane, and its value depends only on the rate at which the particle circles in this plane. This is the U(1) symmetry of the standard model of physics. If we rotate the U(1) plane by the same angle at every point in physical spacetime, then the Lagrangian density is left unchanged. As always, Noether's Theorem requires that there be a conservation law associated with this symmetry. We find that it is conservation of electrical charge.
Why herein is a marvellous thing: The gauge symmetry of the electromagnetic field yields conservation of electric charge. So does the U(1) symmetry of charged particles. Is it not reasonable to surmise that gauge symmetry is simply an artifact of U(1) symmetry? It is. But the connection is deep, and profoundly beautiful.
Up until now, the U(1) symmetry we have discussed has been a global symmetry. The U(1) plane is assumed to be rotated by the same angle at every point in spacetime. This yields conservation of charge, and would do so whether or not there was such a thing as the electromagnetic field. Now suppose we go a step further, and assume local U(1) symmetry. In other words, we will rotate the U(1) plane by an angle that varies smoothly from point to point in spacetime.
Not surprisingly, local U(1) symmetry is harder to achieve than global U(1) symmetry. Since global U(1) symmetry is a special case of local U(1) symmetry, we must still have conservation of electrical charge. But the changes of U(1) from point to point introduce an additional term into the Lagrangian density. And now for the epiphany: This term is identical in form to a gauge transformation of the electromagnetic 4-potential A. In other words, local U(1) symmetry of the Lagrangian density of particle physics is possible only if electric charge is conserved and if there is an electromagnetic field interacting with the charged particles. Particle physicists say that local U(1) symmetry induces the electromagnetic field, which they speak of as the gauge field of the U(1) symmetry.
When I was studying formal logic, my professor gave us an explanation of the correctness proof of the predicate calculus based on an analogy to original sin. Let me likewise try to explain U(1) symmetry in terms of the Christian doctrine of the Atonement.
In the beginning, God created the universe (the Lagrangian density), and saw that it was good. In this universe, there was originally perfect symmetry between justice and mercy (the U(1) plane of the Lagrangian density.) But then God gave Adam free will (introduced a local U(1) transformation), and Adam ate the forbidden fruit and broke the symmetry between justice and mercy (introduced spurious terms into the Lagrangian density.) To reconcile justice and mercy (to establish local U(1) symmetry in the Lagrangian density), God sent His Son, the Light of the World (the electromagnetic field) to make an atonement for sin (cancel the spurious terms in the Lagrangian.)
A silly analogy, perhaps. But there are reasons why I am both a physicist and a believer.
In future posts of this series, I will discuss the remaining symmetries of the standard model of particle physics, SU(2) and SU(3), and such additional concepts as spontaneous symmetry breaking and the Higgs mechanism. As you might anticipate, SU(2) and SU(3) introduce additional gauge fields into the standard model (the weak and color forces), but with some surprising twists. (No pun intended.)
1It may surprise some readers to hear me include boost symmetry, which comes from the Special Theory of Relativity, among the classical symmetries. But at Caltech, where I received my Ph.D., Einstein's General Theory of Relativy is taught as the crowning achievement of classical physics - with considerable historical and philosophical justification. Quantum mechanics, rather than relativity, is regarded there as the starting point for modern physics.
2Strictly speaking, U(1) x SU(2) x SU(3) is one of several slightly different symmetries that are consistent with the standard model of particle physics. We needn't be concerned with this here.
3For the mathematically inclined: The extra term must be the 4-gradient of a scalar function, but the choice of scalar function is completely arbitrary (so long as it satisfies the usual smoothness conditions).