Internal Symmetry

Prerequisites: Symmetry, Group Theory. Full path (3 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

see also Symmetry and Gauge Symmetry

Intuitive

You are sitting in a room with a friend and a ping-pong ball (perfectly spherical and perfectly white— the ping-pong ball, not the friend). The conversation gets around to Newtonian mechanics. You toss the ball to your friend. Both of you agree that, given the speed and direction of the toss, F = m A and the formula for the gravitational attraction at the surface of the earth ( F = −mg k, if the positive z-direction is up), you could calculate the motion of the ball, at least if air resistance is neglected. But then you ask your friend: “As the ball was traveling toward you, was it spinning?” “Not a fair question”, he responds. After all, the ball is perfectly spherical and perfectly white. How is your friend supposed to know if it’s spinning? And, besides, it doesn’t matter anyway. The trajectory of the ball is determined entirely by the motion of its center of mass and we’ve already calculated that. Any internal spinning of the ball is irrelevant to its motion through space. Of course, this internal spinning might well be relevant in other contexts, e.g., if the ball interacts (collides) with another ping-pong ball traveling through the room. Moreover, if we believe in the conservation of angular momentum, any changes in the internal spin state of the ball would have to be accounted for by some force being exerted on it, such as its interaction with the atmosphere in the room, and we have, at least for the moment, neglected such interactions in our calculations. It would seem proper then to regard any intrinsic spinning of the ball about some axis as part of the “internal structure” of the ball, not relevant to its motion through space, but conceivably relevant in other situations.

The phase of a charged particle moving in an electromagnetic field (e.g., a monopole field) is quite like the internal spinning of our ping-pong ball. We have seen that a phase change alters the wavefunction of the charge only by a factor of modulus one and so does not effect the probability of finding the particle at any particular location, i.e., does not effect its motion through space. Nevertheless, when two charges interact (in, for example, the Aharonov-Bohm experiment), phase differences are of crucial significance to the outcome. The gauge field (connection), which mediates phase changes in the charge along various paths through the electromagnetic field, is the analogue of the room’s atmosphere, which is the agency (“force”) responsible for any alteration in the ball’s internal spinning.

The current dogma in particle physics is that elementary particles are distinguished, one from another, precisely by this sort of internal structure. A proton and a neutron, for example, are regarded as but two states of a single particle, differing only in the value of an “internal quantum number” called isotopic spin. In the absence of an electromagnetic field with which to interact, they are indistinguishable. Each aspect of a particle’s internal state is modeled, at each point in the particle’s history, by some sort of mathematical object (a complex number of modulus one for the phase, a pair of complex numbers whose squared moduli sum to one for isotopic spin, etc.) and a group whose elements transform one state into another (U (1) for the phase and, for isotopic spin, the group SU (2) of complex 2 × 2 matrices that are unitary and have determinant one). A bundle is built in which to “keep track” of the particle’s internal state (generally over a 4-dimensional manifold which can accommodate the particle’s “history”). Finally, connections on the bundle are studied as models of those physical phenomena that can mediate changes in the internal state. Not all connections are of physical interest, of course, just as not all 1-forms represent realistic electromagnetic potentials. Those that are of interest satisfy a set of partial differential equations called the Yang-Mills equations, developed by Yang and Mills [YM] in 1954 as a nonlinear generalization of Maxwell’s equations.

page 22ff in Topology, Geometry and Gauge fields by Naber

For the case of regular spin, we had to take spin-space seriously because it was associated with a concrete, measurable, physical quantity—angular momentum. This was only mildly uncomfortable because, although spinspace has the somewhat hard-to-stomach property that you have to turn all the way around twice to get back to your original condition, it’s otherwise pretty much like regular space. Isospin space, however, is completely abstract; it bears no relation whatsoever (other than through analogy) to anything we can grasp with our faculties of perception. How could rotations in such a space possibly have anything to do with the physical world? And yet the physical manifestation of the invariance of the strong force with respect to rotations in this space, the conservation of isospin, is a solidly established fact in the world of experimental science. So, what then is isospin-space from a physical point of view? Physicists usually describe it as an internal symmetry space, but what’s that, really? It’s your old buddy again, telling you that your car’s carburetion system “works on a vacuum principle.” How’s that going to help you to understand and fix the thing? It isn’t. Regarding the physical interpretation of the notion of isospin space, again your guess is as good as mine. Perhaps its experimental manifestations are hinting at some new and deeper truth about the universe that lies just beyond the current limits of our comprehension. Perhaps not. But one thing, however, is true: The introduction of the idea of internal symmetry spaces, of which isospin space was the first example, was an essential step forward in our understanding of the universe and the nature of the laws that govern it." Deep Down Things by Schumm>

„What the heck is an internal space?” you ask. Good question. The best answer I have is “useful.” It’s what we invented to quantify the observed behavior of particles, a mathematical tool that helps us make predictions. “Yes, but is it real?” you want to know. Uh-oh. Depends on whom you ask. Some of my colleagues indeed believe that the math of our theories, like those internal spaces, is real. Personally, I prefer to merely say it describes reality, leaving open whether or not the math itself is real. How math connects to reality is a mystery that plagued philosophers long before there were scientists, and we aren’t any wiser today. But luckily we can use the math without solving the mystery.

Lost in Math by Sabine Hossenfelder

Concrete

A spacetime symmetry moves points: a translation or rotation changes $x$, and fields are carried along, for example $\phi'(x)=\phi(\Lambda^{-1}x)$. An internal symmetry leaves every spacetime point where it is and only mixes the field components at the same point, $$\phi_i(x)\to\phi_i'(x) = U_{ij}\,\phi_j(x),$$ where $U$ is a matrix representing an element of some group $G$. If $U$ is the same at every point, the symmetry is called global. If $U$ is allowed to depend on $x$, it is a local symmetry.

U(1) phase. For a complex scalar field with metric signature $(+,-,-,-)$, $$\mathcal{L} = \partial_\mu\phi^*\partial^\mu\phi - m^2\phi^*\phi,$$ the Lagrangian is unchanged under $\phi\to e^{i\alpha}\phi$, $\phi^*\to e^{-i\alpha}\phi^*$ with constant $\alpha$, because only the combinations $\phi^*\phi$ and $\partial_\mu\phi^*\partial^\mu\phi$ appear. For an infinitesimal $\alpha$ we have $\delta\phi = i\alpha\phi$ and $\delta\phi^* = -i\alpha\phi^*$, and Noether's theorem gives the current $$j^\mu = \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\frac{\delta\phi}{\alpha}+\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi^*)}\frac{\delta\phi^*}{\alpha} = i\left(\phi\,\partial^\mu\phi^*-\phi^*\partial^\mu\phi\right).$$ Using the equation of motion $\partial_\mu\partial^\mu\phi=-m^2\phi$ one checks directly that $\partial_\mu j^\mu=0$. The conserved charge $Q=\int d^3x\, j^0$ is, up to an overall sign that is a matter of convention (the normalization of a Noether current is not fixed), the number of particles minus the number of antiparticles. When the field is coupled to electromagnetism, the electric charge is this number times the charge $q$ of one particle.

Isospin. Take two Dirac fields with equal mass, combined into a doublet $\Psi=(\psi_1,\psi_2)^T$: $$\mathcal{L} = \bar\Psi(i\gamma^\mu\partial_\mu - m)\Psi.$$ This is invariant under $\Psi\to U\Psi$ for any constant $U\in SU(2)$ (in fact under $U(2)$), because $\bar\Psi\to\bar\Psi U^\dagger$ and $U^\dagger U=1$. Writing $U=\exp(i\alpha_a\tau_a/2)$ with the Pauli matrices $\tau_a$, Noether's theorem gives three conserved currents $j^\mu_a=\bar\Psi\gamma^\mu\frac{\tau_a}{2}\Psi$. Proton and neutron, or the up and down quarks, form such a doublet to the extent that their mass difference and electromagnetic interactions can be neglected. In this sense isospin is an approximate internal symmetry.

The practical consequences of an exact internal symmetry are conserved charges, particles grouped into multiplets with equal mass and spin, and selection rules for which processes are allowed. Requiring a symmetry restricts which terms may appear in a Lagrangian, which is how such symmetries are used to build models.

Abstract

In classical field theory, the fields are sections of a vector bundle $E=P\times_G V$ associated to a principal $G$-bundle $P$ over spacetime $M$ through a representation $\rho:G\to GL(V)$. For a trivial bundle $E=M\times V$, a global internal symmetry acts by $\phi(x)\mapsto\rho(g)\phi(x)$ with $g\in G$ fixed. It commutes with the action of spacetime diffeomorphisms or isometries, since it acts only on the fibre $V$ and not on the base.

In quantum field theory, an internal symmetry is a unitary representation $U(g)$ on the Hilbert space that commutes with the Poincaré group. For the generators $Q_a$ this means $$[Q_a,Q_b]=if_{abc}Q_c,\qquad [Q_a,P^\mu]=0,\qquad [Q_a,M^{\mu\nu}]=0.$$ Because $Q_a$ commutes with $P^2$ and with the Pauli-Lubanski invariant $W^2$, every irreducible multiplet of $G$ consists of particles with the same mass and spin. Conversely, the Coleman-Mandula theorem states that, under its assumptions (a Lorentz-invariant S-matrix in four dimensions with a mass gap, finitely many particle types below any mass, and nontrivial analytic scattering), the Lie algebra of symmetries of the S-matrix is the direct sum of the Poincaré algebra and an internal algebra, which is a compact semisimple Lie algebra plus abelian factors. Spacetime and internal symmetries can only mix nontrivially if one allows a graded Lie algebra, which leads to supersymmetry.

An internal symmetry of the Lagrangian need not be realized on the spectrum in the way described above. If the vacuum is not invariant, the symmetry is spontaneously broken and, for a continuous global symmetry, Goldstone's theorem gives massless scalars. When the symmetry parameters are promoted to functions on spacetime, the transformations become automorphisms of the principal bundle $P$ and the theory requires a connection on $P$, which is the subject of gauge symmetry.

Why is it interesting?

Internal symmetries are powerful that we use, for example, to derive the correct Lagrangians describing fundamental interactions.

Path to this page

1. Linear Algebra
2. Group Theory
3. Symmetry