Meaning of $SU(n)$ and $SO(n)$ symmetry groups in particle physics

Question

I am currently self studying introductory particle physics using An Introduction to Elementary Particles by David J. Griffiths. I was reading Chapter 4 of this textbook which deals with the topic of Symmetries. There were some concepts in this chapter which I could not fully grasp and I hope someone can help me out here.

The Background:

In Section 4.1, the author introduces the $SO(n)$ and $SU(n)$ groups. The former is defined to be the group of all orthogonal $n\times n$ matrices with determinant $1$. The latter is defined to be the group of all unitary $n\times n$ matrices with determinant $1$. Therefore my understanding is that these groups contain matrices as their elements with specific properties.

I have not been exposed to group theory before (my formal education level is of a third year undergraduate pursuing an astrophysics specialist) and hence this is the first time I have read about these groups. However it is clear to me now what $SO(n)$ and $SU(n)$ denote mathematically. However I am familiar with the algebraic structure of a 'group' in mathematics.

In the subsequent section 4.3 the author develops the theory of isospin, where he states that 'strong interactions are invariant under rotations in isospin space' which by Noether's Theorem means that 'isospin is conserved under strong interactions'.

My Questions:

In the very next sentence it is stated that In the language of group theory, Heisenberg asserted that the strong interactions are invariant under an internal symmetry group $SU(2)$.

• What does the phrase internal symmetry group $SU(2)$ mean? As per my understanding $SU(2)$ is a group of $2\times 2$ unitary matrices with determinant $1$. What does it mean when we talk about internal symmetry groups and how is related to the definition of $SU(n)$ given above?

A few paragraphs later, similar terms are used when the author talks about regarding the nucleons, $\Lambda$'s, $\Sigma$'s and $\Xi$'s as a supermultiplet which belongs to the same representation of some enlarged symmetry group. The author then says that the Eightfold Way was Gell-Mann's solution to this problem and that the symmetry group is $SU(3)$. Furthermore it is said a few sentences later that the introduction of the charm, bottom and top quark expanded the flavor symmetry from $SU(4)$ to $SU(5)$ to $SU(6)$ respectively.

• This repeated use of the term 'symmetry groups' is something I do not fully understand. How does the author conclude that a multiplet belongs to a symmetry group $SU(n)$? What does it mean to be a part of a symmetry group $SU(n)$ or to have $SU(n)$ symmetry? Again, how does this relate to the definition of $SU(n)$ mentioned above?

The two bulleted points are the main questions I have regarding the concepts in the material I am reading. I would be grateful if someone could clarify my queries.

Answer 1

1. The phrase "internal symmetry" is used to distinguish such types of symmetries from space time symmetries (space-time translations, rotations, Lorentz boosts). In the case of the isospin group, the proton and the neutron fields (or in modern language, the $u$ and $d$ quark fields) are "rotated" by $SU(2)$ transformations.

2. Symmetry group (in physics) simply means that a certain Lagrangian (or better, the associated action integral) is invariant under the corresponding group transformations. In a relativistic quantum field theory, the action must be invariant under the Poincare group (space-time symmetries). Depending on the type of interaction under investigation, the action may also be invariant under additional transformations of the fields with respect to some "internal" symmetry group.

3. A "multiplet" belongs to a certain irreducible representation of your (internal) symmetry group. This is a mathematical problem. You are probably familiar with the classification of the irreducible representations of the Lie algebra of $SU(2)$ (weights $j =0, 1/2, 1, 3/2, \ldots$ corresponding to angular momentum or isospin as possible physical interpretations).

Answer 2

Symmetries are called "internal" to distinguish them from spatial symmetries.

The initial idea behind the use of symmetries was motivated by degeneracies in the spectrum of the Hamiltonian. If you have elements of some symmetry group $G=\{g_1,g_2,\ldots,\}$, and $g_iH=Hg_i$ for every element in the group, then all elements in $G$ will take an eigenstate of $H$ to another eigenstate of $H$ with the same energy. Those states connected by group elements form a multiplet.

This is already clear for the hydrogen atom. The energies $E_n=-13.6\text{eV}/n^2$ does not depend on the angular momentum $\ell$ of the system, and indeed there is a (continuous) group SO(3) so that any element commutes with $H$: this is obvious since $H$ is rotationally invariant. Note that even if elements in SO(3) are defined from $3\times 3$ orthogonal matrices, one can easily construct matrices of size $(2L+1)\times (2L+1)$ using the usual angular momentum operators $J_z, J_\pm$ that will commute with $H$. By exponentiating these matrices we obtain a set of $(2L+1)\times (2L+1)$ matrices that multiply in exactly the same way as the original $3\times 3$ orthogonal matrices.

There is a similar idea for the n-dimensional harmonic oscillator, but the symmetry group here is $U(n)$: see this question.

Crystals also have a variety of (discrete) spatial symmetries and indeed some of the first applications in physics of group theory were to understand the vibration of crystals. The pioneering work of Hans Bethe

H. Bethe, Splitting of terms in Crystals, Ann.Physik 3 (1929) 133-206

remains a classic reference.

Heisenberg noted that the proton and neutron have approximately the same mass and (neglecting the electric charge), appear to interact in the same way in the nucleus. Thus he postulated that a proton and a neutron were like the spin-up and spin-down states in systems where interactions did not depend on spin, i.e. he assumed that $SU(2)$ transformations operated on neutrons and protons just like they operate on spin-up and spin-down states, and so these $SU(2)$ transformations between neutrons and protons should commute with the nuclear Hamiltonian. This symmetry has nothing to do with actual physical space but is a symmetry in some abstract space where basis states are $\vert n\rangle\mapsto (1,0)^\top$ and $\vert p\rangle\mapsto (0,1)^\top$.

The idea of such an ``internal'' symmetry'' has since been generalized beyond $SU(2)$ to include various groups. The value of such internal symmetries is in how they connect and predict various quantities (masses, cross-sections, etc.).

Note that the idea has been further generalized to ``dynamical symmetry''. In such situations, the exponential of Hamiltonian is an element in a group (i.e. the Hamiltonian is an element of the corresponding algebra) and no longer commutes with all elements of the group. However, given a multiplet of the group, the Hamiltonian will act only inside this multiplet so that multiplets now block diagonalize $H$ rather than directly giving eigenstates of $H$.