# SU(N) symmetry in harmonic oscillator

• tom.stoer
In summary, the states of an isotropic oscillator can be labeled by the occupation numbers n1, n2,... nN. The symmetric part of the n-fold tensor product (N ⊗ N ⊗ ... ⊗ N)sym is a reducible representation of SU(N).
tom.stoer
Starting with the D-dim. harmonic oscillator and generators of SU(D)

$$T^a;\quad [T^a,T^b] = if^{abc}T^c$$

one can construct conserved charges

$$Q^a = a^\dagger_i\,(T^a)_{ik}\,a_k;\quad [Q^a,Q^b] = if^{abc}Q^c$$

satisfying the same algebra and commuting with the Hamiltonian

$$H = a^\dagger_i\,1_{ik}\,a_k + \frac{N}{2} = \sum_{i=1}^D a^\dagger_i a_i + \frac{N}{2};\qquad [H,Q^a] = 0$$

where I introcuced a new generator for U(1).

That means that the states

$$|n_1, n_2, \ldots n_N\rangle;\quad \sum_{i=1}^D n_i = N$$

are related to integer- or vector-representations of SU(D).

The degeneracy for fixed N is calculated as

$$\text{dim}\,N_D = \left(\array{N+D-1 \\ N}\right)$$

Questions:
- how are SU(D) vector representations labelled (compared to |lm> for SU(2))
- what are their dimensions (compared to 2l+1 for l-rep. in SU(2))
- and how do the dimensions sum up to dim ND

Thanks

Last edited:
Checking this explicitly for the simplest cases SU(2) and SU(3) it becomes strange already:

SU(2)

N=0: |00> is the trivial 1-rep. of SU(2)

N=1: |10>, |01> is the 2-rep. of SU(2); but this is a spinor rep. with l = 1/2 and 2l+1 = 2

N=2: |20>, |11>, |02> is the 3-rep. of SU(2)

SU(3)

N=0: |000> is the trivial 1-rep. of SU(3)

N=1: |100>, |010>, |001> is a 3-rep. of SU(3); again this is not a vector rep.; and I know there are two 3-reps, 3 and 3*

N=2: |200>, ... |110>, ... is a 6-dim. rep. of SU(3) which seems to be a sum of the two 3-reps 6 = 3 + 3*, correct?

N=3: here I find dim 33 = 10; now this can either be the 10-dim. irreducible rep., or just 1+3+6

So my first conclusion from SU(2) is that I have to include half-integer reps as well; this is strange, but OK. But for SU(3) it's unclear how to arrange rep's and how to count them.

Tom, the general theory of unitary representations of SU(n) is developed in Chapter 13 of Wu Ki Tung's Group Theory book and is derivable from the general theory of representations of GL(n) as it's done in the book.

For the 3-dim isotropic oscillator a thorough investigation from a group theoretical perspective is made by Wybourne in his book in Chapter 20.

Last edited:
This is equivalent to the problem of populating N cells with bosons. The states are labeled by the occupation numbers n1, n2,... nN where n = ∑ ni. The representation of SU(N) is the symmetric part of the n-fold tensor product (N ⊗ N ⊗ ... ⊗ N)sym, a reducible representation.

## 1. What is SU(N) symmetry in the context of a harmonic oscillator?

SU(N) symmetry in a harmonic oscillator refers to the symmetry of the Hamiltonian, or the energy operator, under rotations in a complex N-dimensional space. This symmetry arises when the Hamiltonian is invariant under transformations that mix the different energy levels of the oscillator.

## 2. How does SU(N) symmetry affect the energy levels of a harmonic oscillator?

When a harmonic oscillator exhibits SU(N) symmetry, the energy levels are degenerate, meaning that multiple energy levels have the same energy. This is because the symmetry allows for different states to have the same energy, making the energy levels indistinguishable.

## 3. Can SU(N) symmetry be broken in a harmonic oscillator?

Yes, SU(N) symmetry can be broken in a harmonic oscillator. This can occur when there are external perturbations, such as an external magnetic field, that break the rotational symmetry of the Hamiltonian. As a result, the energy levels will no longer be degenerate and the symmetry will be broken.

## 4. How is SU(N) symmetry related to angular momentum in a harmonic oscillator?

In a harmonic oscillator, SU(N) symmetry is closely related to the concept of angular momentum. This is because rotations in a complex N-dimensional space correspond to the different components of angular momentum in three-dimensional space. Therefore, SU(N) symmetry can be thought of as a generalization of angular momentum in higher dimensional spaces.

## 5. What are some applications of SU(N) symmetry in physics?

SU(N) symmetry has many applications in physics, particularly in the study of quantum systems. It is often used to describe the behavior of particles, such as in the study of quarks and gluons in quantum chromodynamics. In addition, SU(N) symmetry has also been applied to other areas of physics, such as condensed matter physics and cosmology.

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