QM: What are Tensor Operators & How to Understand Georgi's Definition?

[Kontilera]

Hello! I´m trying to read Georgi's book on Lie algebras in particle physics but am confused about the start of chapter 4.

Georgi writes that "A tensor operator is a set of operators that transforms under commutation with the generators of some Lie algebra like an irreducible representation of the algebra. [...] A tensor operator transforming under the spin-s representation of SU(2) consists of a set of operators, O^s_l, for l=1 to l = 2s+1(or -s to s), such that:$$[J_a, O^s_l] = O^s_m[J^s_a]_{ml}"$$

I thought I understood 90-95% sofar in the book but I really don't see what he tries to define here.. Could somebody maybe help me and introduce the concept in Georgis way but with some more words and an example i will recognize from QM? I don't recognize what he is trying to construct from my QM courses.

 

[Bill_K]

In QM an example of a vector operator is the position operator x, a set of three operators x, y, z. Together they form a vector because under rotations they transform into each other. This fact can be expressed by writing out their commutators with the operators J_i that generate infinitesimal rotations:

[J_i, x_j] = i ε_ijk x_k

Another example of a tensor operator is the electric quadrupole moment, which is a set of five operators Q_m, m = -2, ... +2. Given a charge distribution ρ(x,y,z), you define its electric quadrupole moment by

Q_m = ∫ρ r² Y_2m dV

where Y_2m is a spherical harmonic. Under rotations the five operators Q_m go into each other, and one can express this by writing out all the commutators of J_i with Q_m.

 

[Kontilera]

Of course! Didnt think of the three dimensional position operator as a set of operators. :)
The equation of transformation however doesn't seem to intuitive.. how should I think about the indices in the equation I qouted above?

For x_j, j is obviously the dimension of our space so for O^s_l, l specifies a operator in our set? What about s, is it just referring to which representation? :/

 

[king vitamin]

I've been studying this book too so someone can (and should) correct any misunderstandings/misstatements below.

The index s labels the representation. For SU(2) as Georgi defines his indices, s=0,1/2,1,... labels the representation. Since the dimension of these representations is 2s+1, then the lower indices on the operators run from 1 to 2s+1.

In the example of the position operator, it clearly transforms under the s=1 representation, or the adjoint representation. Notice that i ε_ijk is exactly how he defines the adjoint representation, in terms of the structure constants of SU(2) in chapter 2.

For a more intuitive (imo) introduction to tensor operators, check out chapter 3.10 in Sakurai. It shows how transforming under the commutator as Georgi defines it is equivalent to V -> (J^-1)VJ.