Multiplication of operators defined by commutation relations

Question

I'm confused about how usual multiplication of operators is uniquely defined. Usually, we only now for physical reasons the anti-/commutation properties of operators so that $AB-BA$ is fixed.

Take for example spins, we know that spin operators must represent $su(2)$ so that we have three basis vectors $\sigma_{i}$ in our vector space and the Lie bracket $[\sigma_a,\sigma_b]=2i\epsilon_{abc}\sigma_c$. How is it that we are allowed to use terms like $\sigma_x\sigma_y$ in our Hilbert space representation? We know that this is $i\sigma_z$ if Pauli matrices are used, but this multiplication rule is not defined by requiring only a given Lie bracket to be fulfilled, is it? Might it be that there is some representation of $su(2)$ where the multiplication table is a different one?

Another way to ask me question: Why are we allowed to multiply operators if only commutation relations are known? How can the multiplication table of operators be defined through a specific Lie bracket? Is it uniquely defined?

Answer 1

• Multiplication of generators is a feature of the representation, not the Lie algebra.

Indeed, you are asking about the unital associative universal enveloping algebra of generators which defines the Casimir invariants with distinctive eigenvalues specifying the representation of the Lie algebra, and the Lie group, crudely the exponentials of these generators detailed in your group theory text, no doubt.

We know that... if Pauli matrices are used, ... this multiplication rule is not defined by requiring only a given Lie bracket to be fulfilled, is it? Might it be that there is some representation of 𝑠𝑢(2) where the multiplication table is a different one? How can the multiplication table of operators be defined through a specific Lie bracket? Is it uniquely defined?

I will remind you of what you already know from basic QM: different representations obey different multiplication relations. For example, for the 2d irrep of su(2), $\vec \sigma /2$, you know $(\sigma_x+i\sigma_y)^2$=0, but the analog fails for the 3d irrep, $$ (S_x+iS_y)^2/2= \begin{bmatrix}0&1&0\\0&0&1\\ 0&0&0 \end{bmatrix} ^2= \begin{bmatrix}0&0&1\\0&0&0\\ 0&0&0 \end{bmatrix}\neq 0, $$ and all higher ones. Also from basic QM, you recall that $\sum_i (\sigma_i/2)^2 = \tfrac{3}{4}\mathbb{I}_2$, but $\sum_i (S_i)^2 = 2\mathbb{I}_3$.

So, for example, different irreps of group elements of SU(2) present as very different polynomials in the universal enveloping algebra.

Answer 2

Algebraic structures can be defined somewhat abstractly. The Lie bracket is not necessarily represented as $[A,B] = AB-BA$, though this is the standard choice for operators as it satisfies the various properties required of a Lie bracket (anti-symmetry, bilinearity, and the Jacobi identity). If one wants to represent a Lie bracket as $[A,B] = AB-BA$, then the product of elements must be defined. For operators, this is clear as $AB$ represents the composition of linear maps which (barring domains of definition if the operators are unbounded) is well-defined.

If however you do not explicitly work with operators, you can still define $AB$, but to do so, you need an algebra structure. Algebras are effectively vector spaces with a notion of multiplication (of vectors). These can exist completely abstractly. You could define the free algebra over the generators $\sigma_x$, $\sigma_y$, and $\sigma_z$ and then quotient out by the equivalence relations you'd like them to have (in this case $[\sigma_x, \sigma_y] = i\sigma_z$ and so on). This then gives you an algebra consisting of the set of words in $\sigma_x$, $\sigma_y$, and $\sigma_z$ with the appropriate commutation relations. (You even naturally get a Lie algebra from this!) There's not really a direct physics link here other than $\mathfrak{su}(2)$ being at play, but it is totally legitimate mathematically.

If you want something more concrete and directly related to physics, you can then look at representations of your algebra. Typically, we pick linear maps on (or between) Hilbert spaces as our representations. The term "representation" has two somewhat dual meanings: it can mean "what the algebra looks like" in terms of linear maps (matrices, bounded operators, or unbounded operators) or the map that takes you from the algebra to the space of linear maps. Let $\mathcal{A}$ be your algebra and $\rho:\mathcal{A}\to\mathcal{L}(V)$ ($\mathcal{L}(V)$ is the set of linear maps on the vector space $V$), then the representation of $\sigma_x$, $\sigma_y$, and $\sigma_z$ would be $\rho(\sigma_x)$, etc. which could be the Pauli matrices if $V$ is finite-dimensional and complex.