8.4 The Casimir operator

Concept links · terms present in this machine draft; source roles are unverified: vector space · eigenvalue · Lie algebra · Lie algebra representation · irreducible representation · complexification · Schur’s lemma

For both $S U (2)$ and $S O (3)$ , we have found that all representations can be constructed out of function spaces, with the Lie algebra acting as first-order diferential operators. It turns out that there is also a very interesting secondorder diferential operator that comes from these Lie algebra representations, known as the Casimir operator. For the case of $S O (3)$

Definition (Casimir operator for $S O ( 3 )

) $. T h e C a s imi r o p er a t or f or t h er e p r ese n t a t i o n o f$ S O ( 3 ) $o n f u n c t i o n so n$ S ^ { 2 }$ is the second-order diferential operator
$L^{2} \equiv L_{1}^{2} + L_{2}^{2} + L_{3}^{2}$

(the symbol $L^{2}$ is not intended to mean that this is the square of an operator $L)$

A straightforward calculation using the commutation relations satisfied by the $L_{j}$ shows that

[L^{2}, ρ^{'} (X)] = 0

for any $X \in so (3)$ . Knowing this, a version of Schur’s lemma says that $L^{2}$ will act on an irreducible representation as a scalar $(i.e.$ , all vectors in the representation are eigenvectors of $\overset{ˉ}{L}^{2}$ , with the same eigenvalue). This eigenvalue can be used to characterize the irreducible representation.

The easiest way to compute this eigenvalue turns out to be to act with $L^{2}$ on a highest weight vector. First one rewrites $L^{2}$ in terms of raising and lowering

operators

L_{-} L_{+} = (L_{1} - i L_{2}) (L_{1} + i L_{2}) = L_{1}^{2} + L_{2}^{2} + i [L_{1}, L_{2}] = L_{1}^{2} + L_{2}^{2} - L_{3}

L^{2} = L_{1}^{2} + L_{2}^{2} + L_{3}^{2} = L_{-} L_{+} + L_{3} + L_{3}^{2}

For the representation $ρ$ of $S O (3)$ on functions on $S^{2}$ constructed above, we know that on a highest weight vector of the irreducible representation $ρ_{l}$ (restriction of $ρ$ to the $2 l + 1$ dimensional irreducible subspace of functions that are linear combinations of the $Y_{l}^{m} (θ, ϕ))$ , we have the two eigenvalue equations

L_{+} f = 0, L_{3} f = l f

with solution the functions proportional to $Y_{l}^{l} (θ, ϕ)$ . Just from these conditions and our expression for $L^{2}$ we can immediately find the scalar eigenvalue of $L^{2}$ since

L^{2} f = L_{-} L_{+} f + (L_{3} + L_{3}^{2}) f = (0 + l + l^{2}) f = l (l + 1) f

We have thus shown that our irreducible representation $ρ_{l}$ can be characterized as the representation on which $L^{2}$ acts by the scalar $l (l + 1)$

In summary, we have two diferent sets of partial diferential equations whose solutions provide a highest weight vector for and thus determine the irreducible representation $ρ_{l} :$

L_{+} f = 0, L_{3} f = l f

which are first-order equations, with the first using complexification and something like a Cauchy-Riemann equation, and

L^{2} f = l (l + 1) f, L_{3} f = l f

where the first equation is a second-order equation, something like a Laplace equation.

That a solution of the first set of equations gives a solution of the second set is obvious. Much harder to show is that a solution of the second set gives a solution of the first set. The space of solutions to

L^{2} f = l (l + 1) f

for l a non-negative integer includes as we have seen the $2 l + 1$ dimensional vector space of linear combinations of the $Y_{l}^{m} (θ, ϕ)$ (there are no other solutions, although we will not show that). Since the action of $S O (3)$ on functions commutes with the operator $L^{2}$ , this $2 l + 1$ dimensional space will provide a representation, the irreducible one of spin $l$ .

The second-order diferential operator $L^{2}$ in the $ρ$ representation on functions can explicitly be computed, it is

L^{2} = L_{1}^{2} + L_{2}^{2} + L_{3}^{2} = (i (sin ϕ \frac{\partial}{\partial θ} + cot θ cos ϕ \frac{\partial}{\partial ϕ}))^{2} + (i (- cos ϕ \frac{\partial}{\partial θ} + cot θ sin ϕ \frac{\partial}{\partial ϕ}))^{2} + (- i \frac{\partial}{\partial ϕ})^{2} = - (\frac{1}{s i n θ} \frac{\partial}{\partial θ} (sin θ \frac{\partial}{\partial θ}) + \frac{1}{s i n ^{2} θ} \frac{\partial ^{2}}{\partial ϕ ^{2}}) (8.4)

We will re-encounter this operator in chapter 21 as the angular part of the Laplace operator on $R^{3}$

For the group $S U (2)$ we can also find irreducible representations as solution spaces of diferential equations on functions on $C^{2}$ . In that case, the diferential equation point of view is much less useful, since the solutions we are looking for are just the homogeneous polynomials, which are more easily studied by purely algebraic methods.

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 89、90、91、92、93、94、95、96、97、98、99、100、101、102、103、104、105、106、107、108

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：f30e5e19104ce525fa7316f25353a05960c606085ccb5d251e1897c8bfa8c58c

OCR 产物 SHA-256：f30e5e19104ce525fa7316f25353a05960c606085ccb5d251e1897c8bfa8c58c

数学物理阅读

书架

8.4 The Casimir operator

8.4 The Casimir operator

关系图谱