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Rotational scalars on multiplicity-resolved spin blocks

Statement

Let $H$ be a finite direct sum over $C$ of spin blocks $V_{j}$ , with $n_{j}$ explicitly indexed copies of each occurring spin $j$ . Write $e_{j, a, m}$ for magnetic label $m = - j, - j + 1, \dots, j$ in copy $a = 1, \dots, n_{j}$ , and let $J_{x}, J_{y}, J_{z}, J_{\pm}$ act by the specified spin matrices on every copy. Give $H$ the sum of the block inner products, so these basis vectors are orthonormal, and equip it with the induced norm $∥ \sum_{j, a, m} z_{j, a, m} e_{j, a, m} ∥ = (\sum_{j, a, m} ∣ z_{j, a, m} ∣^{2})^{1 / 2}$ . Define the axis rotations $U_{k} (t) = \exp (- i t J_{k})$ for $k = x, y, z$ and real $t$ by the operator exponential series.

If a complex-linear operator $T : H \to H$ commutes with all these axis rotations, then there are unique complex matrices $A^{(j)} = (A_{b a}^{(j)})_{b, a = 1}^{n_{j}}$ such that $T e_{j, a, m} = \sum_{b = 1}^{n_{j}} A_{b a}^{(j)} e_{j, b, m}$ for every $j, a, m$ . Thus $T$ is the identity in the magnetic index and may mix copies of equal spin; in particular, on a spin occurring once it is a scalar multiple of the identity.

More generally, let $D_{1}, D_{2}, D_{3}, F : H \to H$ be complex-linear maps, and suppose a family of invertible complex-linear maps $U : H \to H$ includes all the axis rotations. For each $U$ suppose there is a real orthogonal $3 \times 3$ matrix $R$ such that $U D_{i} U^{- 1} = \sum_{k = 1}^{3} R_{k i} D_{k}$ for $i = 1, 2, 3$ . If $F$ commutes with every such $U$ , then $T = \sum_{i = 1}^{3} D_{i} F D_{i}$ commutes with every such $U$ and has the block form above. An isotropic coefficient $δ_{i j}$ in a contraction alone does not imply the covariance and commutation hypotheses just stated.

Statement references

Finite spin blocks, spherical tensors and Clebsch–Gordan phases

The induced length is a norm

The exponential series of a bounded operator

Linear map between vector spaces over the same field

The orthogonal group O(3) and the rotation group SO(3)

Facts & Assumptions

Given: The finite indexed complex direct sum with its displayed norm, spin matrices and complex-linear operator $T : H \to H$ , or the complex-linear maps $D_{i}, F : H \to H$ and family of invertible complex-linear maps $U : H \to H$ , under the hypotheses of the Statement.

Goal: Establish the displayed block form and the conditional contraction conclusion.

[F1]

On each copy $J_{z} e_{j, a, m} = m e_{j, a, m}$ and $J_{-} e_{j, a, m} = \sqrt{(j + m) (j - m + 1)} e_{j, a, m - 1}$ , with the endpoint vector zero. The coefficient is nonzero for $m > - j$ . The Casimir $C = J_{x}^{2} + J_{y}^{2} + J_{z}^{2}$ acts on that copy as $j (j + 1) I$ . Finite spin blocks, spherical tensors and Clebsch–Gordan phases Finite spin matrices

[F2]

The spin convention supplies the direct-sum complex inner product. Its induced length, displayed in the Statement, is a norm. Order the finite basis by $j$ , then $a$ , then $m$ ; the finite-dimensional completeness corollary therefore makes $H$ a complex Banach space. The boundedness corollary, with domain and codomain both $H$ , makes every complex-linear endomorphism of $H$ bounded, including $T$ and $- i J_{k}$ . Finite spin blocks, spherical tensors and Clebsch–Gordan phases The induced length is a norm Every finite-dimensional normed space is Banach A linear map from a finite-dimensional normed space is bounded

[F3]

For a bounded operator on a complex Banach space, the exponential series is norm differentiable and $(\exp (t A) - I) / t \to A$ as real $t \to 0$ . It defines invertible operators with $\exp (t A)^{- 1} = \exp (- t A)$ . By [F2] this applies to $A = - i J_{k}$ on $H$ . The exponential series of a bounded operator

[F4]

For bounded endomorphisms of $H$ , use the induced operator norm $∥ S ∥_{o p} = \sup_{∥ v ∥ \leq 1} ∥ S v ∥$ . It is a norm, and composition satisfies $∥ S B ∥_{o p} \leq ∥ S ∥_{o p} ∥ B ∥_{o p}$ . These facts permit multiplication by a fixed bounded map in an operator-norm limit. The operator norm as the least bound and as the unit-sphere or unit-ball supremum The operator norm is a norm on the space of bounded linear operators Composition satisfies ‖ST‖≤‖S‖ ‖T‖

[F5]

All maps in the contraction act complex-linearly from $H$ to $H$ . The inverse of an invertible complex-linear map is complex-linear: apply that map to $a U^{- 1} x + b U^{- 1} y$ and use its injectivity. Sums and compositions of these maps are complex-linear, and composition distributes over finite linear combinations by the defining linearity equation. Linear map between vector spaces over the same field

[F6]

For a real $3 \times 3$ matrix, orthogonality means $R^{T} R = I_{3}$ . The orthogonal-group supplier also derives $R^{- 1} = R^{T}$ and $R R^{T} = I_{3}$ ; in entries the latter equation is $\sum_{i = 1}^{3} R_{k i} R_{ℓ i} = δ_{k ℓ}$ , where $δ_{k ℓ} = 1$ for $k = ℓ$ and $0$ otherwise. The orthogonal group O(3) and the rotation group SO(3)

Proof

1.1givenF2F3F4algebra

Put $A = - i J_{k}$ and $Q_{k} (t) = (U_{k} (t) - I) / t$ for $t \neq 0$ . Since $T$ commutes with $U_{k} (t)$ , it commutes with $Q_{k} (t)$ . By [F2] $T$ is bounded, and by [F3] $Q_{k} (t) \to A$ in operator norm. Thus $∥ T A - A T ∥_{o p} = ∥ T (A - Q_{k} (t)) - (A - Q_{k} (t)) T ∥_{o p} \leq 2 ∥ T ∥_{o p} ∥ A - Q_{k} (t) ∥_{o p} \to 0$ by [F4]. Hence $[T, J_{k}] = 0$ for $k = x, y, z$ . Consequently $T$ commutes with $J_{-} = J_{x} - i J_{y}$ and with $C$ .

2.1F1step 1.1algebra

For a basis vector $e_{j, a, m}$ , the equation $C T e_{j, a, m} = j (j + 1) T e_{j, a, m}$ and the direct-sum basis expansion show that only spin labels $ℓ$ satisfying $ℓ (ℓ + 1) = j (j + 1)$ can occur. As $j, ℓ \geq 0$ , this equality implies $(ℓ - j) (ℓ + j + 1) = 0$ , hence $ℓ = j$ . Commutation with $J_{z}$ similarly leaves only magnetic label $m$ . Thus $T e_{j, a, m} = \sum_{b} c_{b a}^{(j, m)} e_{j, b, m}$ .

3.1F1step 2.1algebra

For $m > - j$ , applying $T J_{-} = J_{-} T$ to $e_{j, a, m}$ and using the same nonzero ladder coefficient in every copy gives $c_{b a}^{(j, m - 1)} = c_{b a}^{(j, m)}$ for all $a, b$ . Iterating through the finite ladder, including the single-label case $j = 0$ , makes these coefficients independent of $m$ . Their values are unique because the displayed vectors form a basis. For $n_{j} = 1$ the resulting matrix is one complex scalar.

4.1givenF1F5F6step 3.1algebra∎

Under the additional contraction hypotheses, [F5] makes $T$ a complex-linear endomorphism of $H$ . Conjugation distributes over the finite sum and compositions by [F5]; using $U F U^{- 1} = F$ gives $U T U^{- 1} = \sum_{i, k, ℓ} R_{k i} R_{ℓ i} D_{k} F D_{ℓ} = \sum_{k} D_{k} F D_{k} = T$ , since [F6] gives $\sum_{i} R_{k i} R_{ℓ i} = δ_{k ℓ}$ . The axis rotations belong to the family, so step 3.1 applies. The coefficient $δ_{i j}$ by itself specifies only the summation: for example, take $D_{1} = J_{z}$ , $D_{2} = D_{3} = 0$ , and $F = I$ on a spin-one block. The resulting $T = J_{z}^{2}$ has unequal eigenvalues on the magnetic basis and hence is not scalar, showing why covariance cannot be omitted.

Depends on

Used by · 1

Isotropic Lamb shift preserves magnetic degeneracy

Dependency tree · 2 levels

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