LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Real coefficient arrays for the signed spin-four isometry

Statement

In $V_{4}$ set $u_{0} = \sqrt{1 / 3} e_{4, - 4} + \sqrt{2 / 3} e_{4, 2}$ and $u_{1} = \sqrt{2 / 3} e_{4, - 2} - \sqrt{1 / 3} e_{4, 4}$ . Let $e_{0}, e_{1}$ be the standard orthonormal basis of $C^{2}$ , and define $V : C^{2} \to V_{4}$ on every input by $V (z_{0} e_{0} + z_{1} e_{1}) = z_{0} u_{0} + z_{1} u_{1}$ for $z_{0}, z_{1} \in C$ . This is the unique complex-linear map with $V e_{μ} = u_{μ}$ for $μ = 0, 1$ , and it is an isometry: $∥ V x ∥ = ∥ x ∥$ for every $x \in C^{2}$ , where $∥ w ∥ = \sqrt{⟨ w, w ⟩}$ for the standard coordinate inner products. Under $j = m + 4$ , its real coefficient arrays are $α_{0} = 1 / \sqrt{3}$ , $α_{6} = \sqrt{2 / 3}$ , $β_{2} = \sqrt{2 / 3}$ , $β_{8} = - 1 / \sqrt{3}$ , and zero elsewhere. In particular $\sum_{j = 0}^{8} α_{j} β_{j} = 0$ and $\sum α_{j}^{2} = \sum β_{j}^{2} = 1$ .

Facts & Assumptions

[F1]

The magnetic basis of a spin block is orthonormal and indexed from minus spin to spin. Finite spin blocks, spherical tensors and Clebsch–Gordan phases

[F2]

A map between vector spaces over $C$ is complex-linear when it preserves $a x + b y$ for every pair of vectors $x, y$ and scalars $a, b \in C$ . Linear map between vector spaces over the same field

[F3]

In the specified orthonormal bases of finite coordinate spaces, $⟨ x, y ⟩ = \sum_{i} x_{i} \overline{y_{i}}$ , and the adjoint of a linear map is represented by its conjugate-transpose matrix. Here $V^{*}$ denotes this adjoint and $I_{2}$ the identity matrix on $C^{2}$ . Finite operator and ket conventions

Proof

Given: The vectors, coordinate spaces and map in the Statement.

Goal: Prove complex-linearity, uniqueness, isometry and the coefficient identities.

1.1F1algebra

The substitution $j = m + 4$ sends magnetic labels $- 4, 2, - 2, 4$ to $0, 6, 2, 8$ . Reading the two displayed vectors in the orthonormal basis gives exactly the listed arrays, including the negative coefficient at $j = 8$ . Their supports ${0, 6}$ and ${2, 8}$ are disjoint, so every summand of $\sum α_{j} β_{j}$ vanishes.

1.2F2F3givenalgebra

Each $x \in C^{2}$ has unique coordinates $x = z_{0} e_{0} + z_{1} e_{1}$ , so the stated formula defines $V$ on its entire domain. If $y = w_{0} e_{0} + w_{1} e_{1}$ and $a, b \in C$ , then $V (a x + b y) = (a z_{0} + b w_{0}) u_{0} + (a z_{1} + b w_{1}) u_{1} = a V x + b V y$ . Thus $V$ is complex-linear. Any other complex-linear map $W$ with the same two basis values satisfies $W x = z_{0} W e_{0} + z_{1} W e_{1} = z_{0} u_{0} + z_{1} u_{1} = V x$ , proving uniqueness.

2.1F1F3step 1.1step 1.2algebra

The squared norm of each column is $1 / 3 + 2 / 3 = 1$ , and their inner product is zero by disjoint support. Therefore $\sum_{j = 0}^{8} α_{j}^{2} = \sum_{j = 0}^{8} β_{j}^{2} = 1$ . The conjugate-transpose product of the two-column matrix with itself has these two sums on the diagonal and zero off the diagonal, so $V^{*} V = I_{2}$ .

3.1F1F3step 1.1step 1.2step 2.1algebra∎

For arbitrary $x = z_{0} e_{0} + z_{1} e_{1}$ , the four supported coordinates give $∥ V x ∥^{2} = ∣ z_{0} ∣^{2} (1 / 3 + 2 / 3) + ∣ z_{1} ∣^{2} (2 / 3 + 1 / 3) = ∣ z_{0} ∣^{2} + ∣ z_{1} ∣^{2} = ∥ x ∥^{2}$ . Taking nonnegative square roots proves norm preservation, hence the stated isometry. Together with steps 1.1 and 2.1 this proves all three coefficient identities.

Depends on

Used by · 3

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.