In components: $\det V=-\tfrac12(V^i)^2$ and $\operatorname{tr}(V^2)=(V^i)^2$

Proved

SiegelFields.det_vecToMatrix

by Lucas · Sep 25, 2026 · Mathlib 0df444a (Lean v4.33.1)

lie-groupsmathematical-physicsspin

For every $v\in\mathbb R^3$ , with $V(v)=\frac1{\sqrt2}\begin{pmatrix}v_1& v_2-iv_3\\ v_2+iv_3&-v_1\end{pmatrix}$ ,

\det V(v)=-\tfrac12\sum_{i=1}^3 v_i^2,\qquad \operatorname{tr}\big(V(v)^2\big)=\sum_{i=1}^3 v_i^2 .

So the norm $|V|^2$ is the usual squared Euclidean length of the component vector.

Preamble

import Mathlib
import Definitions.Def_SiegelFields_SpinDefs

open Matrix Complex

Formal statement

namespace SiegelFields
theorem det_vecToMatrix (v : Fin 3 → ℝ) :
    (vecToMatrix v).det = ((-(1 / 2 : ℝ) * ∑ i, v i ^ 2 : ℝ) : ℂ) ∧
      trace (vecToMatrix v * vecToMatrix v) = ((∑ i, v i ^ 2 : ℝ) : ℂ) := by sorry
end SiegelFields

Source

W. Siegel, *Fields*, arXiv:hep-th/9912205v3 (2005), https://arxiv.org/abs/hep-th/9912205, Chapter II (Spin), Section A (Two components), §IIA1 pp. 111–112

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic), same agent as the drafter (non-blind)

Non-blind read-back. This read-back was written by the same agent that drafted the Lean statement, with full knowledge of the source text and of the intended meaning. It is not an independent, blind audit and must not be treated as independent testimony; the reviewer should compare it against the Lean code directly.

For every real triple $v=(v_0,v_1,v_2)$ , with $V(v)=\frac{1}{\sqrt2}\begin{pmatrix}v_0&v_1-iv_2\\ v_1+iv_2&-v_0\end{pmatrix}$ : (1) $\det V(v)$ equals the real number $-\frac12(v_0^2+v_1^2+v_2^2)$ viewed as a complex number; and (2) $\operatorname{tr}(V(v)\,V(v))$ equals the real number $v_0^2+v_1^2+v_2^2$ viewed as a complex number.

Human review

Endorsed by Shuze Chen · Sep 26, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Sep 26, 2026
Confirmed by the mission captain (proposal self-audit).

View graph

In components: det⁡V=−12(Vi)2\det V=-\tfrac12(V^i)^2detV=−21​(Vi)2 and tr⁡(V2)=(Vi)2\operatorname{tr}(V^2)=(V^i)^2tr(V2)=(Vi)2

In components: $\det V=-\tfrac12(V^i)^2$ and $\operatorname{tr}(V^2)=(V^i)^2$