ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-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.

A nontrivial D=2 Lorentz character lies over zero momentum

Example

In $D = 2$ , the connected Lorentz group is the additive rapidity group $K = L ≅ R$ . At zero momentum, $K_{0} = K$ . For any fixed $s \neq 0$ , the character

θ_{s} (r) = e^{i s r}

therefore gives the one-dimensional Poincaré representation

Π_{s} (a, r) = θ_{s} (r) = e^{i s r}, a \in R^{2}, r \in K .

It is irreducible, nontrivial, and positive-energy. Thus zero momentum includes nonvacuum Lorentz-cover representations.

Facts & Assumptions

Given: $D = 2$ , the connected boost group $K = L ≅ R$ , zero momentum, and $s \neq 0$ .

[F1]

Global Lorentz stabilizers and their connected-cover preimages: $D = 2$ has no nontrivial connected Lorentz cover and the zero-momentum stabilizer is all of $K = L ≅ R$ .

[F2]

Positive-energy Wigner–Mackey classification for connected covers: at $p = 0$ , any irreducible strongly continuous unitary representation of $K_{0} = K$ gives an irreducible strongly continuous positive-energy induction.

[F3]

Connected Poincaré covers and positive translation energy: $χ_{0} (a) = 1$ for all translations, and a representation is positive-energy when its translation spectral PVM is concentrated on $C_{+}$ .

[A1]

AC is assumed through [F2]. The Axiom of Choice

Proof

Given: $s \neq 0$ , $p = 0$ , and the semidirect product group $R^{2} ⋊ K$ .

1.1F1F2F3A1given

The function $θ_{s} (r) = e^{i s r}$ is a continuous one-dimensional unitary character of the additive group $K = R$ , hence is irreducible. Since $p = 0$ , [F1] gives $K_{p} = K$ , and [F2] yields its induced representation over the zero orbit. Induction from the whole group stabilizer identifies with the inducing representation itself, so the formula is $Π_{s} (a, r) = χ_{0} (a) θ_{s} (r) = e^{i s r}$ .

2.1F2F3step 1.1algebra∎

The representation is nontrivial: at $r = π / s$ it has value $e^{i π} = - 1$ . Its translation subgroup acts identically because $χ_{0} = 1$ , and [F2] guarantees positive energy since the inducing momentum $0$ lies in $C_{+}$ . This proves that a nontrivial Lorentz character lies over zero momentum.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

35 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Xavier Bekaert and Nicolas Boulanger, The unitary representations of the Poincaré group in any spacetime dimension, v2 (2021) (standard reference, not scraped)
Niklas Beisert, Quantum Field Theory I, Lecture Notes, ETH Zürich, 2025 HS (standard reference, not scraped)