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.

D=3 null translations have two signs and independent central phases

Example

For $D = 3$ , use the universal Lorentz cover $K_{\infty} = \tilde{L}$ . Its null little group is the direct product $R \times Z$ , with deck generator $z = (0, 1)$ . For $t \in R$ and $α \in R / Z$ , define

θ_{t, α} (u, j) = e^{i t u} e^{2 π i α j} .

These are one-dimensional irreducible null little-group representations. For every $ρ > 0$ , the two signs $t = ρ$ and $t = - ρ$ are inequivalent, and $z$ acts by the independently chosen phase $e^{2 π i α}$ . Thus all phases occur on $K_{\infty}$ ; on a finite cover $K_{N}$ , the allowed phases are the $N$ th roots of unity, $N α \in Z$ , and either sign remains available.

Facts & Assumptions

Given: $D = 3$ , a nonzero future null momentum, and the universal or finite connected Lorentz cover.

[F1]

Global Lorentz stabilizers and their connected-cover preimages: the $D = 3$ null stabilizer is $R \times Z$ on $K_{\infty}$ , with $z = (0, 1)$ ; on $K_{N}$ it is $R \times (Z / N Z)$ .

[F2]

Null little-group unitary duals, including low dimensions: every irreducible strongly continuous unitary representation of a $D = 3$ null little group is $θ_{t, α} (u, j) = e^{i t u} e^{2 π i α j}$ ; the real parameter has both signs, and finite $N$ requires $N α \in Z$ .

[F3]

Central kernel characters control descent of Poincaré inductions: the character descends from $K_{\infty}$ to $K_{N}$ exactly when it is trivial on $z^{N}$ , which is the condition $N α \in Z$ independent of $t$ .

[F4]

Positive-energy Wigner–Mackey classification for connected covers: inducing any such irreducible null little-group character over a future null momentum gives an irreducible positive-energy representation.

[A1]

AC is assumed through [F2], [F3], and [F4]. The Axiom of Choice

Proof

Given: The groups and parameters in the Example.

1.1F1givenalgebra

The displayed formula is multiplicative because $R \times Z$ is a direct product: $θ_{t, α} ((u, j) + (v, k)) = e^{i t (u + v)} e^{2 π i α (j + k)} = θ_{t, α} (u, j) θ_{t, α} (v, k)$ . It is continuous in $u$ , and $Z$ is discrete. A one-dimensional unitary representation is irreducible.

2.1F1F2step 1.1algebra

For $ρ > 0$ , evaluating the characters with $t = ρ$ and $t = - ρ$ at $(u, j) = (π / (2 ρ), 0)$ gives respectively $i$ and $- i$ , so they are inequivalent. This is consistent with the orbit classification because the connected rotation factor is $S O (1) = {1}$ and cannot reverse the real translation character.

2.2F1F2step 1.1algebra

The deck generator is $(0, 1)$ , so $θ_{t, α} (z) = e^{2 π i α}$ and the value is independent of $t$ . As $α$ ranges over $R / Z$ , these are all of $T$ . On $K_{N}$ , the relation $z^{N} = 1$ forces and is forced by $e^{2 π i N α} = 1$ , equivalently $N α \in Z$ , giving precisely the $N$ possible phases.

3.1F2F3F4A1step 2.1step 2.2∎

For every $t = \pm ρ$ and allowed $α$ , [F4] gives a positive-energy irreducible Poincaré induction. The independent central-phase calculation in step 2.2 and the sign calculation in step 2.1 show that neither the finite-cover descent condition [F3] nor the phase parameter identifies the two signs. This proves the stated D=3 distinction.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

33 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)