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The two D=2 future null rays give inequivalent representations

Example

In $D = 2$ , use the base Lorentz group $L = S O_{0} (1, 1) ≅ R$ . The two nonzero future null momenta

p_{+} = (1, 1), p_{-} = (1, - 1)

lie on distinct Lorentz orbits. Their stabilizers are trivial, so they give two induced positive-energy representations

π_{\pm} = {Ind}_{R^{2}}^{R^{2} ⋊ L} χ_{p_{\pm}} .

The representations $π_{+}$ and $π_{-}$ are inequivalent despite having the same mass squared $η (p_{\pm}, p_{\pm}) = 0$ .

Facts & Assumptions

Given: $D = 2$ , the connected boost group $L$ , and the mostly-plus form $η$ .

[F1]

Lorentz momentum orbits are regular homogeneous Borel spaces: in $D = 2$ the nonzero null orbits are the four rays distinguished by the signs of $p^{0}$ and $p^{1}$ .

[F2]

Global Lorentz stabilizers and their connected-cover preimages: $L ≅ R$ is the only connected Lorentz cover in $D = 2$ , and every nonzero null stabilizer is trivial.

[F3]

Positive-energy Wigner–Mackey classification for connected covers: each distinct orbit and its irreducible little-group representation give an inequivalent irreducible representation; a future null orbit gives positive energy.

[A1]

AC is assumed through [F3]. The Axiom of Choice

Proof

Given: $p_{+} = (1, 1)$ and $p_{-} = (1, - 1)$ .

1.1F1givenalgebra

For $q = (p^{0}, p^{1})$ , the connected boost of rapidity $s$ acts on light-cone coordinates by $(p^{0} + p^{1}, p^{0} - p^{1}) \mapsto (e^{s} (p^{0} + p^{1}), e^{- s} (p^{0} - p^{1}))$ . For $p_{+}$ the second coordinate is zero and the first positive; for $p_{-}$ the first coordinate is zero and the second positive. These zero-coordinate patterns are preserved by every finite boost, so no element of $L$ carries $p_{+}$ to $p_{-}$ . Both have $p^{0} > 0$ and $η (p, p) = 0$ , hence are future null.

2.1F1F2F3A1step 1.1algebra∎

Their little groups are trivial by [F2], and the unique representation of the trivial group is its one-dimensional identity representation. Applying [F3] to the two distinct future null orbits yields irreducible positive-energy inductions, and its orbit-equivalence criterion makes $π_{+}$ and $π_{-}$ inequivalent. Their squared masses are both zero by direct substitution into $η (p, p) = - (p^{0})^{2} + (p^{1})^{2}$ .

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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)