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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.24945 |
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| _version_ | 1866910172479225856 |
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| author | Bianchi, Eugenio Chen, Chaosong Gamonal, Mauricio |
| author_facet | Bianchi, Eugenio Chen, Chaosong Gamonal, Mauricio |
| contents | We study the analytic properties and three equivalent representations of the Toller matrices $T^{(\pm)}$ which appear in the causal formulation of spinfoam transition amplitudes for 4d Lorentzian quantum gravity. These are polynomially bounded functions on the Lorentz group which satisfy the relation $T^{(+)}+T^{(-)}=D$, where the Wigner matrix $D$ provides a unitary irreducible representation of $SL(2,C)$. Ruhl's definition of $T^{(\pm)}$ in terms of analyticity and asymptotic properties is shown to be equivalent to the recently introduced Feynman $i\varepsilon$ prescription in spinfoams. We show that, equivalently, they can be represented as an integral over eigenvalues of the boost operator, which results in a sum over residues. The latter reproduces the Wick rotation relating Euclidean $Spin(4)$ to Lorentzian $SL(2,C)$ spinfoams studied by Dona, Gozzini and Nicotra. We provide explicit expressions in terms of hypergeometric functions and specialize them to the $γ$-simple representations relevant for spinfoams. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24945 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Toller matrices and the Feynman $i\varepsilon$ in spinfoams Bianchi, Eugenio Chen, Chaosong Gamonal, Mauricio General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics We study the analytic properties and three equivalent representations of the Toller matrices $T^{(\pm)}$ which appear in the causal formulation of spinfoam transition amplitudes for 4d Lorentzian quantum gravity. These are polynomially bounded functions on the Lorentz group which satisfy the relation $T^{(+)}+T^{(-)}=D$, where the Wigner matrix $D$ provides a unitary irreducible representation of $SL(2,C)$. Ruhl's definition of $T^{(\pm)}$ in terms of analyticity and asymptotic properties is shown to be equivalent to the recently introduced Feynman $i\varepsilon$ prescription in spinfoams. We show that, equivalently, they can be represented as an integral over eigenvalues of the boost operator, which results in a sum over residues. The latter reproduces the Wick rotation relating Euclidean $Spin(4)$ to Lorentzian $SL(2,C)$ spinfoams studied by Dona, Gozzini and Nicotra. We provide explicit expressions in terms of hypergeometric functions and specialize them to the $γ$-simple representations relevant for spinfoams. |
| title | Toller matrices and the Feynman $i\varepsilon$ in spinfoams |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2604.24945 |