Toller matrices T^(±) in causal spinfoam amplitudes satisfy T^(+) + T^(-) = D and admit equivalent definitions via analyticity, iε prescription, and boost-eigenvalue integrals that reproduce the Euclidean-to-Lorentzian Wick rotation.
Lorentzian spin foam amplitudes: graphical calculus and asymptotics
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abstract
The amplitude for the 4-simplex in a spin foam model for quantum gravity is defined using a graphical calculus for the unitary representations of the Lorentz group. The asymptotics of this amplitude are studied in the limit when the representation parameters are large, for various cases of boundary data. It is shown that for boundary data corresponding to a Lorentzian simplex, the asymptotic formula has two terms, with phase plus or minus the Lorentzian signature Regge action for the 4-simplex geometry, multiplied by an Immirzi parameter. Other cases of boundary data are also considered, including a surprising contribution from Euclidean signature metrics.
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γ-duality in the EPRL spinfoam model determines the relation between parity-even and parity-odd terms in an effective gravity theory, allowing the Barbero-Immirzi parameter to be measured from inflationary tensor observables.
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Toller matrices and the Feynman $i\varepsilon$ in spinfoams
Toller matrices T^(±) in causal spinfoam amplitudes satisfy T^(+) + T^(-) = D and admit equivalent definitions via analyticity, iε prescription, and boost-eigenvalue integrals that reproduce the Euclidean-to-Lorentzian Wick rotation.
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Spinfoams, $\gamma$-duality and parity violation in primordial gravitational waves
γ-duality in the EPRL spinfoam model determines the relation between parity-even and parity-odd terms in an effective gravity theory, allowing the Barbero-Immirzi parameter to be measured from inflationary tensor observables.