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Thermal State with Quadratic Interaction

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Abstract

We consider the perturbative construction, proposed in Fredenhagen and Lindner (Commun Math Phys 332:895, 2014), for a thermal state \(\Omega _{\beta ,\lambda V\{f\}}\) for the theory of a real scalar Klein–Gordon field \(\phi \) with interacting potential \(V\{f\}\). Here, f is a space-time cut-off of the interaction V, and \(\lambda \) is a perturbative parameter. We assume that V is quadratic in the field \(\phi \) and we compute the adiabatic limit \(f\rightarrow 1\) of the state \(\Omega _{\beta ,\lambda V\{f\}}\). The limit is shown to exist; moreover, the perturbative series in \(\lambda \) sums up to the thermal state for the corresponding (free) theory with potential V. In addition, we exploit the same methods to address a similar computation for the non-equilibrium steady state (NESS) Ruelle (J Stat Phys 98:57–75, 2000) recently constructed in Drago et al. (Commun Math Phys 357:267, 2018).

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Acknowledgements

The author is grateful to Claudio Dappiaggi, Federico Faldino, Klaus Fredenhagen, Thomas-Paul Hack and Nicola Pinamonti, for enlightening discussions and comments on a preliminary version of this paper. This work was supported by the National Group of Mathematical Physics (GNFMINdAM).

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Authors and Affiliations

  1. Dipartimento di Fisica, Università degli Studi di Pavia, Via Bassi, 6, 27100, Pavia, Italy

    Nicolò Drago

  2. Istituto Nazionale di Fisica Nucleare – Sezione di Pavia, Via Bassi, 6, 27100, Pavia, Italy

    Nicolò Drago

  3. Unità di Ricerca dell’Università di Pavia, Dipartimento di Matematica, Istituto Nazionale d’Alta Matematica “Francesco Severi”, Via Adolfo Ferrata, 5, 27100, Pavia, PV, Italy

    Nicolò Drago

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  1. Nicolò Drago
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Correspondence to Nicolò Drago.

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Communicated by Karl-Henning Rehren.

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Drago, N. Thermal State with Quadratic Interaction. Ann. Henri Poincaré 20, 905–927 (2019). https://doi.org/10.1007/s00023-018-0739-6

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