## Abstract

We consider a model quantum field theory with a scalar quantum field in de Sitter space-time in a Bohmian version with a field ontology, i.e., an actual field configuration \(\varphi (\mathbf{x},t)\) guided by a wave function on the space of field configurations. We analyze the asymptotics at late times (\(t\rightarrow \infty \)) and provide reason to believe that for more or less any wave function and initial field configuration, every Fourier coefficient \(\varphi _\mathbf{k}(t)\) of the field is asymptotically of the form \(c_\mathbf{k}\sqrt{1+\mathbf{k}^2 \exp (-2Ht)/H^2}\), where the limiting coefficients \(c_\mathbf{k}=\varphi _\mathbf{k}(\infty )\) are independent of *t* and *H* is the Hubble constant quantifying the expansion rate of de Sitter space-time. In particular, every field mode \(\varphi _\mathbf{k}\) possesses a limit as \(t\rightarrow \infty \) and thus “freezes.” This result is relevant to the question whether Boltzmann brains form in the late universe according to this theory, and supports that they do not.

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## Notes

It is hard to give an exact condition on the wave function and the initial field configuration under which (4) is valid. But our reasoning suggests that it is very commonly valid, at least as commonly as Bohmian trajectories of non-relativistic oscillators are smooth—a property usually taken for granted in the physics literature.

Here is the difficulty. It is clear that the relation (8) needs to be replaced by \(\mathrm {d}t= a\,\mathrm {d}\eta \), (10) by \(y=a\varphi \), and factors of \(1/\eta \) by \(-a'/a\) in (11), (13), and (22). But the present result depends on the simple explicit solution (30)–(32) to the coupled ordinary differential equations (21)–(23), and with \(1/\eta \) in (22) replaced by \(-a'/a\) I do not know an analogous explicit solution.

## References

Albrecht, A., Sorbo, L.: Can the universe afford inflation? Phys. Rev. D 70, 063528 (2004). hep-th/0405270

Bell, J.S.: Beables for quantum field theory. Phys. Rep. 137, 49-54 (1986) (Reprinted on p. 173 in Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press (1987))

Boddy, K.K., Carroll, S.M., Pollack, J.: Why Boltzmann brains dont fluctuate into existence from the De Sitter vacuum. In Chamcham, K., Barrow, J., Silk, J., Saunders, S. (eds.) The Philosophy of Cosmology. Cambridge University Press (2016). arXiv:1505.02780

Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. I and II. Phys. Rev. 85, 166-193 (1952)

Colin, S., Struyve, W.: A Dirac sea pilot-wave model for quantum field theory. J. Phys. A Math. Theor. 40, 7309-7342 (2007). hep-lat/0701085

Dürr, D., Goldstein, S., Norsen, T., Struyve, W., Zanghì, N.: Can Bohmian mechanics be made relativistic? Proc. R. Soc. A 470, 20130699 (2014). arXiv:1307.1714

Dürr, D., Goldstein, S., Tumulka, R., Zanghì, N.: Bohmian mechanics and quantum field theory. Phys. Rev. Lett. 93, 090402 (2004). quant-ph/0303156

Dürr, D., Goldstein, S., Tumulka, R., Zanghì, N.: Bell-type quantum field theories. J. Phys. A Math. Gener. 38, R1-R43 (2005). quant-ph/0407116

Goldstein, S.: Bohmian mechanics. In Zalta, E.N. (ed.) Stanford Encyclopedia of Philosophy. Published online by Stanford University (2001) http://plato.stanford.edu/entries/qm-bohm

Gott, J.R.: Implications of the Copernican principle for our future prospects. Nature 363, 315-319 (1993)

Goldstein, S., Struyve, W., Tumulka, R.: The Bohmian approach to the problems of cosmological quantum fluctuations. In: Ijjas, A., Loewer, B. (eds.) Guide to the Philosophy of Cosmology. Oxford University Press, (2016). arXiv:1508.01017

Hawking, S.W., Ellis, G.F.R.: The Large-Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973)

Hiley, B.J., Aziz Mufti, A.H.: The ontological interpretation of quantum field theory applied in a cosmological contex. In: Ferrero, M., van der Merwe, A. (eds.) Fundamental Theories of Physics, vol. 73, pp. 141-156. Kluwer, Dordrecht (1995)

Kiefer, C., Polarski, D.: Why do cosmological perturbations look classical to us? Adv. Sci. Lett. 2, 164-173 (2009). arXiv:0810.0087

Pinto-Neto, N., Santos, G., Struyve, W.: The quantum-to-classical transition of primordial cosmological perturbations. Phys. Rev. D 85, 083506 (2012). arXiv:1110.1339

Polarski, D., Starobinsky, A.A.: Semiclassicality and decoherence of cosmological perturbations. Class. Quantum Gravity 13, 377-392 (1996). gr-qc/9504030v2

Ryssens, W.: On the Quantum-to-Classical Transition of Primordial Perturbations. Master thesis, Department of Physics and Astronomy, Katholieke Universiteit Leuven (2012)

Struyve, W.: Pilot-wave theory and quantum fields. Rep. Prog. Phys. 73, 106001 (2010). arXiv:0707.3685

Struyve, W.: Pilot-wave approaches to quantum field theory. J. Phys. Conf. Ser. 306, 012047 (2011). arXiv:1101.5819

Struyve, W., Valentini, A.: De Broglie-Bohm guidance equations for arbitrary Hamiltonians. J. Phys. A Math. Theor. 42, 035301 (2009). arXiv:0808.0290

## Acknowledgments

The author acknowledges support from the John Templeton Foundation, Grant No. 37433. I thank Shelly Goldstein and Ward Struyve for helpful discussions and two anonymous referees for their suggestions.

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Tumulka, R. Long-time asymptotics of a Bohmian scalar quantum field in de Sitter space-time.
*Gen Relativ Gravit* **48**, 2 (2016). https://doi.org/10.1007/s10714-015-1995-0

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DOI: https://doi.org/10.1007/s10714-015-1995-0