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Mean-Field Limits of Particles in Interaction with Quantized Radiation Fields

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Book cover Macroscopic Limits of Quantum Systems (MaLiQS 2017)

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Abstract

We report on a novel strategy to derive mean-field limits of quantum mechanical systems in which a large number of particles weakly couple to a second-quantized radiation field. The technique combines the method of counting and the coherent state approach to study the growth of the correlations among the particles and in the radiation field. As an instructional example, we derive the Schrödinger–Klein–Gordon system of equations from the Nelson model with ultraviolet cutoff and possibly massless scalar field. In particular, we prove the convergence of the reduced density matrices (of the nonrelativistic particles and the field bosons) associated with the exact time evolution to the projectors onto the solutions of the Schrödinger–Klein–Gordon equations in trace norm. Furthermore, we derive explicit bounds on the rate of convergence of the one-particle reduced density matrix of the nonrelativistic particles in Sobolev norm.

Dedicated to Herbert Spohn on the occasion of his 70th birthday.

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Notes

  1. 1.

    These approaches usually embed the N particle states of \(\mathcal {H}_p^{(N)}\) in a bosonic Fock space for the particles \(\mathcal {F}_p\) and consider the Hilbert space \(\mathcal {F}_p \otimes \mathcal {F}\).

  2. 2.

    For the sake of clarity, we want to stress that only the number of the nonrelativistic particles is fixed while field bosons are created and destroyed during the time evolution.

  3. 3.

    Note that \(\Psi _N^{(n)}\) is symmetric in the variables \(k_1, \ldots k_n\). For notational convenience, we will use the shorthand notation \(\Psi _N^{(n)}(X_N,K_n) = \Psi _N^{(n)}(x_1, \ldots , x_N, k_1, \ldots k_n)\).

  4. 4.

    Here, \(\hat{k}_j\) means that \(k_j\) is left out in the argument of the function.

  5. 5.

    We use \(\mathcal {FT}[\Phi ](k,t) = (2 \pi )^{-3/2} \int _{\mathbb {R}^3} d^3x \, e^{-ikx} \Phi (x,t)\) to denote the Fourier transform of \(\Phi (x,t)\) and \(\left( \kappa * \Phi \right) (x,t) = \int d^3k \, e^{ikx} \tilde{\kappa }(k) \mathcal {FT}[\Phi ](k,t)\).

  6. 6.

    Here, \(W^{-1}(\sqrt{N} \alpha _0) = W(- \sqrt{N} \alpha _0)\) is the inverse of the unitary Weyl operator \(W(\sqrt{N} \alpha _0)\), see Sect. 9.

  7. 7.

    To ease the presentation, we have chosen for given t the scaling parameter N large enough such that \(0 \le \beta (t) \le 1\) and \(0 \le \beta _2(t) \le 1\) (see Sects. 8.2 and 8.3).

  8. 8.

    For the precise definition of the initial data, we refer to [2, (I.3) and Theorem 3].

  9. 9.

    One should note that the high frequency modes of the radiation field do not interact with the nonrelativistic particles and evolve according to the free dynamics.

  10. 10.

    Occasionally, we use the bra–ket notation \(p_j^{\varphi _t} = | \varphi _t(x_j) \rangle \langle \varphi _t(x_j) | = | \varphi _t \rangle \langle \varphi _t |_j\).

  11. 11.

    This is a simple consequence of \(W(\sqrt{N} \alpha _t)\) being unitary and \(W^*(\sqrt{N} \alpha _t) a(k) = a(k) W^*(\sqrt{N} \alpha _t) + \sqrt{N} W^*(\sqrt{N} \alpha _t) \alpha _t(k)\), see (129).

  12. 12.

    We sometimes apply the shorthand notation \(\beta (t) = \beta (\Psi _{N,t},\varphi _t,\alpha _t)\).

  13. 13.

    More information is given for instance in [12, p. 9].

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Acknowledgements

We would like to thank Dirk André Deckert, Marco Falconi and David Mitrouskas for helpful discussions. N.L. gratefully acknowledges financial support by the Cusanuswerk and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 694227). The article appeared in slightly different form in one of the author’s (N.L.) Ph.D. thesis [48].

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Authors and Affiliations

  1. IST Austria (Institute of Science and Technology Austria), Am Campus 1, 3400, Klosterneuburg, Austria

    Nikolai Leopold

  2. Duke Kunshan University, Duke Avenue 8, Kunshan, 215316, China

    Peter Pickl

  3. Ludwig-Maximilians-Universität München, Theresienstraße 39, 80333, München, Germany

    Peter Pickl

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  1. Nikolai Leopold
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Correspondence to Nikolai Leopold .

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Editors and Affiliations

  1. Institut für Theoretische, Physik Universität Leipzig, Leipzig, Germany

    Daniela Cadamuro

  2. Zentrum Mathematik, M5, Technische Universität München, Garching, Germany

    Maximilian Duell

  3. Zentrum Mathematik, M5, Technische Universität München, Garching, Germany

    Wojciech Dybalski

  4. UMPA UMR 5669 CNRS, École normale supérieure, Lyon, France

    Sergio Simonella

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Leopold, N., Pickl, P. (2018). Mean-Field Limits of Particles in Interaction with Quantized Radiation Fields. In: Cadamuro, D., Duell, M., Dybalski, W., Simonella, S. (eds) Macroscopic Limits of Quantum Systems. MaLiQS 2017. Springer Proceedings in Mathematics & Statistics, vol 270. Springer, Cham. https://doi.org/10.1007/978-3-030-01602-9_9

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