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Higher-Order Approximation of Coherent State Dynamics in Self-Interacting Quantum Field Theories

Zied Ammari, Julien Malartre, Maher Zerzeri

Abstract

We study the propagation of coherent states in self-interacting bosonic quantum field theories in the semi-classical (mean-field) regime. Relying on Hepp's method and a detailed analysis of the associated classical and quantum field dynamics, non-linear and linear respectively, we construct an asymptotic expansion of arbitrary order for the quantum evolution of coherent states. The results are first established for the spatially cutoff $P(φ)_2$ model, under standard assumptions ensuring essential self-adjointness of the Hamiltonian and well-posedness of the classical flow, and are then extended to a class of non-polynomial analytic interactions. This work refines and generalizes earlier results, which identified only the leading-order term of the expansion.

Higher-Order Approximation of Coherent State Dynamics in Self-Interacting Quantum Field Theories

Abstract

We study the propagation of coherent states in self-interacting bosonic quantum field theories in the semi-classical (mean-field) regime. Relying on Hepp's method and a detailed analysis of the associated classical and quantum field dynamics, non-linear and linear respectively, we construct an asymptotic expansion of arbitrary order for the quantum evolution of coherent states. The results are first established for the spatially cutoff model, under standard assumptions ensuring essential self-adjointness of the Hamiltonian and well-posedness of the classical flow, and are then extended to a class of non-polynomial analytic interactions. This work refines and generalizes earlier results, which identified only the leading-order term of the expansion.
Paper Structure (21 sections, 31 theorems, 235 equations)

This paper contains 21 sections, 31 theorems, 235 equations.

Table of Contents

  1. Background and main results
  2. Formalism of Quantum Field Theory
  3. Polynomial Wick quantization
  4. Main results
  5. The spatially cutoff P(φ)₂ model
  6. Introduction of the model
  7. The classical field equation for the $P(\phi)_2$ model
  8. The quadratic dynamics for the $P(\phi)_2$ model
  9. Existence of the unitary propagator
  10. Quadratic propagation of a polynomial observable
  11. Complete asymptotic expansion for the $P(\phi)_2$ model
  12. Differentiation of \ref{['premiere_apparition_Theta(t)']}
  13. Choice of the parameters
  14. End of the proof
  15. Analytic interactions
  16. ...and 6 more sections

Key Result

Theorem 1.1

There exists a probability space $(Q,\mathfrak{T},\mu)$ and a unitary map fulfilling the following conditions: Furthermore, for all $V \in \Gamma_{\mathrm{s}}(\mathcal{Z})$, we have and, for all $u \in \mathcal{Z}_0$, $\varepsilon > 0$,

Theorems & Definitions (52)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Example 1.9
  • Remark 2.1
  • ...and 42 more