[Paper Review] On the derivation of the Boltzmann equation in quantum field theory: Flat spacetime

Stefan Hollands, Gregor Leiler|arXiv (Cornell University)|Mar 8, 2010

Noncommutative and Quantum Gravity Theories28 references3 citations

TL;DR

This paper provides a mathematically rigorous derivation of the Boltzmann equation in quantum field theory within flat spacetime, using a hermitian scalar field with polynomial self-interaction in 2D Minkowski space. It derives a non-perturbative pre-Boltzmann equation and shows that, in the long-time, low-density limit, the resulting equation resembles the Boltzmann equation but includes additional rescattering terms that must be retained for consistency when loop corrections are included.

ABSTRACT

In this paper, we analyze in a mathematically rigorous fashion the validity of the Boltzmann transport equation within quantum field theory. We work within the specific model of a hermitian, scalar field with polynomial self-interaction in two-dimensional Minkowski space. Our main results are as follows: Firstly, that one can obtain a non-perturbative, exact integro-differential equation for the number densities, which we called the pre-Boltzmann equation. We secondly take the long-time-dilute-medium limit of this equation, to obtain a simpler equation. This limiting equation is qualitatively similar to the Boltzmann equation, but it involves additional re-scattering terms. These terms disappear if perform a perturbation expansion in the coupling constant and ignore the loop corrections (Born approximation). If loop corrections are included, then we argue that for consistency, one must also keep corresponding rescattering terms which are normally ignored. Our analysis is hence of potential relevance for physical applications of the Boltzmann equation wherein loop effects are essential, such as in the standard scenario of baryogensis in the Early Universe. Our analysis is performed in the context of flat spacetime, but in such a way that the main ingredients can be transferred, straightforwardly to the case of a curved spacetime of Robertson-Walker-type. Our main technical tools are methods from constructive quantum field theory, as well as a general method called "projection technique". This turns out to give convergent expansions, and a rather elegant way of organizing the combinatorics of the various quantum field theoretic expansions in the analysis.

Motivation & Objective

• To rigorously derive the Boltzmann equation from quantum field theory in flat spacetime, addressing foundational assumptions in its standard heuristic derivation.
• To identify and analyze the role of rescattering terms that arise in the long-time, low-density limit when loop corrections are included.
• To clarify the conditions under which the Boltzmann equation is valid in quantum field theory, particularly in scenarios involving loop effects such as baryogenesis.
• To develop a framework applicable to curved spacetime, especially Robertson-Walker backgrounds, by generalizing the projection technique and number density definitions.
• To demonstrate that consistent inclusion of loop corrections necessitates corresponding rescattering terms, which are typically omitted in the Born approximation.

Proposed method

• Employing the projection technique from constructive quantum field theory to systematically organize perturbative and non-perturbative expansions in the coupling constant.
• Deriving a non-perturbative integro-differential equation for number densities, termed the pre-Boltzmann equation, valid for the φ^p-model in 2D Minkowski space.
• Taking the long-time and low-density limit of the pre-Boltzmann equation, parameterized by a small scaling parameter ε, to extract the leading-order effective dynamics.
• Using a modified S-matrix approach that accounts for medium effects on scattering amplitudes, including self-energy corrections and dispersion shifts.
• Applying a WKB-type adiabatic approximation to mode functions to extend the formalism to curved spacetime, particularly Robertson-Walker cosmologies.
• Analyzing the structure of loop corrections and their interplay with rescattering terms, showing that neglecting the latter breaks consistency when loops are included.

Experimental results

Research questions

• RQ1 Under what conditions can the Boltzmann equation be rigorously derived from quantum field theory in flat spacetime?
• RQ2 What is the role of rescattering terms in the effective dynamics when loop corrections are included in the scattering matrix elements?
• RQ3 How does the inclusion of loop corrections affect the validity of the Born approximation in the Boltzmann equation?
• RQ4 Can the formalism be generalized to curved spacetime, particularly in expanding cosmological backgrounds?
• RQ5 To what extent do medium effects, such as modified dispersion relations, emerge from the self-consistent solution of the pre-Boltzmann equation?

Key findings

• A non-perturbative, exact pre-Boltzmann equation is derived for the φ^p-model in two-dimensional Minkowski space, capturing all quantum field theoretic effects in the number density evolution.
• In the long-time, low-density limit, the resulting equation qualitatively resembles the Boltzmann equation but includes additional rescattering terms that are absent in the standard derivation.
• These rescattering terms vanish in the Born approximation but must be retained when loop corrections are included, ensuring consistency in the effective dynamics.
• The analysis shows that loop corrections in the matrix elements cannot be consistently treated without corresponding rescattering contributions, challenging the standard practice of neglecting them.
• The framework is extendable to curved spacetime, particularly Robertson-Walker cosmologies, by adapting the projection technique and defining number densities via adiabatic mode functions.
• The formalism predicts a self-consistent modification of the dispersion relation, E(p) = ω_p + O(ε), reflecting medium effects in non-dilute regimes, which are typically ignored in the standard Boltzmann approach.