Abstract
In modern physics, we are increasingly concerned with the dynamics of dense systems of ultra-relativistic particles—the density frontier.
Keywords
- Particle Number Density
- Boltzmann Transport Equation
- Gradient Expansion
- Thermal Field Theory
- Closed Time Path
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In modern physics, we are increasingly concerned with the dynamics of dense systems of ultra-relativistic particles—the density frontier. These quantum-field-theoretic many-body systems are encountered across a number of disciplines, including high-energy particle physics, astro-particle physics, cosmology, nuclear physics, plasma physics and condensed matter physics. For this reason, it is pertinent to study the theoretical frameworks that allow us to make predictions about the dynamics of these systems.
At the interface of high-energy and nuclear physics, one such system is the deconfined phase of quantum chromo dynamics (QCD), known as the quark gluon plasma (QGP) [1, 2]. The existence of this exotic phase of matter has been inferred from the observation of jet quenching (see for instance [3]) in Pb-Pb collisions by the ATLAS [4], CMS [5] and ALICE [6] experiments at CERN’s Large Hadron Collider (LHC) in Geneva.
In astro-particle physics, we are interested in the internal structure of compact objects such as neutron stars [7]. In addition, aspects of these theoretical frameworks are relevant to the description of the thermal states seen by accelerating observers via the Unruh effect [8, 9]. In recent years, there has also been increasing interest in the study of holographic thermalization in strongly coupled plasmas [10, 11] as well as applications of the AdS/CFT correspondence to obtain thermal propagators [12], and to study thermal noise [13] and quantum quenches [14].
In cosmology, predictions about the evolution of the early universe rely upon our understanding of the dynamics of the exotic states of matter that were present in the high energy densities that followed the Big Bang. The Wilkinson Microwave Anisotropy Probe (WMAP) [15, 16] measured a baryon-to-photon ratio at the present epoch of \(\eta =n_{B}/n_{\gamma }=6.116^{+0.197}_{-0.249}\times 10^{-10}\), where \(n_{B}=n_b-n_{\bar{b}}\) is the difference in the number densities of baryons and anti-baryons. This observed Baryon Asymmetry of the Universe (BAU)—the asymmetry between the present-day numbers of baryons and anti-baryons—is consistent with the predictions of Big-Bang Nucleosynthesis (BBN) [17]. The generation of this asymmetry requires the presence in the early universe of out-of-equilibrium processes and the violation of baryon number (\(B\)), charge (\(C\)) and charge-parity (\({ CP}\)). These are the so-called Sakharov conditions [18]. One such set of processes is prescribed by the baryogenesis via leptogenesis scenario [19–21] in which an initial excess in lepton number (\(L\)), provided by the decay of heavy right-handed Majorana neutrinos, is converted to a baryon number excess through the \(B\,+\,L\)-violating sphaleron interactions [22] of the Weinberg–Salam electroweak theory [23, 24]. The description of these phenomena requires a consistent approach to the non-equilibrium dynamics of particle number densities. Such a treatment may also be relevant to reheating and preheating [25–27] at the end of the inflationary epoch and the generation of dark matter relic densities [28, 29].
The classical evolution of particle number densities is described by the Boltzmann transport equation, see for instance [30–40]. Semi-classical approaches are achieved by substituting the classical Boltzmann distributions with quantum-statistical Bose–Einstein or Fermi–Dirac distribution functions, in the case of bosons and fermions, respectively. These semi-classical approaches have been successfully employed to model a variety of physical phenomena. However, in such approaches, finite-width and off-shell effects, which would be present in a complete quantum-field-theoretic formulation, must be introduced in an effective manner.
The first framework for calculating ensemble expectation values (EEVs) of field operators was provided by Matsubara [41] in the so-called imaginary time formalism (ITF) of thermal field theory [42]. This formalism relies upon the interpretation of the canonical density operator as an evolution operator in negative imaginary time. Consistent real-time Green’s functions may then be obtained by appropriate, but subtle, analytic continuation. The ITF, however, remains limited to the description of processes occurring at thermodynamic equilibrium.
The calculation of EEVs directly in real time is achieved using so-called real time formalisms [43, 44]. One such formulation is provided by thermo field dynamics, which is based upon \(c^*\)-algebras and the construction of coherent thermal states by the doubling of physical degrees of freedom [45–48]. For non-equilibrium systems, one generally uses the closed time path (CTP) [49–51] or in-in formalism due to Schwinger and Keldysh. A consistent non-perturbative expansion of the in-in generating functional is provided by the Cornwall–Jackiw–Tomboulis (CJT) effective action [52, 53], as applied to the CTP formalism by Calzetta and Hu [54]. The equivalence of the real and imaginary time formalisms in thermodynamic equilibrium has been discussed at length in the literature [55–62].
The CTP formalism has been used extensively in the derivation so-called quantum-corrected or quantum Boltzmann equations [63–94]. These transport equations are obtained from systems of Kadanoff–Baym equations [95, 96], which were originally applied in the non-relativistic regime [97, 98]. In comparison to semi-classical approaches, these quantum transport equations have the advantage that finite-width and off-shell effects are fully incorporated  [99]. Such equations have also been discussed in the context of flavour mixing [100–102] and have recently been applied to the glasma [103]. However, existing approaches often rely upon the truncation of a gradient expansion [104] in time derivatives in order to obtain tractable expressions. In this case, one necessarily makes assumptions about various time-scales in the evolution and, in general, we must throw away the fast quantum-mechanical behaviour. In addition, one must often make quasi-particle ansaetze for the forms of the propagators in order to extract meaningful observables. Other authors propose approaches based on functional renormalization techniques [105] or directly on expansions of the von Neumann (or quantum Liouville equation) [106, 107]. In [108], the authors discuss an alternative approximation scheme based on the Wentzel–Kramers–Brillouin (WKB) method.
Comprehensive introductions to thermal field theory may be found in [109–119]. The context of non-equilibrium dynamics is discussed in [120–122]. For applications to gauge theories, relevant to the description of QCD plasmas, the reader is directed to [123]. Lastly, an introduction to applications in condensed matter physics is provided in [124].
In this thesis, we develop a new perturbative approach to non-equilibrium thermal quantum field theory, applicable to the description of the out-of-equilibrium dynamics of ultra-relativistic many-body systems. We derive non-homogeneous free propagators and manifestly time-dependent Feynman rules, yielding systematic diagrammatic expansions in which space-time translational invariance is explicitly broken. We show that these perturbation series are free of the mathematically ill-defined pinch singularities, previously thought to spoil such perturbative approaches to non-equilibrium field theory. We show that the absence of these pinch singularities is ensured by the systematic treatment of finite-time effects and the dependence upon the time of observation. After arriving at a physically meaningful definition of the particle number density, which does not require quasi-particle ansaetze, we derive master time evolution equations for statistical distribution functions. These transport equations are valid to all orders in perturbation theory and to all orders in a gradient expansion. However, having established a well-defined underlying perturbation theory, these master time evolution equations may be truncated in a perturbative loopwise sense, whilst retaining all orders in the time behaviour. As a result, in this new formalism, we do not need to rely on the separation of time scales or the truncation of a gradient expansion in order to obtain tractable expressions. Moreover, we argue that the truncation of such a gradient expansion is inappropriate for the early-time evolution of these non-equilibrium systems.
Finally, we study the thermalization of a simple scalar model. We show that the systematic incorporation of finite-time effects and the consistent treatment of generalized decay kinematics lead to the appearance of processes that would otherwise be kinematically disallowed. These evanescent processes result from the microscopic violation of energy conservation at early times. This energy violation is a consequence of Heisenberg’s uncertainty principle, since, for non-equilibrium systems, we necessarily make observations over a finite interval of time. Moreover, we show that the evanescent regime is fundamental to the statistical evolution of these systems. In addition, we investigate the spectral evolution of the width of a heavy real scalar field. This width is shown to exhibit oscillations with time-dependent frequencies. This non-Markovian evolution of so-called memory effects is expected in truly out-of-equilibrium dynamics.
Throughout our exposition, we highlight the correspondence of our approach with known results in the thermodynamic equilibrium limit. In addition, we discuss the compatibility with the semi-classical Boltzmann transport equation and the truncated gradient expansion of the Kadanoff–Baym equations in the Markovian energy-conserving limit.
This thesis is divided into two parts. Part I contains a review of material pertinent to the subsequent formulation of our approach to non-equilibrium quantum field theory. We revisit the central ideas of classical mechanics, thermodynamics and statistical mechanics; their first quantization in the form of quantum statistical mechanics; and finally their second quantization, with reference to the approaches of equilibrium thermal field theory. In Part II, we develop the perturbative formulation of thermal quantum field theory that is the original contribution of this work.
The content of this thesis is briefly outlined as follows; more detailed summaries are provided in the introductions to Parts I and II and at the start of each chapter where appropriate. In Chap. 3, we set out from Hamilton’s principle and the resulting formulation of classical mechanics. Following a brief review of key ideas from thermodynamics and statistical mechanics, we arrive at the classical Boltzmann transport equation, the quantum-field-theoretic analogue of which is the aim of this thesis. Moving to quantum mechanics, we proceed in Chap. 4 to a review of quantum statistical mechanics, with particular reference to the quantum harmonic oscillator. Here, we introduce many of the formal ideas required for our later discussions. Subsequently, in Chap. 5, we introduce the path-integral representation of the quantum harmonic oscillator and derive the form of its propagators. In Chap. 6, we outline pertinent details of the ITF and introduce the thermal propagators ubiquitous to thermal field theory. Finally, in Chap. 7, we highlight relevant details of the quantum field theory of real and complex scalar fields.
Part II sets out from a description of the CTP formalism in the zero-temperature limit, provided in Chap. 9. In Chap. 10, we outline the generalization to non-homogeneous backgrounds and derive the non-homogeneous free propagators of our perturbative formulation. As a check of the consistency of our approach, Chap. 11 describes the thermodynamic equilibrium limit and the correspondence with known results from the ITF. In Chap. 12, we describe the potential origin of so-called pinch singularities and show that these mathematically ill-defined terms are not present in this new approach. Subsequently, in Chap. 13, we arrive at a physically meaningful definition of the number density of particles, based on the Noether charge. With this definition, we proceed in Chap. 14 to derive master time evolution equations for statistical distribution functions. In Chap. 15, we discuss the techniques required to compute loop integrals that are built from the non-homogeneous Feynman rules of this perturbative formalism. We then apply the results of the previous chapters in the context of the thermalization of a simple scalar model in Chap. 16, highlighting the merits of our approach. Finally, in Chap. 17, we conclude our discussions.
A number of appendices are also included at the end of this thesis. Appendix A lists the properties of and relations between the various propagators used throughout our analysis. In Appendix B, we illustrate the form of the most general Guassian-like non-homogeneous density operator. For comparison with the results of Chap. 14, we derive the familiar form of the Kadanoff–Baym equations in Appendix C. Lastly, in Appendix D, we give a brief overview of the Monte Carlo integration used to perform the numerical analysis in this thesis.
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Millington, P. (2014). Introduction. In: Thermal Quantum Field Theory and Perturbative Non-Equilibrium Dynamics. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-01186-8_1
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