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Effective models of strong and electroweak interactions are
extensively applied in particle physics phenomenology, and in many
instances can compete with large-scale numerical simulations of
Standard Model physics. These contexts include but are not limited
to providing indications for phase transitions and the nature of
elementary excitations of strong and electroweak matter. A
precondition for obtaining high-precision predictions is the
application of some advanced functional techniques to the effective
models, where the sensitivity of the results to the accurate choice
of the input parameters is under control and the insensitivity to
the actual choice of ultraviolet regulators is ensured. The
credibility of such attempts ultimately requires a clean
renormalization procedure and an error estimation due to a
necessary truncation in the resummation procedure. In this concise
primer we discuss systematically and in sufficient technical depth
the features of a number of approximate methods, as applied to
various effective models of chiral symmetry breaking in strong
interactions and the BEH-mechanism of symmetry breaking in the
electroweak theory. After introducing the basics of the functional
integral formulation of quantum field theories and the derivation
of different variants of the equations which determine the n-point
functions, the text elaborates on the formulation of the optimized
perturbation theory and the large-N expansion, as applied to the
solution of these underlying equations in vacuum. The optimisation
aspects of the 2PI approximation is discussed. Each of them is
presented as a specific reorganisation of the weak coupling
perturbation theory. The dimensional reduction of high temperature
field theories is discussed from the same viewpoint. The
renormalization program is described for each approach in detail
and particular attention is paid to the appropriate interpretation
of the notion of renormalization in the presence of the Landau
singularity. Finally, results which emerge from the application of
these techniques to the thermodynamics of strong and electroweak
interactions are reviewed in detail.
Field theory, relying on the concept of continuous space and time
while confronted with the quantum physical nature of observable
quantities, still has some fundamental challenges to face. One such
challenge is to understand the emergence of complexity in the
behavior of interacting elementary fields, including among other
things nontrivial phase structures of elementary matter at high
energy density or an atypical emergence of statistical properties,
e.g., when an apparent temperature is proportional to a constant
acceleration in a homogeneous gravitational field. Most modern
textbooks on thermal field theory are mainly concerned with how the
field theory formalism should be used if a finite temperature is
given. In contrast, this short primer explores how the phenomenon
of temperature emerges physically for elementary fields - inquiring
about the underlying kinetic field theory and the way energy
fluctuations and other noise should be handled - and it
investigates whether and how this harmonizes with traditional field
theory concepts like spectral evolution, the Keldysh formalism, and
phase transitions.
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