statistical field theory: Statistical Field Theory.jpg

Preface
This book is an introduction to statistical field theory, an important subject of
theoretical physics that has undergone formidable progress in recent years. Most of
the attractiveness of this field comes from its profound interdisciplinary nature and its
mathematical elegance; it sets outstanding challenges in several scientific areas, such
as statistical mechanics, quantum field theory, and mathematical physics.
Statistical field theory deals, in short, with the behavior of classical or quantum
systems consisting of an enormous number of degrees of freedom. Those systems have
different phases, and the rich spectrum of the phenomena they give rise to introduces
several questions: What is their ground state in each phase? What is the nature of
the phase transitions? What is the spectrum of the excitations? Can we compute
the correlation functions of their order parameters? Can we estimate their finite size
effects? An ideal guide to the fascinating area of phase transitions is provided by a
remarkable model, the Ising model.
There are several reasons to choose the Ising model as a pathfinder in the field of
critical phenomena. The first one is its simplicity – an essential quality to illustrate
the key physical features of the phase transitions, without masking their derivation
with worthless technical details. In the Ising model, the degrees of freedom are simple
boolean variables σi, whose values are σi = ±1, defined on the sitesi of a d-dimensional
lattice. For these essential features, the Ising model has always played an important
role in statistical physics, both at the pedagogical and methodological levels.
However, this is not the only reason of our choice. The simplicity of the Ising
model is, in fact, quite deceptive. Despite its apparent innocent look, the Ising model
has shown an extraordinary ability to describe several physical situations and has a
remarkable theoretical richness. For instance, the detailed analysis of its properties involves
several branches of mathematics, quite distinguished for their elegance: here we
mention only combinatoric analysis, functions of complex variables, elliptic functions,
the theory of nonlinear differential and integral equations, the theory of the Fredholm
determinant and, finally, the subject of infinite dimensional algebras. Although this is
only a partial list, it is sufficient to prove that the Ising model is an ideal playground
for several areas of pure and applied mathematics.
Equally rich is its range of physical aspects. Therefore, its study offers the possibility
to acquire a rather general comprehension of phase transitions. It is time to
say a few words about them: phase transitions are remarkable collective phenomena,
characterized by sharp and discontinous changes of the physical properties of a statistical
system. Such discontinuities typically occur at particular values of the external
parameters (temperature or pressure, for instance); close to these critical values, there
is a divergence of the mean values of many thermodynamical quantities, accompanied
by anomalous fluctuations and power law behavior of correlation functions. From an
experimental point of view, phase transitions have an extremely rich phenomenology,
ranging from the superfluidity of certain materials to the superconductivity of others,
viii Preface
from the mesomorphic transformations of liquid crystals to the magnetic properties of
iron. Liquid helium He4, for instance, shows exceptional superfluid properties at temperatures
lower than Tc = 2.19K, while several alloys show phase transitions equally
remarkable, with an abrupt vanishing of the electrical resistance for very low values
of the temperature.
The aim of the theory of phase transitions is to reach a general understanding of
all the phenomena mentioned above on the basis of a few physical principles. Such
a theoretical synthesis is made possible by a fundamental aspect of critical phenomena:
their universality. This is a crucial property that depends on two basic features:
the internal symmetry of the order parameters and the dimensionality of the lattice.
In short, this means that despite the differences that two systems may have at their
microscopic level, as long as they share the two features mentioned above, their critical
behaviors are surprisingly identical.1 It is for these universal aspects that the theory
of phase transitions is one of the pillars of statistical mechanics and, simultaneously,
of theoretical physics. As a matter of fact, it embraces concepts and ideas that have
proved to be the building blocks of the modern understanding of the fundamental
interactions in Nature. Their universal behavior, for instance, has its natural demonstration
within the general ideas of the renormalization group, while the existence
itself of a phase transition can be interpreted as a spontaneously symmetry breaking
of the hamiltonian of the system. As is well known, both are common concepts in
another important area of theoretical physics: quantum field theory (QFT), i.e. the
theory that deals with the fundamental interactions of the smallest constituents of the
matter, the elementary particles.
The relationship between two theories that describe such different phenomena may
appear, at first sight, quite surprising. However, as we will see, it will become more
comprehensible if one takes into account two aspects: the first one is that both theories
deal with systems of infinite degrees of freedom; the second is that, close to the phase
transitions, the excitations of the systems have the same dispersion relations as the
elementary particles.2 Due to the essential identity of the two theories, one should
not be surprised to discover that the two-dimensional Ising model, at temperature T
slightly away from Tc and in the absence of an external magnetic field, is equivalent
to a fermionic neutral particle (a Majorana fermion) that satisfies a Dirac equation.
Similarly, at T = Tc but in the presence of an external magnetic field B, the twodimensional
Ising model may be regarded as a quantum field theory with eight scalar
particles of different masses.
The use of quantum field theory – i.e. those formalisms and methods that led to
brilliant results in the study of the fundamental interactions of photons, electrons, and
all other elementary particles – has produced remarkable progress both in the understanding
of phase transitions and in the computation of their universal quantities. As
will be explained in this book, our study will significantly benefit from such a possibility:
since phase transitions are phenomena that involve the long distance scales of
1This becomes evident by choosing an appropriate combination of the thermodynamical variables
of the two systems.
2The explicit identification between the two theories can be proved by adopting for both the path
integral formalism.
Preface ix
the systems – the infrared scales – the adoption of the continuum formalism of field
theory is not only extremely advantageous from a mathematical point of view but also
perfectly justified from a physical point of view. By adopting the QFT approach, the
discrete structure of the original statistical models shows itself only through an ultraviolet
microscopic scale, related to the lattice spacing. However, it is worth pointing
out that this scale is absolutely necessary to regularize the ultraviolet divergencies of
quantum field theory and to implement its renormalization.
The main advantage of QFT is that it embodies a strong set of constraints coming
from the compatibility of quantum mechanics with special relativity. This turns into
general relations, such as the completeness of the multiparticle states or the unitarity of
their scattering processes. Thanks to these general properties, QFT makes it possible to
understand, in a very simple and direct way, the underlying aspects of phase transitions
that may appear mysterious, or at least not evident, in the discrete formulation of the
corresponding statistical model.
There is one subject that has particularly improved thanks to this continuum
formulation: this is the set of two-dimensional statistical models, for which one can
achieve a classification of the fixed points and a detailed characterization of their classes
of universality. Let us briefly discuss the nature of the two-dimensional quantum field
theories.
Right at the critical points, the QFTs are massless. Such theories are invariant
under the conformal group, i.e. the set of geometrical transformations that implement
a scaling of the length of the vectors while preserving their relative angle. But, in two
dimensions conformal transformations coincide with mappings by analytic functions
of a complex variable, characterized by an infinite-dimensional algebra known as a
Virasoro algebra. This enables us to identify first the operator content of the models
(in terms of the irreducible representations of the Virasoro algebra) and then to
determine the exact expressions of the correlators (by solving certain linear differential
equations). In recent years, thanks to the methods of conformal field theory, physicists
have reached the exact solutions of a huge number of interacting quantum theories,
with the determination of all their physical quantities, such as anomalous dimensions,
critical exponents, structure constants of the operator product expansions, correlation
functions, partition functions, etc.
Away from criticality, quantum field theories are, instead, generally massive. Their
analysis can often be carried out only by perturbative approaches. However, there are
some favorable cases that give rise to integrable models of great physical relevance.
The integrable models are characterized by the existence of an infinite number of conserved
charges. In such fortunate circumstances, the exact solution of the off-critical
models can be achieved by means of S-matrix theory. This approach makes it possible
to compute the exact spectrum of the excitations and the matrix elements of the
operators on the set of these asymptotic states. Both these data can thus be employed
to compute the correlation functions by spectral series. These expressions enjoy remarkable
convergence properties that turn out to be particularly useful for the control
of their behaviors both at large and short distances. Finally, in the integrable cases,
it is also possible to study the exact thermodynamical properties and the finite size
effects of the quantum field theories. Exact predictions for many universal quantities
x Preface
can also be obtained. For the two-dimensional Ising model, for instance, there are
two distinct integrable theories, one corresponding to its thermal perturbation (i.e.
T = Tc, B = 0), the other to the magnetic deformation (B = 0, T = Tc). In the last
case, a universal quantity is given, for instance, by the ratio of the masses of the lowest
excitations, expressed by the famous golden ratio m2/m1 = 2 cos(π/5) = (√5 + 1)/2.
In addition to their notable properties, the exact solution provided by the integrable
theories is an important step towards the general study of the scaling region close to
the critical points. In fact, they permit an efficient perturbative scheme to study nonintegrable
effects, in particular to follow how the mass spectrum changes by varying
the coupling constants. Thanks to this approach, new progress has been made in
understanding several statistical models, in particular the class of universality of the
Ising model by varying the temperature T and the magnetic field B. Non-integrable
field theories present an extremely interesting set of new physical phenomena, such
as confinement of topological excitations, decay processes of the heavier particles, the
presence of resonances in scattering processes, or false vacuum decay, etc. The analytic
control of such phenomena is one of the most interesting results of quantum field theory
in the realm of statistical physics.
This book is a long and detailed journey through several fields of physics and
mathematics. It is based on an elaboration of the lecture notes for a PhD course,
given by the author at the International School for Advances Studies (Trieste). During
this elaboration process, particular attention has been paid to achieving a coherent
and complete picture of all surveyed topics. The effort done to emphasize the deep
relations among several areas of physics and mathematics reflects the profound belief
of the author in the substantial unity of scientific knowledge.
This book is designed for students in physics or mathematics (at the graduate
level or in the last year of their undergraduate courses). For this reason, its style is
greatly pedagogical; it assumes only some basis of mathematics, statistical physics,
and quantum mechanics. Nevertheless, we count on the intellectual curiosity of the
reader.

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感谢楼主分享

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