The conformal bootstrap is a
non-perturbative
In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. An example is the function
: f(x) = e^,
which does not have a Taylor series at ''x'' = 0. Every coefficient of the Taylor ...
mathematical method to constrain and solve
conformal field theories
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometime ...
, i.e. models of
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
or
statistical physics
Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approxim ...
that exhibit similar properties at different levels of resolution.
Overview
Unlike more traditional techniques of
quantum field theory, conformal bootstrap does not use the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
of the theory. Instead, it operates with the general axiomatic parameters, such as the
scaling dimension
In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is sca ...
s of the local operators and their
operator product expansion
In quantum field theory, the operator product expansion (OPE) is used as an axiom to define the product of fields as a sum over the same fields. As an axiom, it offers a non-perturbative approach to quantum field theory. One example is the vert ...
coefficients. A key axiom is that the product of local operators must be expressible as a sum over local operators (thus turning the product into an
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
); the sum must have a non-zero radius of convergence. This leads to decompositions of correlation functions into structure constants and
conformal blocks.
The main ideas of the conformal bootstrap were formulated in the 1970s by the Soviet physicist
Alexander Polyakov[
] and the Italian physicists
Sergio Ferrara,
Raoul Gatto and
Aurelio Grillo.
Other early pioneers of this idea were
Gerhard Mack and
Ivan Todorov.
In two dimensions, the conformal bootstrap was demonstrated to work in 1983 by
Alexander Belavin
Alexander "Sasha" Abramovich Belavin (russian: Алекса́ндр Абра́мович Бела́вин, born 1942) is a Russian physicist, known for his contributions to string theory.
He is a professor at the Independent University of Moscow a ...
,
Alexander Polyakov and
Alexander Zamolodchikov
Alexander Borisovich Zamolodchikov (russian: Алекса́ндр Бори́сович Замоло́дчиков; born September 18, 1952) is a Russian physicist, known for his contributions to condensed matter physics, two-dimensional conform ...
.
Many
two-dimensional conformal field theories were solved using this method, notably the
minimal models and the
Liouville field theory
In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation.
Liouville theory is defined for all complex values of the ...
.
In higher dimensions, the conformal bootstrap started to develop following the 2008 paper by
Riccardo Rattazzi
Riccardo Rattazzi (born 1964) is an Italian theoretical physicist and a professor at the École Polytechnique Fédérale de Lausanne. His main research interests are in physics beyond the Standard Model and in cosmology.
Career
Riccardo Rattazz ...
,
Slava Rychkov
Vyacheslav Rychkov (called Slava Rychkov, Russian Вячеслав Рычков, transcription Vyacheslav Rychkov; born 27 May 1975 in Samara, Russia ) is a Russian-Italian-French theoretical physicist and mathematician.
Career
In 1996, Rychk ...
,
Erik Tonni and
Alessandro Vichi
Alessandro is both a given name and a surname, the Italian form of the name Alexander. Notable people with the name include:
People with the given name Alessandro
* Alessandro Allori (1535–1607), Italian portrait painter
* Alessandro Baricco ...
.
The method was since used to obtain many general results about conformal and
superconformal field theories in three, four, five and six dimensions. Applied to the conformal field theory describing the
critical point of the three-dimensional
Ising model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
, it produced the most precise predictions for its
critical exponents
Critical or Critically may refer to:
*Critical, or critical but stable, medical states
**Critical, or intensive care medicine
* Critical juncture, a discontinuous change studied in the social sciences.
* Critical Software, a company specializing i ...
.
Current research
The internationa
Simons Collaboration on the Nonperturbative Bootstrapunites researchers devoted to developing and applying the conformal bootstrap and other related techniques in quantum field theory.
History of the name
The modern usage of the term "conformal bootstrap" was introduced in 1984 by Belavin et al.
In the earlier literature, the name was sometimes used to denote a different approach to conformal field theories, nowadays referred to as the
skeleton expansion or the "old bootstrap". This older method is perturbative in nature,
and is not directly related to the conformal bootstrap in the modern sense of the term.
External links
Open problems in conformal bootstrap
References
Conformal field theory
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