The conformal bootstrap is a
non-perturbative mathematical method to constrain and solve
conformal field theories, 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) and ...
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 approxi ...
that exhibit similar properties at different levels of resolution.
Overview
Unlike more traditional techniques of
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
, 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 dimensions 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 verte ...
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 areas of mathematics, 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 mathem ...
); 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
Sergio Ferrara (born May 2, 1945) is an Italian physicist working on theoretical physics of elementary particles and mathematical physics. He is renowned for the discovery of theories introducing supersymmetry as a symmetry of elementary particles ...
,
Raoul Gatto and
Aurelio Grillo Aurelio may refer to:
People Politicians
*Aurelio D. Gonzales Jr. (born 1964), congressman in the Philippines
*Aurélio de Lira Tavares (1905–1998), President of Brazil
*Aurelio Martínez, Honduran politician
*Aurelio Mosquera (1883–1939), Pre ...
.
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 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 conforma ...
.
Many
two-dimensional conformal field theories were solved using this method, notably the
minimal models and the
Liouville field theory.
In higher dimensions, the conformal bootstrap started to develop following the 2008 paper by
Riccardo Rattazzi,
Slava Rychkov,
Erik Tonni and
Alessandro Vichi.
The method was since used to obtain many general results about conformal and
superconformal
In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, supercon ...
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.
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
A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...
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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