The conformal bootstrap is a
non-perturbative
In mathematics and physics, a non-perturbative function (mathematics), function or process is one that cannot be described by perturbation theory. An example is the function
: f(x) = e^,
which does not equal its own Taylor series in any neighbo ...
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 Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
or
statistical physics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
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 Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, 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 scal ...
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 vertex ...
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 a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
); 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 physicists
Alexander Polyakov[
] and
Alexander Migdal and the Italian physicists
Sergio Ferrara
Sergio Ferrara (born 2 May 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 ( ...
, and
Aurelio Grillo.
Other early pioneers of this idea were and .
In two dimensions, the conformal bootstrap was demonstrated to work in 1983 by
Alexander Belavin,
Alexander Polyakov and
Alexander Zamolodchikov
Alexander Borisovich Zamolodchikov (; born September 18, 1952) is a Russian-American theoretical physicist, known for his contributions to conformal field theory, statistical mechanics, string theory and condensed matter physics.
He is widel ...
.
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 th ...
.
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
Rattazzi studi ...
,
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
The given name Eric, Erich, Erikk, Erik, Erick, Eirik, or Eiríkur is derived from the Old Norse name ''Eiríkr'' (or ''Eríkr'' in Old East Norse due to monophthongization).
The first element, ''ei-'' may be derived from the older Proto-Nor ...
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 Baric ...
.
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), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that r ...
, it produced the most precise predictions for its
critical exponents
Critical exponents describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on some of its g ...
.
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
{{quantum-stub