Combinatorics and physics - AbsoluteAstronomy.com

Combinatorial physics or physical combinatorics is the area of interaction between physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

and combinatorics

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

.

"Combinatorial Physics is an emerging area which unites combinatorial and discrete mathematical techniques applied to theoretical physics, especially Quantum Theory.".

"Physical combinatorics might be defined naively as combinatorics guided by ideas or insights from physics".

Combinatorics has always played an important role in
quantum field theory

Quantum field theory

Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

and statistical physics

Statistical physics

Statistical physics is the branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic...

.
However, combinatorial physics only emerged as a
specific field after a seminal work by
Alain Connes

Alain Connes

Alain Connes is a French mathematician, currently Professor at the Collège de France, IHÉS, The Ohio State University and Vanderbilt University.-Work:...

and Dirk Kreimer

Dirk Kreimer

Dirk Kreimer is a German physicist who pioneered the Hopf-algebraic approach to quantum field theory with Alain Connes and other co-authors. He is currently affiliated with IHES and Boston University.-External links:*...

,
showing that the renormalization

Renormalization

In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....

of Feynman diagram

Feynman diagram

Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...

s can be described by a Hopf algebra

Hopf algebra

In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...

.

Combinatorial physics can be characterized by the
use of algebraic concepts to interpret and solve
physical problems involving combinatorics.
It gives rise to a particularly
harmonious collaboration between mathematicians and
physicists.

Among the significant physical results of combinatorial physics
we may mention the reinterpretation of renormalization
as a Riemann-Hilbert problem,
the fact that the Slavnov–Taylor identities of gauge theories

Gauge theory

In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

generate a Hopf ideal,
the quantization of fields and strings

String theory

String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...

and a completely algebraic description of the
combinatorics of quantum field theory. The important example of editing combinatorics and physics is relation between enumeration of Alternating sign matrix

Alternating sign matrix

In mathematics, an alternating sign matrix is a square matrix of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These matrices arise naturally when using Dodgson condensation to compute a determinant...

and Ice-type model

Ice-type model

In statistical mechanics, the ice-type models or six-vertex models are a family of vertex models for crystal lattices with hydrogen bonds. The first such model was introduced by Linus Pauling in 1935 to account for the residual entropy of water ice. Variants have been proposed as models of certain...

. Corresponding ice-type model is six vertex model with domain wall boundary conditions.

See also

Further reading

Combinatorics and statistical physics

Conference proceedings