Combinatorial physics or physical combinatorics is the area of interaction between
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
and
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
.
Overview
:"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
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 ...
and
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 ...
. However, combinatorial physics only emerged as a specific field after a seminal work by
Alain Connes
Alain Connes (; born 1 April 1947) is a French mathematician, known for his contributions to the study of operator algebras and noncommutative geometry. He was a professor at the , , Ohio State University and Vanderbilt University. He was awar ...
and
Dirk Kreimer, showing that the
renormalization
Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
of
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced ...
s can be described by a
Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover ...
.
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
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
generate a Hopf ideal, the
quantization of fields and
strings, and a completely algebraic description of the combinatorics of quantum field theory.
[C. Brouder]
Quantum field theory meets Hopf algebra
''Mathematische Nachrichten
''Mathematische Nachrichten'' (abbreviated ''Math. Nachr.''; English: ''Mathematical News'') is a mathematical journal published in 12 issues per year by Wiley-VCH GmbH. It should not be confused with the ''Internationale Mathematische Nachrichten ...
'' 282 (2009), 1664-1690 An important example of applying combinatorics to physics is the enumeration of
alternating sign matrix in the solution of
ice-type models. The corresponding ice-type model is the six vertex model with domain wall boundary conditions.
See also
*
Mathematical physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
*
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 ...
*
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 ...
*
Percolation theory
In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected ...
*
Tutte polynomial
*
Partition function
*
Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover ...
*
Combinatorics and dynamical systems
*
Quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
References
Further reading
Some Open Problems in Combinatorial Physics G. Duchamp, H. Cheballah
One-parameter groups and combinatorial physics G. Duchamp, K.A. Penson, A.I. Solomon, A.Horzela, P.Blasiak
Combinatorial Physics, Normal Order and Model Feynman Graphs A.I. Solomon, P. Blasiak, G. Duchamp, A. Horzela, K.A. Penson
Hopf Algebras in General and in Combinatorial Physics: a practical introduction G. Duchamp, P. Blasiak, A. Horzela, K.A. Penson, A.I. Solomon
Discrete and Combinatorial PhysicsBit-String Physics: a Novel "Theory of Everything"
H. Pierre Noyes
Combinatorial Physics
Ted Bastin, Clive W. Kilmister, World Scientific, 1995,
Physical Combinatorics and Quasiparticles
Giovanni Feverati, Paul A. Pearce, Nicholas S. Witte
*
Paths, Crystals and Fermionic Formulae
G.Hatayama, A.Kuniba, M.Okado, T.Takagi, Z.Tsuboi
On powers of Stirling matrices
István Mező
*"On cluster expansions in graph theory and physics", N BIGGS — The Quarterly Journal of Mathematics, 1978 - Oxford Univ Press
Enumeration Of Rational Curves Via Torus Actions
Maxim Kontsevich
Maxim Lvovich Kontsevich (, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami. He ...
, 1995
Non-commutative Calculus and Discrete Physics
Louis H. Kauffman, February 1, 2008
Sequential cavity method for computing free energy and surface pressure
David Gamarnik, Dmitriy Katz, July 9, 2008
Combinatorics and statistical physics
*"Graph Theory and Statistical Physics", J.W. Essam, Discrete Mathematics, 1, 83-112 (1971).
Combinatorics In Statistical PhysicsHard Constraints and the Bethe Lattice: Adventures at the Interface of Combinatorics and Statistical Physics
Graham Brightwell
Graham Brightwell is a British mathematician working in the field of discrete mathematics.
A professor at the London School of Economics, he has published nearly 100 papers in pure mathematics, including over a dozen with Béla Bollobás. His r ...
, Peter Winkler
Graphs, Morphisms, and Statistical Physics: DIMACS Workshop Graphs, Morphisms and Statistical Physics, March 19-21, 2001, DIMACS Center
Jaroslav Nešetřil, Peter Winkler, AMS Bookstore, 2001,
Conference proceedings
*Proc. of Combinatorics and Physics, Los Alamos, August 1998
Physics and Combinatorics 1999: Proceedings of the Nagoya 1999 International Workshop
Anatol N. Kirillov, Akihiro Tsuchiya, Hiroshi Umemura, World Scientific, 2001,
Physics and combinatorics 2000: proceedings of the Nagoya 2000 International Workshop
Anatol N. Kirillov, Nadejda Liskova, World Scientific, 2001,
Asymptotic combinatorics with applications to mathematical physics: a European mathematical summer school held at the Euler Institute, St. Petersburg, Russia, July 9-20, 2001
Anatoliĭ, Moiseevich Vershik, Springer, 2002, {{ISBN, 3-540-40312-4
Counting Complexity: An International Workshop On Statistical Mechanics And Combinatorics
10–15 July 2005, Dunk Island, Queensland, Australia
*Proceedings of the Conference on Combinatorics and Physics, MPIM Bonn, March 19–23, 2007
*
Quantum mechanics
*