Combinatorial physics or physical combinatorics is the area of interaction between
physics
Physics is the natural science that studies matter, its 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 which r ...
and
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
.
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 classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
and
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 Mathematics, mathematical tools for dealing with large populations ...
. 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, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vande ...
and
Dirk Kreimer, showing that the
renormalization
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering v ...
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 introduc ...
s can be described by a
Hopf algebra Hopf is a German surname. Notable people with the surname include:
*Eberhard Hopf (1902–1983), Austrian mathematician
*Hans Hopf (1916–1993), German tenor
*Heinz Hopf (1894–1971), German mathematician
*Heinz Hopf (actor) (1934–2001), Swedis ...
.
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
In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems h ...
, the fact that the
Slavnov–Taylor identities
In quantum field theory, a Slavnov–Taylor identity is the non-Abelian generalisation of a Ward–Takahashi identity, which in turn is an identity between correlation functions that follows from the global or gauged symmetries of a theory, and ...
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 (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
generate a Hopf ideal, the
quantization of fields and
strings
String or strings may refer to:
*String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects
Arts, entertainment, and media Films
* ''Strings'' (1991 film), a Canadian anim ...
, 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 The important example of editing combinatorics and physics is relation between enumeration of
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 generalize permutation matrices and a ...
and
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. ...
. Corresponding ice-type model is six vertex model with domain wall boundary conditions.
See also
*
Mathematical physics
Mathematical physics refers to 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 t ...
*
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 Mathematics, mathematical tools for dealing with large populations ...
*
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 ...
*
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
The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory. It is defined for every undirected graph G and contai ...
*
Partition function
*
Hopf algebra Hopf is a German surname. Notable people with the surname include:
*Eberhard Hopf (1902–1983), Austrian mathematician
*Hans Hopf (1916–1993), German tenor
*Heinz Hopf (1894–1971), German mathematician
*Heinz Hopf (actor) (1934–2001), Swedis ...
*
Combinatorics and dynamical systems The mathematical disciplines of combinatorics and dynamical systems interact in a number of ways. The ergodic theory of dynamical systems has recently been used to prove combinatorial theorems about number theory which has given rise to the field of ...
*
Quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
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
H. Pierre Noyes (December 10, 1923 – September 30, 2016) was an American theoretical physicist. He became a member of the faculty at the SLAC National Accelerator Laboratory at Stanford University in 1962. Noyes specialized in several areas o ...
Combinatorial Physics
Ted Bastin
Edward William "Ted" Bastin (8 January 1926 – 15 October 2011) was a physicist and mathematician who held doctorate degrees in mathematics from Queen Mary College, London University and physics from King's College, Cambridge, to which he won an ...
, Clive W. Kilmister
Clive William Kilmister (3 January 1924 – 2 May 2010) was a British mathematician who specialised in the mathematical foundations of physics, especially quantum mechanics and relativity.
Kilmister attended Queen Mary College London for both hi ...
, 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 (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques an ...
, 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.
Currently a professor at the London School of Economics, he has published nearly 100 papers in pure mathematics, including over a dozen with Béla Bollo ...
, Peter Winkler
Graphs, Morphisms, and Statistical Physics: DIMACS Workshop Graphs, Morphisms and Statistical Physics, March 19-21, 2001, DIMACS Center
Jaroslav Nešetřil
Jaroslav (Jarik) Nešetřil (; born March 13, 1946 in Brno) is a Czech mathematician, working at Charles University in Prague. His research areas include combinatorics (structural combinatorics, Ramsey theory), graph theory (coloring problems, s ...
, Peter Winkler
Peter Mann Winkler is a research mathematician, author of more than 125 research papers in mathematics and patent holder in a broad range of applications, ranging from cryptography to marine navigation.Information listed oPeter Winkler's homepagea ...
, 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
*