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Matrix representing the effect of scattering on a physical system
In physics, the S-matrix or scattering matrix is a matrix that relates the initial state and the final state of a physical system undergoing a scattering
S-matrix
Precursor physical model to string theory and quantum chromodynamics
replacing it with abstract mathematical properties of the S-matrix. In S-matrix theory, the S-matrix relates the infinite past to the infinite future in one
S-matrix_theory
Square matrix used to represent a graph or network
computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices
Adjacency_matrix
Equation used in quantum scattering problems
\phi } 's if one uses the S-matrix to swap the two ψ {\displaystyle \psi } 's. Identifying the coefficients of the ϕ {\displaystyle \phi } 's on both
Lippmann–Schwinger_equation
Array of numbers
In mathematics, a matrix (pl.: matrices) is a rectangular array of numbers or other mathematical objects with elements or entries arranged in rows and
Matrix_(mathematics)
Matrix defined using smaller matrices called blocks
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices
Block_matrix
1999 film by the Wachowskis
influences include Plato's cave, 1990's Telnet hacker communities, and William Gibson's cyberpunk novel Neuromancer. The Matrix popularized terms such as the
The_Matrix
Expectation value of time-ordered quantum operators
theory where they can be used to calculate various observables such as S-matrix elements, although they are not themselves observables. This is because
Correlation function (quantum field theory)
Correlation_function_(quantum_field_theory)
Mathematical operation in linear algebra
columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number
Matrix_multiplication
Measure of covariance of components of a random vector
covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the
Covariance_matrix
American media franchise
The Matrix is an American cyberpunk media franchise consisting of four feature films, beginning with The Matrix (1999) and continuing with three sequels
The_Matrix_(franchise)
outgrowth of S-matrix theory, a research program begun by Werner Heisenberg in 1943 following John Archibald Wheeler's 1937 introduction of the S-matrix. Many
History_of_string_theory
2021 film by Lana Wachowski
The Matrix Resurrections is a 2021 American science fiction action film co-produced and directed by Lana Wachowski, who co-wrote the screenplay with David
The_Matrix_Resurrections
Weisskopf, and others. Related theories are U-matrix, S-matrix, by M-matrix, or T-matrix. The term R-matrix is used in connection with the Yang–Baxter equation
R-matrix
Pictorial representation of the behavior of subatomic particles
particles or fields. The transition amplitude is then given as the matrix element of the S-matrix between the initial and final states of the quantum system.
Feynman_diagram
Requirement that quantum states' time evolution operators are unitary transformations
implies the S-matrix is 1, no interaction occur and all states remain unchanged. Unitarity of the S-matrix: S † S = 1 {\displaystyle S^{\dagger }S=1} is then
Unitarity
Square matrix in linear algebra
In linear algebra, a completely-S matrix is a square matrix such that for every principal submatrix R there exists a positive vector u such that Ru >
Completely-S_matrix
Quantum field theory of electromagnetism
have S f i = ⟨ f | U | i ⟩ . {\displaystyle S_{fi}=\langle f|U|i\rangle .} This technique is also known as the S-matrix, and S f i {\displaystyle S_{fi}}
Quantum_electrodynamics
Representation of a matrix as a product
algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions;
Matrix_decomposition
Fictional character in The Matrix
character in The Matrix franchise. He is portrayed by Laurence Fishburne in the first three films, and in the video game The Matrix: Path of Neo, where
Morpheus_(The_Matrix)
Physics theorem
first formulated by Steven Weinberg in 1965, allows calculation of the S-matrix, used in calculating the outcome of collisions between particles, when
Soft_graviton_theorem
For a square matrix, the transpose of the cofactor matrix
classical adjoint adj(A) of a square matrix A is the transpose of its cofactor matrix. It is occasionally known as adjunct matrix, or "adjoint", though that normally
Adjugate_matrix
Norm on a vector space of matrices
such norms are referred to as matrix norms. Matrix norms behave in certain ways like the distance from the zero matrix. They are distinguished from the
Matrix_norm
Probability of a given process occurring in a particle collision
in terms of the S-matrix. Here δ is the Dirac delta function. The computation of the S-matrix is the main goal of the scattering theory
Cross_section_(physics)
Range of physical processes in physics
scattering of fundamental particles is described by the Scattering Matrix or S-Matrix, introduced and developed by John Archibald Wheeler and Werner Heisenberg
Scattering
Topics referred to by the same term
also be referred to as an S-matrix in the context of radio frequency transmissions. Scattering parameters as S-matrix or S-parameters for how an electrical
S_(disambiguation)
Locality condition in quantum field theory
counter-example against this being true at all energy scales. The S-matrix S β α {\displaystyle S_{\beta \alpha }} describes the scattering amplitude of a process
Cluster_decomposition
Matrix representing a Euclidean rotation
rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix R = [
Rotation_matrix
American theoretical physicist (1911–2008)
Resonating Group Structure", Wheeler introduced the S-matrix—short for scattering matrix—"a unitary matrix of coefficients connecting the asymptotic behavior
John_Archibald_Wheeler
2003 film by the Wachowskis
The Matrix Revolutions is a 2003 American science fiction action film written and directed by the Wachowskis. It is the third film in The Matrix film series
The_Matrix_Revolutions
Theory in particle physics
theory from assumptions on the spectrum of particles. It is a form of S-matrix theory. The term is derived from bootstrapping as in 'pulling oneself up
Bootstrap_model
American physicist
his time at Berkeley and the Harvard Society of Fellows, he worked on S-matrix theory, the determination of the quark–gluon coupling constant, and parity
Ramamurti_Shankar
Equations for correlation functions in QFT
tower of SDEs. In his paper "The S-Matrix in Quantum electrodynamics", Dyson derived relations between different S-matrix elements, or more specific "one-particle
Schwinger–Dyson_equation
Matrix with a multiplicative inverse
algebra, an invertible matrix (non-singular, non-degenerate or regular) is a square matrix that has an inverse. In other words, if a matrix is invertible, it
Invertible_matrix
Algebra used in certain conformal field theories
terms of a sum of products of S {\displaystyle S} -matrix entries and entries of the inverse of the S {\displaystyle S} -matrix, normalized by quantum dimensions
Verlinde_algebra
Soviet theoretical physicist (1908–1968)
electrodynamics, the two-component theory of neutrinos, and Landau's equations for S-matrix singularities. He received the 1962 Nobel Prize in Physics for his development
Lev_Landau
American theoretical physicist (1924–2019)
one of the pioneers of string theory. Chew was known as a leader of the S-matrix approach to the strong interaction and the associated bootstrap principle
Geoffrey_Chew
Values which describe behavior of a linear electric circuit
Scattering parameters or S-parameters are the elements of a scattering matrix or S-matrix which describe the steady state response of linear electrical
Scattering_parameters
2021 video game and technology demonstration
PlayStation 5 and Xbox Series X and Series S, serving as a marketing tie-in for the 2021 film The Matrix Resurrections. Released during the 2021 Game
The_Matrix_Awakens
Binary operation
scattering matrix S B {\displaystyle S^{B}} , then the combined system A B {\displaystyle AB} has scattering matrix S A B = S A ⋆ S B {\displaystyle S^{AB}=S^{A}\star
Redheffer_star_product
Mathematical ring whose elements are matrices
abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set
Matrix_ring
Conjecture in theoretical physics
Lagrangian that respect the desired symmetries is the simplest way to implement S-matrix theory, and its low energy limit leads to an effective field theory. As
Folk_theorem_(physics)
Matrix operation generalizing exponentiation of scalar numbers
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems
Matrix_exponential
Matrix representation of a graph
theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix, or discrete Laplacian, is a matrix representation of a
Laplacian_matrix
2003 film by the Wachowskis
The Matrix Reloaded is a 2003 American science fiction action film written and directed by the Wachowskis. It is the sequel to The Matrix (1999) and the
The_Matrix_Reloaded
Stochastic matrix representing links between entities
A Google matrix is a particular stochastic matrix that is used by Google's PageRank algorithm. The matrix represents a graph with edges representing links
Google_matrix
Methods and tools to measure electricity
sensor Electric field Electrical power by the means of electricity meter S-matrix by the means of network analyzer (electrical) Electrical power spectrum
Electrical_measurements
Matrix equal to its conjugate-transpose
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose
Hermitian_matrix
Matrices named after Élie Cartan
S {\displaystyle DS} , where D {\displaystyle D} is a diagonal matrix, and S {\displaystyle S} is a symmetric matrix. For example, the Cartan matrix for
Cartan_matrix
Fictional character in The Matrix
Matrix franchise. She is portrayed by Jada Pinkett Smith. She serves as a supporting character in the three sequels of the original film, The Matrix Reloaded
Niobe_(The_Matrix)
No-go theorem pertaining the triviality of space-time and internal symmetries
supersymmetry was already underway. Consider a theory that can be described by an S-matrix and that satisfies the following conditions The symmetry group is a Lie
Coleman–Mandula_theorem
Real square matrix whose columns and rows are orthogonal unit vectors
In linear algebra, an orthogonal matrix or orthonormal matrix Q, is a real-valued square matrix whose columns and rows are orthonormal vectors. One way
Orthogonal_matrix
Unitary matrix containing information on the weak interaction
physics, the Cabibbo–Kobayashi–Maskawa matrix, CKM matrix, quark mixing matrix, or KM matrix is a unitary matrix that contains information on the strength
Cabibbo–Kobayashi–Maskawa matrix
Cabibbo–Kobayashi–Maskawa_matrix
Matrix decomposition
(also known as eigenvalue decomposition or EVD) is a factorization of a matrix A {\displaystyle A} into a canonical form given by A = Q D Q − 1 {\displaystyle
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
American mathematical physicist
his work in quantum mechanics, particularly the development of axiomatic S-matrix theory, the proofs of strong nonlocality properties,[citation needed] and
Henry_Stapp
Property of scattering amplitudes
backwards in time. Crossing states that the same formula that determines the S-matrix elements and scattering amplitudes for particle A {\displaystyle \mathrm
Crossing_(physics)
Subatomic particles consisting of quarks and gluons
confirmed as observed. Exotic hadrons can be searched for by looking for S-matrix poles with quantum numbers forbidden to ordinary hadrons. Experimental
Exotic_hadron
Theorem of matrix algebra of invariance properties under basis transformations
invariant under a change of basis. Namely, if S {\displaystyle S} is a symmetric matrix, then for any invertible matrix P {\displaystyle P} , the numbers of positive
Sylvester's_law_of_inertia
Study of the analytic properties of scattering amplitudes
or in the related S {\displaystyle S} -matrix. In the case of the Coulomb potential considered above this S {\displaystyle S} -matrix is given by the following
Regge_theory
Finds likely sequence of hidden states
observations prob ← T × S matrix of zeroes prev ← empty T × S matrix for each state s in states do prob[0][s] = init[s] * emit[s][obs[0]] for t = 1 to T
Viterbi_algorithm
Matrices similar to diagonal matrices
linear algebra, a square matrix A {\displaystyle A} is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists
Diagonalizable_matrix
Matrix used to describe the transitions of a Markov chain
It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix. The stochastic matrix was first developed by Andrey Markov
Stochastic_matrix
German physicist (1901–1976)
scattering matrix, or S-matrix, in elementary particle physics. The first two papers were published in 1943 and the third in 1944. The S-matrix described
Werner_Heisenberg
Connection between correlation functions and the S-matrix
Lehmann–Symanzik–Zimmermann (LSZ) reduction formula is a method to calculate S-matrix elements (the scattering amplitudes) from the time-ordered correlation
LSZ_reduction_formula
Mathematics concept
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose
Hadamard_matrix
Mathematical concept in algebra
In linear algebra, a nilpotent matrix is a square matrix N such that N k = 0 {\displaystyle N^{k}=0\,} for some positive integer k {\displaystyle k}
Nilpotent_matrix
Property of gauge theories in particle physics
scale decreases. (Alternatively, and perhaps contrarily, in applying an S-matrix, asymptotically free refers to free particles states in the distant past
Asymptotic_freedom
Polynomial of the elements of a matrix
In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant,
Permanent_(mathematics)
Theorem of quantum field theory
functions and matrix elements of the scattering matrix (S-matrix) are free of ultraviolet divergencies. Green's functions and scattering matrix are the fundamental
Bogoliubov–Parasyuk_theorem
Method used in mathematical physics
it is equal to the QED perturbative S-matrix; and (iv) exhibits the same symmetries as the QED perturbative S-matrix. Let us refer to such a regularization
Regularization_(physics)
Model of neutrino oscillation
matrix (PMNS matrix), Maki–Nakagawa–Sakata matrix (MNS matrix), lepton mixing matrix, or neutrino mixing matrix is a unitary mixing matrix that contains
Pontecorvo–Maki–Nakagawa–Sakata matrix
Pontecorvo–Maki–Nakagawa–Sakata_matrix
n matrix, and B is an n by s matrix over R, then ρ(AB) ≥ ρ(A) + ρ(B) – n where ρ is the inner rank of a matrix. The inner rank of an m by n matrix is
Sylvester_domain
Radio technology devices
The S-matrix for an ideal, symmetric hybrid coupler reduces to; S = 1 2 [ 0 − i − 1 0 − i 0 0 − 1 − 1 0 0 − i 0 − 1 − i 0 ] {\displaystyle \mathbf {S} ={\frac
Power dividers and directional couplers
Power_dividers_and_directional_couplers
Theory of the strong nuclear interactions
because space and time break down. This was the more radical approach of S-matrix theory. James Bjorken proposed that pointlike partons would imply certain
Quantum_chromodynamics
On the number of spanning trees in a graph
Laplacian matrix L of G is the difference between the graph's degree matrix (the diagonal matrix of vertex degrees) and its adjacency matrix (a (0,1)-matrix with
Kirchhoff's_theorem
Computer printing process
Dot matrix printing, sometimes called impact matrix printing, is a computer printing process in which ink is applied to a surface using a relatively low-resolution
Dot_matrix_printing
Compact station wagon by Toyota, 2002–2014
The Toyota Matrix, officially named Toyota Corolla Matrix, is a compact hatchback manufactured by Toyota Motor Manufacturing Canada in Cambridge, Ontario
Toyota_Matrix
Identity in abelian theories due to gauge invariance
they do not contribute to S-matrix elements. The Ward identity is a specialization of the Ward–Takahashi identity to S-matrix elements, which describe
Ward–Takahashi_identity
Formulation of quantum mechanics
probabilities of all physically possible outcomes must add up to one) of the S-matrix is obscure in the formulation. The path-integral approach has proven to
Path-integral_formulation
2003 video game
Enter the Matrix is a 2003 action-adventure video game developed by Shiny Entertainment and published by Infogrames under the Atari brand name. The first
Enter_the_Matrix
Fictional character
also known as The One) is a fictional character and the protagonist of The Matrix franchise, created by the Wachowskis. He was portrayed as a cybercriminal
Neo_(The_Matrix)
Theorem in theoretical physics
the S-matrix. Supergravity S-matrix Haag, R.; Łopuszański, J.T.; Sohnius, M. (1975). "All possible generators of supersymmetries of the S-matrix". Nuclear
Haag–Łopuszański–Sohnius theorem
Haag–Łopuszański–Sohnius_theorem
Matrix equal to its transpose
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, A is symmetric ⟺ A = A T . {\displaystyle A{\text{
Symmetric_matrix
Age-structured model of population growth
Leslie matrix is a discrete, age-structured model of population growth named after Patrick H. Leslie and used in population ecology. The Leslie matrix (also
Leslie_matrix
Expansion of the time evolution operator
\end{aligned}}} The S-matrix may be obtained by writing this in the Heisenberg picture, taking the in and out states to be at infinity: ⟨ Ψ o u t ∣ S ∣ Ψ i n ⟩ =
Dyson_series
Mathematical description of quantum state
for a given s {\textstyle s} -spin particles can be represented as a finite ( 2 s + 1 ) 2 {\textstyle (2s+1)^{2}} matrix which acts on 2 s + 1 {\textstyle
Wave_function
Swiss mathematician and physicist
physics, including the exchange particle model of fundamental forces, causal S-matrix theory, and the renormalization group. His idiosyncratic style and publication
Ernst_Stueckelberg
Equivalence under a change of basis (linear algebra)
matrices A and B are called similar if there exists an invertible n-by-n matrix P such that B = P − 1 A P . {\displaystyle B=P^{-1}AP.} Two matrices are
Matrix_similarity
Matrix with exactly one 1 per row and column
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column
Permutation_matrix
Any P-matrix is a Q-matrix. Conversely, if a matrix is a Z-matrix and a Q-matrix, then it is also a P-matrix. P-matrix Z-matrix Karamardian, S. (1976)
Q-matrix
if A = S B S ¯ − 1 {\displaystyle A=SB{\bar {S}}^{-1}\,} for some invertible n × n {\displaystyle n\times n} matrix S {\displaystyle S} , where S ¯ {\displaystyle
Matrix_consimilarity
Fibrillar network lying on nuclear membrane
In biology, the nuclear matrix is the network of fibres found throughout the inside of a cell nucleus after a specific method of chemical extraction. According
Nuclear_matrix
matrix is a matrix with entries from a Boolean algebra. When the two-element Boolean algebra is used, the Boolean matrix is called a logical matrix.
Boolean_matrix
Special kind of square matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal
Triangular_matrix
Two-dimensional matrix barcode
A Data Matrix is a two-dimensional code consisting of black and white "cells" or dots arranged in either a square or rectangular pattern, also known as
Data_Matrix
Matrix-valued random variable
probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled
Random_matrix
Structural support for biological cells
In biology, the extracellular matrix (ECM), also called the intercellular matrix, is a network consisting of extracellular macromolecules and minerals
Extracellular_matrix
Indian scientist and academic (1939–2025)
are naturally described in the S-matrix rather than Hamiltonian formalism, this investigation relied on the S-matrix formulation of Statistical Mechanics
Ramamurti_Rajaraman
divisor matrix (sometimes abbreviated as GCD matrix) is a matrix that may also be referred to as Smith's matrix. The study was initiated by H.J.S. Smith
GCD_matrix
Theory proposed by Roger Penrose
Radu; Spradlin, Marcus; Volovich, Anastasia (2004-07-30). "Tree-level S matrix of Yang–Mills theory". Physical Review D. 70 (2) 026009. arXiv:hep-th/0403190
Twistor_theory
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