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# Standard Model (mathematical formulation)

This article describes the mathematics of the Standard Model of particle physics, a gauge quantum field theory containing the internal symmetries of the unitary product group. [1]

145 relations: Abelian group, Abstract index notation, Angular momentum, Antiparticle, Baryon, Baryon number, Bottom quark, Bra–ket notation, Cabibbo–Kobayashi–Maskawa matrix, Charge (physics), Charm quark, Chirality (physics), Circle group, Classical field theory, Classical mechanics, Color charge, Color confinement, Conservation law, Conserved current, Coupling constant, CP violation, Creation and annihilation operators, Dirac equation, Direct product of groups, Down quark, Dyson series, Effective field theory, Electric charge, Electromagnetic four-potential, Electron, Electron neutrino, Electroweak interaction, Energy, Fermi's interaction, Feynman diagram, Feynman slash notation, Field (physics), First quantization, Flavour (particle physics), Four-momentum, Four-vector, Fourier series, Fundamental interaction, Gamma matrices, Gauge boson, Gauge theory, Gell-Mann–Nishijima formula, General relativity, Generator (mathematics), Global symmetry, ... Expand index (95 more) »

## Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written (the axiom of commutativity).

## Abstract index notation

Abstract index notation is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis.

## Angular momentum

In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum.

## Antiparticle

Corresponding to most kinds of particles, there is an associated antimatter antiparticle with the same mass and opposite charge (including electric charge).

## Baryon

A baryon is a composite subatomic particle made up of three quarks (as distinct from mesons, which are composed of one quark and one antiquark).

## Baryon number

In particle physics, the baryon number is a strictly conserved additive quantum number of a system.

## Bottom quark

The bottom quark or b quark, also known as the beauty quark, is a third-generation quark with a charge of − ''e''.

## Bra–ket notation

In quantum mechanics, bra–ket notation is a standard notation for describing quantum states, composed of angle brackets and vertical bars.

In the Standard Model of particle physics, the Cabibbo–Kobayashi–Maskawa matrix (CKM matrix, quark mixing matrix, sometimes also called KM matrix) is a unitary matrix which contains information on the strength of flavour-changing weak decays.

## Charge (physics)

In physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics.

## Charm quark

The charm quark or c quark (from its symbol, c) is the third most massive of all quarks, a type of elementary particle.

## Chirality (physics)

A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality).

## Circle group

In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, i.e., the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C&times;, the multiplicative group of all nonzero complex numbers.

## Classical field theory

A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter.

## Classical mechanics

In physics, classical mechanics and quantum mechanics are the two major sub-fields of mechanics.

## Color charge

Color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD).

## Color confinement

Color confinement, often simply called confinement, is the phenomenon that color charged particles (such as quarks) cannot be isolated singularly, and therefore cannot be directly observed.

## Conservation law

In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time.

## Conserved current

In physics a conserved current is a current, j^\mu, that satisfies the continuity equation \partial_\mu j^\mu.

## Coupling constant

In physics, a coupling constant or gauge coupling parameter is a number that determines the strength of the force exerted in an interaction.

## CP violation

In particle physics, CP violation (CP standing for charge parity) is a violation of the postulated CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge conjugation symmetry) and P-symmetry (parity symmetry).

## Creation and annihilation operators

Creation and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems.

## Dirac equation

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928.

## Direct product of groups

In group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted.

## Down quark

The down quark or d quark (symbol: d) is the second-lightest of all quarks, a type of elementary particle, and a major constituent of matter.

## Dyson series

In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative series, and each term is represented by Feynman diagrams.

## Effective field theory

In physics, an effective field theory is a type of approximation to (or effective theory for) an underlying physical theory, such as a quantum field theory or a statistical mechanics model.

## Electric charge

Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field.

## Electromagnetic four-potential

An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived.

## Electron

The electron is a subatomic particle, symbol or, with a negative elementary electric charge.

## Electron neutrino

The electron neutrino is a subatomic lepton elementary particle which has no net electric charge.

## Electroweak interaction

In particle physics, the electroweak interaction is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction.

## Energy

In physics, energy is a property of objects which can be transferred to other objects or converted into different forms, but cannot be created or destroyed.

## Fermi's interaction

In particle physics, Fermi's interaction (also the fermi theory of beta decay) is an explanation of the beta decay, proposed by Enrico Fermi in 1933.

## Feynman diagram

In theoretical physics, Feynman diagrams are pictorial representations of the mathematical expressions describing the behavior of subatomic particles.

## Feynman slash notation

In the study of Dirac fields in quantum field theory, Richard Feynman invented the convenient Feynman slash notation (less commonly known as the Dirac slash notation).

## Field (physics)

In physics, a field is a physical quantity that has a value for each point in space and time.

## First quantization

A first quantization of a physical system is a semi-classical treatment of quantum mechanics, in which particles or physical objects are treated using quantum wave functions but the surrounding environment (for example a potential well or a bulk electromagnetic field or gravitational field) is treated classically.

## Flavour (particle physics)

In particle physics, flavour or flavor refers to a species of an elementary particle.

## Four-momentum

In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime.

## Four-vector

In the theory of relativity, a four-vector or 4-vector is a vector in Minkowski space, a four-dimensional real vector space.

## Fourier series

In mathematics, a Fourier series is a way to represent a (wave-like) function as the sum of simple sine waves.

## Fundamental interaction

Fundamental interactions, also known as fundamental forces, are the interactions in physical systems that don't appear to be reducible to more basic interactions.

## Gamma matrices

In mathematical physics, the gamma matrices, \, also known as the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cℓ1,3(R).

## Gauge boson

In particle physics, a gauge boson is a force carrier, a bosonic particle that carries any of the fundamental interactions of nature.

## Gauge theory

In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations.

## Gell-Mann–Nishijima formula

The Gell-Mann–Nishijima formula (sometimes known as the NNG formula) relates the baryon number B, the strangeness S, the isospin I3 of hadrons to the charge Q. It was originally given by Kazuhiko Nishijima and Tadao Nakano in 1953, and led to the proposal of strangeness as a concept, which Nishijima originally called "eta-charge" after the eta meson.

## General relativity

General relativity, also known as the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.

## Generator (mathematics)

In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts.

## Global symmetry

In physics, a global symmetry is a symmetry that holds at all points in the spacetime under consideration, as opposed to a local symmetry which varies from point to point.

No description.

## Graviton

In physics, the graviton is a hypothetical elementary particle that mediates the force of gravitation in the framework of quantum field theory.

## Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.

In particle physics, a hadron (ἁδρός, hadrós, "stout, thick") is a composite particle made of quarks held together by the strong force (in a similar way as molecules are held together by the electromagnetic force).

## Heisenberg picture

In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory.

In mathematics, specifically in functional analysis, each bounded linear operator on a complex Hilbert space has a corresponding adjoint operator.

## Hierarchy problem

In theoretical physics, the hierarchy problem is the large discrepancy between aspects of the weak force and gravity.

## Higgs boson

The Higgs boson or Higgs particle is an elementary particle in the Standard Model of particle physics.

## Higgs mechanism

In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons.

## Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

## Interaction picture

In quantum mechanics, the interaction picture (also known as the Dirac picture) is an intermediate representation between the Schrödinger picture and the Heisenberg picture.

## Invariant mass

The invariant mass, rest mass, - Lawrence S. Lerner - Science - 1997 intrinsic mass, proper mass, or in the case of bound systems simply mass, is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations.

## Journal of Physics G

Journal of Physics G: Nuclear and Particle Physics is a peer-reviewed journal that publishes theoretical and experimental research into nuclear physics, particle physics and particle astrophysics, including all interface areas between these fields.

## Klein–Gordon equation

The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic version of the Schrödinger equation.

## Lagrangian (field theory)

Lagrangian field theory is a formalism in classical field theory.

## Landau pole

In physics, the Landau pole or the Moscow zero is the momentum (or energy) scale at which the coupling constant (interaction strength) of a quantum field theory becomes infinite.

## Lepton

A lepton is an elementary, half-integer spin (spin) particle that does not undergo strong interactions, but is subject to the Pauli exclusion principle.

## Lepton number

In particle physics, the lepton number is the number of leptons minus the number of antileptons.

## Lie group

In mathematics, a Lie group is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

## Local symmetry

In physics, a local symmetry is symmetry of some physical quantity, which smoothly depends on the point of the base manifold.

## Lorentz group

In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all (nongravitational) physical phenomena.

## Lorentz transformation

In physics, the Lorentz transformation (or transformations) is named after the Dutch physicist Hendrik Lorentz.

## Majorana equation

The Majorana equation is a relativistic wave equation similar to the Dirac equation but includes the charge conjugate ψc of a spinor ψ.

## Mass

In physics, mass is a property of a physical body which determines the strength of its mutual gravitational attraction to other bodies, its resistance to being accelerated by a force, and in the theory of relativity gives the mass–energy content of a system.

## Minimal subtraction scheme

In quantum field theory, the minimal subtraction scheme, or MS scheme, is a particular renormalization scheme used to absorb the infinities that arise in perturbative calculations beyond leading order, introduced independently by and.

## Momentum

In classical mechanics, linear momentum or translational momentum (pl. momenta; SI unit kg m/s, or equivalently, N s) is the product of the mass and velocity of an object.

## Momentum operator

In quantum mechanics, momentum (like all other physical variables) is defined as an operator, which "acts on" or pre-multiplies the wave function to extract the momentum eigenvalue from the wave function: the momentum vector a particle would have when measured in an experiment.

## Muon

The muon (from the Greek letter mu (μ) used to represent it) is an elementary particle similar to the electron, with electric charge of −1 e and a 2, but with a much greater mass.

## Muon neutrino

The muon neutrino is a subatomic lepton elementary particle which has the symbol and no net electric charge.

## Neutrino

A neutrino (or, in Italian) is an electrically neutral elementary particle with half-integer spin.

## Neutrino oscillation

Neutrino oscillation is a quantum mechanical phenomenon whereby a neutrino created with a specific lepton flavor (electron, muon or tau) can later be measured to have a different flavor.

## Noether's theorem

Noether's (first) theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law.

## Noncommutative standard model

In theoretical particle physics, the non-commutative Standard Model, mainly due to the French mathematician Alain Connes, uses his noncommutative geometry to devise an extension of the Standard Model to include a modified form of general relativity.

## Observable

In physics, particularly in quantum physics, a system observable is a measurable operator, or gauge, where the property of the system state can be determined by some sequence of physical operations.

## Old quantum theory

The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics.

## On shell renormalization scheme

In quantum field theory, and especially in quantum electrodynamics, the interacting theory leads to infinite quantities that have to be absorbed in a renormalization procedure, in order to be able to predict measurable quantities.

## Open problem

In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (no solution for it is known).

## Operator (physics)

In physics, an operator is a function over the space of physical states.

## Particle Data Group

The Particle Data Group (or PDG) is an international collaboration of particle physicists that compiles and reanalyzes published results related to the properties of particles and fundamental interactions.

## Particle number operator

In quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles.

## Particle physics

Particle physics is the branch of physics that studies the nature of the particles that constitute matter (particles with mass) and radiation (massless particles).

## Pauli matrices

In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian and unitary.

## Phase factor

For any complex number written in polar form (such as reiθ), the phase factor is the complex exponential factor (eiθ).

No description.

## Physics beyond the Standard Model

Physics beyond the Standard Model (BSM) refers to the theoretical developments needed to explain the deficiencies of the Standard Model, such as the origin of mass, the strong CP problem, neutrino oscillations, matter–antimatter asymmetry, and the nature of dark matter and dark energy.

## Poincaré group

The Poincaré group, named after Henri Poincaré, is the group of Minkowski spacetime isometries.

## Pontecorvo–Maki–Nakagawa–Sakata matrix

In particle physics, the Pontecorvo–Maki–Nakagawa–Sakata matrix (PMNS matrix), Maki–Nakagawa–Sakata matrix (MNS matrix), lepton mixing matrix, or neutrino mixing matrix, is a unitary matrixThe PMNS matrix is not unitary in the seesaw model.

## Positron

The positron or antielectron is the antiparticle or the antimatter counterpart of the electron.

## Principle of least action

In non-relativistic physics, the principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system by stating a system follows the path where the average difference between the kinetic energy and potential energy is minimized or maximized over any time period.

## QCD matter

Quark matter or QCD matter refers to any of a number of theorized phases of matter whose degrees of freedom include quarks and gluons.

## Quantum chromodynamics

In theoretical physics, quantum chromodynamics (QCD) is the theory of strong interactions, a fundamental force describing the interactions between quarks and gluons which make up hadrons such as the proton, neutron and pion.

## Quantum electrodynamics

In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics.

## Quantum field theory

In theoretical physics, quantum field theory (QFT) is a theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics.

## Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, or quantum theory), including quantum field theory, is a fundamental branch of physics concerned with processes involving, for example, atoms and photons.

## Quantum state

In quantum physics, quantum state refers to the state of a quantum system.

## Quark

A quark is an elementary particle and a fundamental constituent of matter.

## Quark model

In particle physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons.

## Raising and lowering indices

In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type.

## 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.

## Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

## Rotational symmetry

Generally, an object with 'rotational symmetry' also known in biological contexts as 'radial symmetry', is an object that looks the same after a certain amount of rotation.

## Row and column vectors

In linear algebra, a column vector or column matrix is an m &times; 1 matrix, that is, a matrix consisting of a single column of m elements, Similarly, a row vector or row matrix is a 1 &times; m matrix, that is, a matrix consisting of a single row of m elements Throughout, boldface is used for the row and column vectors.

## Schrödinger equation

In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a physical system changes with time.

## Schrödinger picture

In physics, the Schrödinger picture (also called the Schrödinger representation) is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are constant with respect to time.

## Seesaw mechanism

In the theory of grand unification of particle physics, and, in particular, in theories of neutrino masses and neutrino oscillation, the seesaw mechanism is a generic model used to understand the relative sizes of observed neutrino masses, of the order of eV, compared to those of quarks and charged leptons, which are millions of times heavier.

## Semidirect product

In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product.

## Special relativity

In physics, special relativity (SR, also known as the special theory of relativity or STR) is the generally accepted physical theory regarding the relationship between space and time.

## Special unitary group

In mathematics, the special unitary group of degree, denoted, is the Lie group of unitary matrices with determinant 1 (i.e., real-valued determinant, not complex as for general unitary matrices).

## Spinor

In geometry and physics, spinors are elements of a (complex) vector space that can be associated with Euclidean space.

## Spontaneous symmetry breaking

Spontaneous symmetry breaking is a mode of realization of symmetry breaking in a physical system, where the underlying laws are invariant under a symmetry transformation, but the system as a whole changes under such transformations, in contrast to explicit symmetry breaking.

## Standard Model

The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, as well as classifying all the subatomic particles known.

## Sterile neutrino

Sterile neutrinos (or inert neutrinos) are hypothetical particles (neutral leptons – neutrinos) that interact only via gravity and do not interact via any of the fundamental interactions of the Standard Model.

## Strange quark

The strange quark or s quark (from its symbol, s) is the third-lightest of all quarks, a type of elementary particle.

## Strong interaction

In particle physics, the strong interaction is the mechanism responsible for the strong nuclear force (also called the strong force, nuclear strong force or colour force), one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation.

## Symmetry (physics)

In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation.

## Symmetry in quantum mechanics

Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, with applications in the mathematical formulation of the standard model and condensed matter physics.

## Tau (particle)

The tau (τ), also called the tau lepton, tau particle or tauon, is an elementary particle similar to the electron, with negative electric charge and a 2.

## Tau neutrino

The tau neutrino or tauon neutrino is a subatomic elementary particle which has the symbol and no net electric charge.

## Top quark

The top quark, also known as the t quark (symbol: t) or truth quark, is an elementary particle and a fundamental constituent of matter.

## Translational symmetry

In geometry, a translation "slides" a thing by a: Ta(p).

## Unitarity gauge

In theoretical physics, the unitarity gauge or unitary gauge is a particular choice of a gauge fixing in a gauge theory with a spontaneous symmetry breaking.

## Unitary group

In mathematics, the unitary group of degree n, denoted U(n), is the group of unitary matrices, with the group operation that of matrix multiplication.

## Up quark

The up quark or u quark (symbol: u) is the lightest of all quarks, a type of elementary particle, and a major constituent of matter.

## Vacuum angle

In quantum gauge theories, in the Hamiltonian formulation, the wave function is a functional of the gauge connection \,A and matter fields \,\phi.

## Vacuum expectation value

In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average, expected value in the vacuum.

## Vector boson

In particle physics, a vector boson is a boson with the spin equal to 1.

## W and Z bosons

The W and Z bosons (together known as the weak bosons or, less specifically, the intermediate vector bosons) are the elementary particles that mediate the weak interaction; their symbols are,, and.

## Wave equation

The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves.

## Wave function

A wave function in quantum mechanics describes the quantum state of an isolated system of one or more particles.

## Weak hypercharge

The weak hypercharge in particle physics is a quantum number relating the electric charge and the third component of weak isospin.

## Weak interaction

In particle physics, the weak interaction is the mechanism responsible for the weak force or weak nuclear force, one of the four known fundamental interactions of nature, alongside the strong interaction, electromagnetism, and gravitation.

## Weak isospin

In particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction.

## Weinberg angle

The Weinberg angle or weak mixing angle is a parameter in the Weinberg–Salam theory of the electroweak interaction, and is usually denoted as &theta;W.

## Yang–Mills theory

Yang–Mills theory is a gauge theory based on the SU(''N'') group, or more generally any compact, semi-simple Lie group.

## Yukawa interaction

In particle physics, Yukawa's interaction, named after Hideki Yukawa, is an interaction between a scalar field ϕ and a Dirac field ψ of the type The Yukawa interaction can be used to describe the nuclear force between nucleons (which are fermions), mediated by pions (which are pseudoscalar mesons).

## References

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