Standard Model of particle physics
Elementary particles of the Standard Model
Background Particle physics Standard Model Quantum field theory Gauge theory Spontaneous symmetry breaking Higgs mechanism
Constituents Electroweak interaction Quantum chromodynamics CKM matrix Standard Model mathematics
Limitations Strong CP problem Hierarchy problem Neutrino oscillations Physics beyond the Standard Model
Scientists Rutherford Thomson Chadwick Bose Sudarshan Davis Jr Anderson Fermi Dirac Feynman Rubbia Gell-Mann Kendall Taylor Friedman Powell Anderson Glashow Iliopoulos Lederman Maiani Meer Cowan Nambu Chamberlain Cabibbo Schwartz Perl Majorana Weinberg Lee Ward Salam Kobayashi Maskawa Mills Yang Yukawa 't Hooft Veltman Gross Pais Pauli Politzer Reines Schwinger Wilczek Cronin Fitch Vleck Higgs Englert Brout Hagen Guralnik Kibble de Mayolo Lattes Zweig
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Quantum field theory
Feynman diagram
History
Background Field theory Electromagnetism Weak force Strong force Quantum mechanics Special relativity General relativity Gauge theory Yang–Mills theory
Symmetries Symmetry in quantum mechanics C-symmetry P-symmetry T-symmetry Lorentz symmetry Poincaré symmetry Gauge symmetry Explicit symmetry breaking Spontaneous symmetry breaking Noether charge Topological charge
Tools Anomaly Background field method BRST quantization Correlation function Crossing Effective action Effective field theory Expectation value Feynman diagram Lattice field theory LSZ reduction formula Partition function Propagator Quantization Regularization Renormalization Vacuum state Wick's theorem
Equations Dirac equation Klein–Gordon equation Proca equations Wheeler–DeWitt equation Bargmann–Wigner equations
Standard Model Quantum electrodynamics Electroweak interaction Quantum chromodynamics Higgs mechanism
Incomplete theories String theory Supersymmetry Technicolor Theory of everything Quantum gravity
Scientists Adler Anderson Anselm Bargmann Becchi Belavin Bell Berezin Bethe Bjorken Bleuer Bogoliubov Brodsky Brout Buchholz Cachazo Callan Coleman Dashen DeWitt Dirac Doplicher Dyson Englert Faddeev Fadin Fayet Fermi Feynman Fierz Fock Frampton Fritzsch Fröhlich Fredenhagen Furry Glashow Gelfand Gell-Mann Goldstone Gribov Gross Gupta Guralnik Haag Heisenberg Hepp Higgs Hagen 't Hooft Iliopoulos Ivanenko Jackiw Jona-Lasinio Jordan Jost Källén Kendall Kinoshita Klebanov Kontsevich Kuraev Landau Lee Lehmann Leutwyler Lipatov Łopuszański Low Lüders Maiani Majorana Maldacena Migdal Mills Møller Naimark Nambu Neveu Nishijima Oehme Oppenheimer Osterwalder Parisi Pauli Peskin Plefka Polyakov Pomeranchuk Popov Proca Rubakov Ruelle Salam Schrader Schwarz Schwinger Segal Seiberg Semenoff Shifman Shirkov Skyrme Sommerfield Stora Stueckelberg Sudarshan Symanzik Thirring Tomonaga Tyutin Vainshtein Veltman Virasoro Ward Weinberg Weisskopf Wentzel Wess Wetterich Weyl Wick Wightman Wigner Wilczek Wilson Witten Yang Yukawa Zamolodchikov Zamolodchikov Zee Zimmermann Zinn-Justin Zuber Zumino
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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 SU(3) × SU(2) × U(1). The theory is commonly viewed as describing the fundamental set of particles – the leptons, quarks, gauge bosons and the Higgs boson.

The Standard Model is renormalizable and mathematically self-consistent,^[1] however despite having huge and continued successes in providing experimental predictions it does leave some unexplained phenomena.^[2] In particular, although the physics of special relativity is incorporated, general relativity is not, and the Standard Model will fail at energies or distances where the graviton is expected to emerge. Therefore, in a modern field theory context, it is seen as an effective field theory.

Quantum field theory

The standard model is a quantum field theory, meaning its fundamental objects are quantum fields which are defined at all points in spacetime. QFT treats particles as excited states (also called quanta) of their underlying quantum fields, which are more fundamental than the particles. These fields are

• the fermion fields, ψ, which account for "matter particles";
• the electroweak boson fields ${\displaystyle W_{1},W_{2},W_{3}}$, and B;
• the gluon field, G_a; and
• the Higgs field, φ.

That these are quantum rather than classical fields has the mathematical consequence that they are operator-valued. In particular, values of the fields generally do not commute. As operators, they act upon a quantum state (ket vector).

Alternative presentations of the fields

As is common in quantum theory, there is more than one way to look at things. At first the basic fields given above may not seem to correspond well with the "fundamental particles" in the chart above, but there are several alternative presentations which, in particular contexts, may be more appropriate than those that are given above.

Fermions

Rather than having one fermion field ψ, it can be split up into separate components for each type of particle. This mirrors the historical evolution of quantum field theory, since the electron component ψ_e (describing the electron and its antiparticle the positron) is then the original ψ field of quantum electrodynamics, which was later accompanied by ψ_μ and ψ_τ fields for the muon and tauon respectively (and their antiparticles). Electroweak theory added ${\displaystyle \psi _{\nu _{\mathrm {e} }},\psi _{\nu _{\mu }}}$, and ${\displaystyle \psi _{\nu _{\tau }}}$ for the corresponding neutrinos. The quarks add still further components. In order to be four-spinors like the electron and other lepton components, there must be one quark component for every combination of flavour and colour, bringing the total to 24 (3 for charged leptons, 3 for neutrinos, and 2·3·3 = 18 for quarks). Each of these is a four component bispinor, for a total of 96 complex-valued components for the fermion field.

An important definition is the barred fermion field ${\displaystyle {\bar {\psi }}}$, which is defined to be ${\displaystyle \psi ^{\dagger }\gamma ^{0}}$, where ${\displaystyle \dagger }$ denotes the Hermitian adjoint of ψ, and γ⁰ is the zeroth gamma matrix. If ψ is thought of as an n × 1 matrix then ${\displaystyle {\bar {\psi }}}$ should be thought of as a 1 × n matrix.

A chiral theory

An independent decomposition of ψ is that into chirality components:

where ${\displaystyle \gamma _{5}}$ is the fifth gamma matrix. This is very important in the Standard Model because left and right chirality components are treated differently by the gauge interactions.

In particular, under weak isospin SU(2) transformations the left-handed particles are weak-isospin doublets, whereas the right-handed are singlets – i.e. the weak isospin of ψ_R is zero. Put more simply, the weak interaction could rotate e.g. a left-handed electron into a left-handed neutrino (with emission of a W⁻), but could not do so with the same right-handed particles. As an aside, the right-handed neutrino originally did not exist in the standard model – but the discovery of neutrino oscillation implies that neutrinos must have mass, and since chirality can change during the propagation of a massive particle, right-handed neutrinos must exist in reality. This does not however change the (experimentally-proven) chiral nature of the weak interaction.

Furthermore, U(1) acts differently on ${\displaystyle \psi _{\mathrm {e} }^{\rm {L}}}$ and ${\displaystyle {\mathrm{\psi <!--\; \psi \; -->}}_{\mathrm{e}}^{}}$Zdroj:https://en.wikipedia.org?pojem=Mathematical_formulation_of_the_Standard_Model
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