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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 . The theory is commonly viewed as containing the fundamental set of particles – the leptons, quarks, gauge bosons and the Higgs particle. The Standard Model is renormalizable and mathematically self-consistent,〔In fact, there are mathematical issues regarding quantum field theories still under debate (see e.g. Landau pole), but the predictions extracted from the Standard Model by current methods are all self-consistent. For a further discussion see e.g. R. Mann, chapter 25.〕 however despite having huge and continued successes in providing experimental predictions it does leave some unexplained phenomena. 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. This article requires some background in physics and mathematics, but is designed as both an introduction and a reference. ==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. These fields are * the fermion field, , which accounts for "matter particles"; * the electroweak boson fields , and ; * the gluon field, ; and * the Higgs field, . 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 the quantum state (ket vector). The dynamics of the quantum state and the fundamental fields are determined by the Lagrangian density (usually for short just called the Lagrangian). This plays a role similar to that of the Schrödinger equation in non-relativistic quantum mechanics, but a Lagrangian is not an equation – rather, it is a polynomial function of the fields and their derivatives, and used with the principle of least action. While it would be possible to derive a system of differential equations governing the fields from the Langrangian, it is more common to use other techniques to compute with quantum field theories. The standard model is furthermore a gauge theory, which means there are degrees of freedom in the mathematical formalism which do not correspond to changes in the physical state. The gauge group of the standard model is , where U(1) acts on and , acts on and , and SU(3) acts on . The fermion field also transforms under these symmetries, although all of them leave some parts of it unchanged. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Standard Model (mathematical formulation)」の詳細全文を読む スポンサード リンク
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