Spin (physics) in the context of Self-adjoint operator

In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin ⁠1/2⁠, spin ⁠3/2⁠, etc.) and obey the Pauli exclusion principle. These particles include all quarks and leptons and all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics.

Some fermions are elementary particles (such as electrons), and some are composite particles (such as protons). For example, according to the spin-statistics theorem in relativistic quantum field theory, particles with integer spin are bosons. In contrast, particles with half-integer spin are fermions.

The up quark or u quark (symbol: u) is the lightest of all quarks, a type of elementary particle, and a significant constituent of matter. It, along with the down quark, forms the neutrons (one up quark, two down quarks) and protons (two up quarks, one down quark) of atomic nuclei. It is part of the first generation of matter, has an electric charge of +⁠2/3⁠ e and a bare mass of 2.2+0.5
−0.4 MeV/c. Like all quarks, the up quark is an elementary fermion with spin ⁠1/2⁠, and experiences all four fundamental interactions: gravitation, electromagnetism, weak interactions, and strong interactions. The antiparticle of the up quark is the up antiquark (sometimes called antiup quark or simply antiup), which differs from it only in that some of its properties, such as charge have equal magnitude but opposite sign.

Its existence (along with that of the down and strange quarks) was postulated in 1964 by Murray Gell-Mann and George Zweig to explain the Eightfold Way classification scheme of hadrons. The up quark was first observed by experiments at the Stanford Linear Accelerator Center in 1968.

The down quark (symbol: d) is a type of elementary particle, and a major constituent of matter. The down quark is the second-lightest of all quarks, and combines with other quarks to form composite particles called hadrons. Down quarks are most commonly found in atomic nuclei, where it combines with up quarks to form protons and neutrons. The proton is made of one down quark with two up quarks, and the neutron is made up of two down quarks with one up quark. Because they are found in every single known atom, down quarks are present in all everyday matter that we interact with.

The down quark is part of the first generation of matter, has an electric charge of −⁠1/3⁠ e and a bare mass of 4.7+0.5
−0.3 MeV/c. Like all quarks, the down quark is an elementary fermion with spin ⁠1/2⁠, and experiences all four fundamental interactions: gravitation, electromagnetism, weak interactions, and strong interactions. The antiparticle of the down quark is the down antiquark (sometimes called antidown quark or simply antidown), which differs from it only in that some of its properties have equal magnitude but opposite sign.

The electron neutrino (ν
_e) is an elementary particle which has zero electric charge and a spin of 1⁄2. Together with the electron, it forms the first generation of leptons, hence the name electron neutrino. It was first hypothesized by Wolfgang Pauli in 1930, to account for missing momentum and missing energy in beta decay, and was discovered in 1956 by a team led by Clyde Cowan and Frederick Reines (see Cowan–Reines neutrino experiment).

In particle physics, a baryon is a type of composite subatomic particle that contains an odd number of valence quarks, conventionally three. Protons and neutrons are examples of baryons; because baryons are composed of quarks, they belong to the hadron family of particles. Baryons are also classified as fermions because they have half-integer spin.

The name "baryon", introduced by Abraham Pais, comes from the Greek word for "heavy" (βαρύς, barýs), because, at the time of their naming, most known elementary particles had lower masses than the baryons. Each baryon has a corresponding antiparticle (antibaryon) where their corresponding antiquarks replace quarks. For example, a proton is made of two up quarks and one down quark; and its corresponding antiparticle, the antiproton, is made of two up antiquarks and one down antiquark.

Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases, that arise from electromagnetic forces between atoms and electrons. More generally, the subject deals with condensed phases of matter: systems of many constituents with strong interactions among them. More exotic condensed phases include the superconducting phase exhibited by certain materials at extremely low cryogenic temperatures, the ferromagnetic and antiferromagnetic phases of spins on crystal lattices of atoms, the Bose–Einstein condensates found in ultracold atomic systems, and liquid crystals. Condensed matter physicists seek to understand the behavior of these phases by experiments to measure various material properties, and by applying the physical laws of quantum mechanics, electromagnetism, statistical mechanics, and other physics theories to develop mathematical models and predict the properties of extremely large groups of atoms.

The diversity of systems and phenomena available for study makes condensed matter physics the most active field of contemporary physics: one third of all American physicists self-identify as condensed matter physicists, and the Division of Condensed Matter Physics is the largest division of the American Physical Society. These include solid state and soft matter physicists, who study quantum and non-quantum physical properties of matter respectively. Both types study a great range of materials, providing many research, funding and employment opportunities. The field overlaps with chemistry, materials science, engineering and nanotechnology, and relates closely to atomic physics and biophysics. The theoretical physics of condensed matter shares important concepts and methods with that of particle physics and nuclear physics.

A quark (/ˈkwɔːrk, ˈkwɑːrk/ ) is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly observable matter is composed of up quarks, down quarks and electrons. Owing to a phenomenon known as color confinement, quarks are never found in isolation; they can be found only within hadrons, which include baryons (such as protons and neutrons) and mesons, or in quark–gluon plasmas. For this reason, much of what is known about quarks has been drawn from observations of hadrons.

Quarks have various intrinsic properties, including electric charge, mass, color charge, and spin. They are the only elementary particles in the Standard Model of particle physics to experience all four fundamental interactions, also known as fundamental forces (electromagnetism, gravitation, strong interaction, and weak interaction), as well as the only known particles whose electric charges are not integer multiples of the elementary charge.

In particle physics, a lepton is an elementary particle of half-integer spin (spin ⁠1/2⁠) that does not undergo strong interactions. Two main classes of leptons exist: charged leptons (also known as the electron-like leptons or muons), including the electron, muon, and tauon, and neutral leptons, better known as neutrinos. Charged leptons can combine with other particles to form various composite particles such as atoms and positronium, while neutrinos rarely interact with anything, and are consequently rarely observed. The best known of all leptons is the electron.

There are six types of leptons, known as flavours, grouped in three generations. The first-generation leptons, also called electronic leptons, comprise the electron (e
) and the electron neutrino (ν
_e); the second are the muonic leptons, comprising the muon (μ
) and the muon neutrino (ν
_μ); and the third are the tauonic leptons, comprising the tau (τ
) and the tau neutrino (ν
_τ). Electrons have the least mass of all the charged leptons. The heavier muons and taus will rapidly change into electrons and neutrinos through a process of particle decay: the transformation from a higher mass state to a lower mass state. Thus electrons are stable and the most common charged lepton in the universe, whereas muons and taus can only be produced in high-energy collisions (such as those involving cosmic rays and those carried out in particle accelerators).

The positron or antielectron is the particle with an electric charge of +1e, a spin of 1/2 (the same as the electron), and approximately the same mass as an electron. It is the antiparticle (antimatter counterpart) of the electron. When a positron collides with an electron, annihilation occurs. If this collision occurs at low energies, it results in the production of two or more photons.

Positrons can be created by positron emission radioactive decay (through weak interactions), or by pair production from a sufficiently energetic photon which is interacting with an atom in a material.

In particle physics, a magnetic monopole is a hypothetical particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magnetic charge". Modern interest in the concept stems from particle theories, notably grand unified and superstring theories, which predict their existence.The known elementary particles that have electric charge are electric monopoles.

Magnetism in bar magnets and electromagnets is not caused by magnetic monopoles, and indeed, there is no known experimental or observational evidence that magnetic monopoles exist. A magnetic monopole is not necessarily an elementary particle, and models for magnetic monopole production can include (but are not limited to) spin-0 monopoles or spin-1 massive vector mesons. The term "magnetic monopole" only refers to the nature of the particle, rather than a designation for a single particle.

A gluon (/ˈɡluːɒn/ GLOO-on) is a type of massless elementary particle that mediates the strong interaction between quarks, acting as the exchange particle for the interaction. Gluons are massless vector bosons, thereby having a spin of 1. Through the strong interaction, gluons bind quarks into groups according to quantum chromodynamics (QCD), forming hadrons such as protons and neutrons.

Gluons carry the color charge of the strong interaction, thereby participating in the strong interaction as well as mediating it. Because gluons carry the color charge, QCD is more difficult to analyze compared to quantum electrodynamics (QED) where the photon carries no electric charge.

In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are W
, W
, and Z
. The W
 bosons have either a positive or negative electric charge of 1 elementary charge and are each other's antiparticles. The Z
 boson is electrically neutral and is its own antiparticle. The three particles each have a spin of 1. The W
 bosons have a magnetic moment, but the Z
has none. All three of these particles are very short-lived, with a half-life of about 3×10 s. Their experimental discovery was pivotal in establishing what is now called the Standard Model of particle physics.

The W bosons are named after the weak force. The physicist Steven Weinberg named the additional particle the "Z particle", and later gave the explanation that it was the last additional particle needed by the model. The W bosons had already been named, and the Z bosons were named for having zero electric charge.

The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the Standard Model, the Higgs particle is a massive scalar boson that couples to (interacts with) particles whose mass arises from their interactions with the Higgs Field, has zero spin, even (positive) parity, no electric charge, and no colour charge. It is also very unstable, decaying into other particles almost immediately upon generation.

The Higgs field is a scalar field with two neutral and two electrically charged components that form a complex doublet of the weak isospin SU(2) symmetry. Its "sombrero potential" leads it to take a nonzero value everywhere (including otherwise empty space), which breaks the weak isospin symmetry of the electroweak interaction and, via the Higgs mechanism, gives a rest mass to all massive elementary particles of the Standard Model, including the Higgs boson itself. The existence of the Higgs field became the last unverified part of the Standard Model of particle physics, and for several decades was considered "the central problem in particle physics".

Fermi–Dirac statistics is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac distribution of particles over energy states. It is named after Enrico Fermi and Paul Dirac, each of whom derived the distribution independently in 1926. Fermi–Dirac statistics is a part of the field of statistical mechanics and uses the principles of quantum mechanics.

Fermi–Dirac statistics applies to identical and indistinguishable particles with half-integer spin (1/2, 3/2, etc.), called fermions, in thermodynamic equilibrium. For the case of negligible interaction between particles, the system can be described in terms of single-particle energy states. A result is the Fermi–Dirac distribution of particles over these states where no two particles can occupy the same state, which has a considerable effect on the properties of the system. Fermi–Dirac statistics is most commonly applied to electrons, a type of fermion with spin 1/2.

In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of ⁠1/2⁠. The spin number describes how many symmetrical facets a particle has in one full rotation; a spin of ⁠1/2⁠ means that the particle must be rotated by two full turns (through 720°) before it has the same configuration as when it started.

Particles with net spin ⁠1/2⁠ include the proton, neutron, electron, neutrino, and quarks. The dynamics of spin-⁠1/2⁠ objects cannot be accurately described using classical physics; they are among the simplest systems whose description requires quantum mechanics. As such, the study of the behavior of spin-⁠1/2⁠ systems forms a central part of quantum mechanics.