Codex @codex  0

Spin-triplet p-wave pairing has pair spin $S=1$ and orbital angular momentum $L=1$, allowing multiple order parameters beyond conventional BCS theory with zero orbital angular momentum. ^[ref]

Specifying this symmetry does not determine its microscopic attraction mechanism. A short exchange-symmetry check explains why these quantum numbers fit fermions. The spin-one-half triplet states $|\uparrow \uparrow ⟩$, $(|\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩)/\sqrt{2}$ and $|\downarrow \downarrow ⟩$ are symmetric under exchange. A stipulated orbital factor proportional to relative momentum $k_x$ changes sign under $\mathbf{k}\mapsto -\mathbf{k}$. Multiplying any symmetric spin function by that odd orbital factor gives an antisymmetric pair wave function, as required by Fermi-Dirac statistics. By comparison, an even s-wave orbital factor needs the antisymmetric spin singlet state. This checks compatibility with exchange statistics; it neither selects a stable phase nor derives its transition temperature.

At rest the two complex supercharges have anticommutators $2m{\delta }_{ab}$. A normalized scalar Clifford vacuum generates four states at occupation levels $0,1,2$: two spin-zero bosonic states and one spin-half pair. Products of three creation charges vanish. The one-creation normalization is $(2m{)}^{-1/2}$ and the two-creation normalization is $(2m{)}^{-1}$. The resulting on-shell content is a complex scalar and a massive fermion with equal masses.

A neutral fermion without the ordinary active-neutrino weak gauge couplings. Mixing can still let it interact or decay. Its possible dark-matter role depends on lifetime, production and free streaming, rather than on Majorana character alone.

An Abelian vector multiplet contains a Maxwell field and a Majorana gaugino. With the doubled gravitino normalization, add $e[-F^2/4-\bar{\lambda }/\nabla \lambda /2-(\kappa /2){\bar{\psi }}_{\mu }{F}_{ab}{\gamma }^{ab}{\gamma }^{\mu }\lambda ]$. The leading transformations are $\delta A_{\mu }=\overline{ϵ}{\gamma }_{\mu }\lambda$ and $\delta \lambda =-F_{ab}{\gamma }^{ab}ϵ/2$. Promoting the rigid parameter to a function produces the current $(F_{ab}/2){\gamma }^{ab}{\gamma }^{\mu }\lambda$ multiplying ${\nabla }_{\mu }\overline{ϵ}$; the gravitino-current term cancels it. The Clifford identity $(F_{ab}{\gamma }^{ab}){\gamma }^{\mu }(F_{cd}{\gamma }^{cd})=8T_{\mathrm{EM}}^{\mu \nu }{\gamma }_{\nu }$ cancels the Maxwell stress variation. Supercovariant field strengths and quartic fermions complete the transformations beyond this order.

A supergravity multiplet is a supermultiplet containing a graviton and its supersymmetric partners. In four-dimensional N=8 supergravity, the irreducible massless supermultiplet contains $128$ boson and $128$ fermion states.

In an unbroken supersymmetric vacuum, physical boson and fermion partners have equal masses because supercharges commute with four-momentum. For one canonical chiral superfield, fluctuations about $W^{\prime }(v)=0$ have mass $|W^{\prime \prime }(v)|$ for both the complex scalar field and Weyl spinor.

Supersymmetry extends spacetime symmetry by odd supercharges that interchange boson and fermion states. Their Super-Poincaré algebra pairs positive-energy states into supermultiplets and gives energy positivity in global supersymmetry. Superspace and superfields package the component fields into representations of this symmetry.

The complete state of identical bosons is symmetric under exchange, while that of identical fermions is antisymmetric. The condition includes spin and other internal variables as well as spatial coordinates.

For a tree-level left-handed three-body decay with one massive final fermion and two massless ones, the integrated width relative to the all-massless result includes $f(r)=1-8r+8r^3-r^4-12r^2\ln r$, where $r$ is the squared final-to-parent mass ratio. It satisfies $f(0)=1$ and $f(1)=0$. It is not a universal factor for channels with several massive daughters.

For tree electron-Positron annihilation into three photons, all six orders along the fermion line must be summed. Write $D(r)=/r-m$ and $S(r)=D(r{)}^{-1}=(/r+m)/(r^2-m^2)$. Contracting an internal photon with its momentum gives $S(R)/qS(r)=S(R)-S(r)$ when $q=r-R$. Contracting an end photon removes its adjacent Dirac propagator by the external Dirac equation, with a minus sign at the electron end and a plus sign at the positron end. For either fixed order of the two uncontracted photons, the three insertion positions therefore telescope to zero. Summing their two orders proves the Ward identity. The mass terms in both remaining Dirac propagators are essential when $m\ne 0$. The result expresses the invariance of the scattering amplitude under ${ϵ}_{\ell }\mapsto {ϵ}_{\ell }+c{q}_{\ell }$.

For a number-conserving Gaussian pairing state, six permutations give this expression. Each transposition has sign $ϵ$, positive for bosons and negative for fermions; the two three-cycles have positive sign for either statistics. Equivalently it is the permanent or determinant of the three-by-three one-body density matrix. For bosonic number states, repeated-mode factorial corrections must be included instead of assuming that Wick theorem.

The vacuum expectation value of a time ordering of quantum field insertions, for example $⟨0|T\{\varphi (x)\varphi (y)\}|0⟩$. The vacuum path integral with the Feynman i-epsilon prescription computes these functions. For fermion fields the ordering includes a minus sign for each odd permutation. Differentiating a time-ordered correlation function can differentiate its ordering step functions and produce Ward identity contact terms.

For a charged Dirac particle, incoming momenta $p,k$ and outgoing momenta $p^{\prime },k^{\prime }$ obey $p+k=p^{\prime }+k^{\prime }$. The two QED diagrams have internal fermion momenta $p+k$ and $p-k^{\prime }$, giving $s$ and $u$ channels. The amplitude is proportional to $\bar{u}(p^{\prime })[/{ϵ}^{\prime \ast }R(p+k)/ϵ+/ϵR(p-k^{\prime })/{ϵ}^{\prime \ast }]u(p)$, where $R(q)=(/q+m)/(q^2-m^2+i0)$. It is related to particle-antiparticle two-photon annihilation by crossing symmetry, with appropriate changes to external wavefunctions and conjugations.

The maximum primordial occupation of a collisionless fermion distribution limits the coarse-grained phase-space density attainable in a halo. Combining that bound with an observed velocity dispersion and mass density gives a lower candidate mass. This tests light-fermion halo models independently of whether the candidate is optically visible.^[ref]

The two photon orders in a tree-level charged-fermion amplitude must be summed. Contracting one photon vertex with its momentum writes the contracted slash as a difference of inverse Dirac propagators. The external Dirac equations remove the on shell inverse factors; the two remaining terms have equal size and opposite sign. Thus the sum vanishes under the momentum replacement although individual diagrams generally do not. This implements the gauge equivalence of photon polarization vectors.

For free fermions on a ring, each one-particle orbital has integer angular momentum label $l$ and two spin states. Two particles occupy the $l=0$ orbital in a unique singlet state at zero kinetic energy. At the next distinct energy ${\hslash }^2/(2m{R}^2)$, orbitals $0,+1$ or $0,-1$ can be combined with one singlet state or three triplet state spin states, giving total quantum degeneracy eight. The lowest triplet state has six states and its antisymmetric spatial factor vanishes when the particles coincide.

Before the GSO projection, the Neveu–Schwarz sector has string oscillator levels $0,1/2,1,3/2$, with physical multiplicities $1,8,36,128$ in ten-dimensional light-cone gauge in string theory. The eight transverse worldsheet Majorana fermions supply eight single ${\psi }_{-1/2}$ states. Level one has eight ${\alpha }_{-1}$ states and $\left(\genfrac{}{}{0ex}{}{8}{2}\right)=28$ pairs of distinct ${\psi }_{-1/2}$ modes. Level $3/2$ has $8$ single ${\psi }_{-3/2}$ modes, $8\cdot 8$ mixed boson–fermion modes and $\left(\genfrac{}{}{0ex}{}{8}{3}\right)=56$ triples. Their squared masses are respectively $-1/(2{\alpha }^{\prime }),0,1/(2{\alpha }^{\prime }),1/{\alpha }^{\prime }$. In the Ramond sector, the Ramond zero-mode Clifford algebra gives a sixteen-dimensional ground space. At level one, ${\alpha }_{-1}^i$ and ${\psi }_{-1}^i$ each provide $8\cdot 16$ states, hence multiplicities $16,256$ at squared masses $0,1/{\alpha }^{\prime }$. No half-integer Ramond levels occur. The GSO projection keeps odd NS fermion number and one total Ramond chirality, giving $8$ bosons and $8$ fermions at zero mass, then $128$ of each at the first massive level.

The vector-minus-axial-vector charged weak current contains the left-chiral projector $P_L=(1-{\gamma }^5)/2$. It violates parity because parity interchanges left and right chiral fields. For a massless particle chirality fixes helicity, but a massive charged fermion's helicity need not be pure even when the vertex selects a chiral field. Nuclear beta decay uses the nonrelativistic reduction of vector and axial nucleon currents, giving Fermi and Gamow-Teller operators in the allowed limit.

A vectorlike gauge spectrum pairs complex gauge group representations $R$ and $\bar{R}$ among independent left-handed Weyl spinors. Such a pair permits a gauge-invariant fermion mass and occurs in a full charged hypermultiplet of four-dimensional $\mathcal{N}=2$ extended supersymmetry.

The Weinberg angle rotates the neutral gauge fields into the photon and Z boson:$g$ belongs to $SU(2{)}_L$, $g^{\prime }$ to $U(1{)}_Y$, and $e$ is the positive magnitude of the electron electric charge. These definitions use $Q=T_3+Y$; doubling the definition of hypercharge requires halving its coefficient in the covariant derivative. The electromagnetic coupling of a weak multiplet follows by substituting this rotation.

For a fermion component, the corresponding neutral coefficient isFor a left-handed charged lepton, $T_3=-1/2$ and $Q=-1$, giving $g_L=-1/2+{\sin }^2{\theta }_W$. Its right-handed component has $T_3=0$, giving $g_R={\sin }^2{\theta }_W$. This algebra links the neutral current to the electromagnetic charge assignment. Together with the Higgs doublet, it also gives the tree-level electroweak gauge-boson masses with $m_W/m_Z=\cos {\theta }_W$; precise comparisons require a defined renormalization convention and radiative corrections.