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Solve the algebraic connection equation for $\hat{\omega }(e,\chi )$, but perform the remaining variations as though the connection were independent. The chain-rule contribution $(\delta S/\delta \omega )\delta \hat{\omega }$ vanishes on that solution. The resulting equations are those of the connection-eliminated second-order action. This is especially useful for local supersymmetry calculations, where the gravitino-induced contorsion is quadratic in fermions and the omitted induced connection variation would otherwise be lengthy.

The antiparticle of a fermion. In a Dirac field, its external wavefunctions are the negative-frequency $v$ spinors, while its physical energy is positive. Fermion-number arrows on Feynman diagrams run opposite an antifermion's physical momentum.

One identical fermion in each of two input modes remains one in each output after a lossless two-mode unitary transformation. The fully occupied two-mode state transforms by the determinant phase. Pauli exclusion prevents double occupation of either specified mode.

Compare uniform three-dimensional ideal gases at the same number density n, with fermion mass $m_F$, spin degeneracy $g_F$, boson mass $m_B$ and a single boson spin component. The Fermi temperature has $k_B{T}_F={\hslash }^2(6{\pi }^{2}n/g_F{)}^{2/3}/(2m_F)$, while Bose saturation gives $k_B{T}_B=2\pi {\hslash }^2[n/\zeta (3/2){]}^{2/3}/m_B$. Eliminating density yieldsThe degeneracy is essential; using a spinless Fermi sea changes the numerical estimate. This is an ideal-gas comparison and does not remove interaction effects in a dense liquid.

The charge-conjugate field of a fermion field transforms in the complex-conjugate internal group representation. With a compatible Dirac adjoint and charge conjugation matrix convention it is ${\psi }^c=C{\bar{\psi }}^T$, up to a conventional constant phase. In four spacetime dimensions charge conjugation exchanges the chirality of the field; a right-handed quark can therefore be represented by a left-handed conjugate field when listing chiral unified matter. In two-dimensional Minkowski spacetime, charge conjugation can instead preserve chirality, permitting a Majorana-Weyl spinor.

A gauge theory whose left- and right-chiral fermions transform differently under its gauge group. The weak charged current gives a physical example. A chiral representation assignment must satisfy the relevant gauge anomaly cancellation conditions to define a consistent quantum theory.

A four-dimensional N=1 chiral multiplet consists off shell of a complex scalar $\varphi$, a Weyl fermion ${\psi }_{\alpha }$, and a complex auxiliary scalar $F$. A chiral superfield packages their supersymmetry transformations. Eliminating the auxiliary scalar leaves two bosonic and two fermionic physical polarizations; an allowed supersymmetric mass gives equal scalar and fermion masses.

Let ${\partial }_{\pm }={\partial }_{\tau }\pm {\partial }_{\sigma }$ and $\mathcal{L}={\partial }_{+}X\cdot {\partial }_{-}X+i{\psi }_1\cdot {\partial }_{+}{\psi }_1+i{\psi }_2\cdot {\partial }_{-}{\psi }_2$. For constant Grassmann-odd parameters, set $\delta X=i({ϵ}_2{\psi }_1-{ϵ}_1{\psi }_2)$, $\delta {\psi }_1=-{ϵ}_2{\partial }_{-}X$, $\delta {\psi }_2={ϵ}_1{\partial }_{+}X$. In the first chiral sector the variation of the fermionic term cancels $i{ϵ}_2{\partial }_{+}{\psi }_1\cdot {\partial }_{-}X$ from the bosonic term, leaving $i{ϵ}_2{\partial }_{-}({\psi }_1\cdot {\partial }_{+}X)$. The second sector gives the other total derivative. This proves off-shell invariance when the boundary flux vanishes. Closure on fermions uses their free equations of motion. A constant parameter must also preserve the chosen boundary conditions and spin structure.

The combined transformation $\psi \mapsto e^{-i\alpha {\gamma }_5/2}\psi$ and $({\varphi }_1+i{\varphi }_2)\mapsto e^{i\alpha }({\varphi }_1+i{\varphi }_2)$ preserves this interaction. The Dirac adjoint transforms with the same axial exponential as $\psi$. For potential $-{\mu }^2{r}^2/2+\lambda r^4/4$, $v=\mu /\sqrt{\lambda }$ gives a radial scalar of squared mass $2\lambda v^2$, a massless Goldstone boson and fermion physical mass $|g|v$. Cubic scalar vertices appear after the shift: $hhh$ has factor $-6i\lambda v$ and $h\pi \pi$ has factor $-2i\lambda v$.

At leading order, for final fermions much lighter than the Z boson, the partial decay width is proportional to $N_c(|g_L{|}^2+|g_R{|}^2)$. Here $N_c$ is the final-state color multiplicity in a decay width, and $g_L=T_{3L}-Q{\sin }^2{\theta }_W$, $g_R=T_{3R}-Q{\sin }^2{\theta }_W$ are the dimensionless left- and right-chirality couplings. The common normalization cancels in ratios.

For an electron, $g_L=-1/2+{\sin }^2{\theta }_W$, $g_R={\sin }^2{\theta }_W$. For one active neutrino flavour, $g_L=1/2$, $g_R=0$. ThereforeThere is no extra flavour factor for a single specified neutrino species. Summing the three active species produces three times that invisible partial width, within the same massless approximation.

Gregor Jotzu and collaborators realized an artificial Haldane model in 2014. They loaded spin-polarized potassium-40 fermions into a laser-made honeycomb lattice. Piezoelectric mirrors shook the optical lattice along two axes; their relative phase controlled time-reversal symmetry breaking. Optical phase control adjusted the sublattice offset.^[ref]

A magnetic field gradient drove Bloch oscillations. Band mapping measured interband transfer to locate valley-gap closures. Opposite-force drift measurements distinguished cancelling from reinforcing Berry curvature contributions; reversing shaking handedness reversed the drift. Interband transfer and harmonic confinement affected this readout, including drift beyond ideal transition lines. This neutral-atom Floquet Hamiltonian realization did not measure electrical sheet conductivity. Magnetic fields were used for preparation and forcing.^[ref]

Fine-grained occupation is conserved along trajectories of a collisionless distribution. Coarse graining cannot increase its maximum. For fermions the primordial occupation is bounded, so a measured halo density and velocity scale restrict the particle mass required to produce that halo.

Exchanging two identical bound atoms exchanges their constituent fermions. An atom with mass number $A$ and atomic number $Z$ contains $A$ nucleons and $Z$ electrons, so its exchange sign is $(-1{)}^{A+Z}$ when its internal state is preserved. An odd count gives a composite fermion; an even count gives a composite boson. This is equivalently the integer-versus-half-integer total-spin distinction in the Spin-statistics theorem. For example, neutral lithium-6 has nine fermionic constituents and is a fermion, while neutral lithium-7 has ten and is a boson. Counting nucleons alone instead classifies the nucleus, not the neutral atom. This effective composite-particle description requires energies and separations where the bound atom remains intact.

Adding a constant to a global superpotential does not change the component action because the chiral superspace integral of that constant vanishes. Neither the F-term scalar potential nor the fermion mass matrix depends on it. This statement does not extend to supergravity, where the superpotential itself enters the gravitational potential.

For two identical fermions of mass $m$ in the same one-dimensional quantum harmonic oscillator, use center of mass $X=(x_1+x_2)/2$ and separation $r=x_1-x_2$. The corresponding oscillator masses are $2m$ and reduced mass $m/2$, with the same frequency $\omega$. If $N,n$ are their occupation numbers, the unperturbed energy is $\hslash \omega (N+n+1)$. The fermionic exchange symmetry requires even $n$ with a spin-one-half singlet state and odd $n$ with a spin-one-half triplet state.

For a delta potential $\lambda \delta (r)$ the first-order shift is $\lambda |{\phi }_n^{\mathrm{rel}}(0){|}^2$. Every odd relative wavefunction vanishes at contact, so these states remain exact unaffected eigenstates of the spatial problem. For the even relative ground state, the oscillator length is $\sqrt{2}\ell$, where $\ell =\sqrt{\hslash /(m\omega )}$, and the shift is $\lambda /(\sqrt{2\pi }\ell )$, independent of $N$. This separates interaction-sensitive singlets from contact-insensitive triplets.

A Cooper pair is a correlated pair of fermions associated with superconductivity. The conventional BCS ground state uses opposite-momentum, opposite-spin pair creation operators $c_{k\uparrow }^{†}{c}_{-k\downarrow }^{†}$. Their total momentum is zero for every relative momentum $k$; zero total momentum does not restrict $k$ to zero.

For orthonormal orbitals $a,b$, the symmetric spatial state $C=(|ab⟩+|ba⟩)/\sqrt{2}$ multiplied by the antisymmetric spin singlet has one electron on each center and no double occupancy. The full state remains antisymmetric under fermion exchange.

For vector coupling to a Dirac fermion, write the Hermitian color connection ${\mathcal{A}}_{\mu }=A_{\mu }^a{T}^a$. Charge conjugation reverses the vector current and transposes its color matrix, requiring ${\mathcal{A}}_{\mu }^C=-{\mathcal{A}}_{\mu }^T$. Combining with parity gives ${\mathcal{A}}_{\mu }^{CP}(x)=-P_{\mu }{}^{\nu }{\mathcal{A}}_{\nu }^T(x_P)$, where $P=\mathrm{diag}(1,-1,-1,-1)$. The transpose acts only on color indices; intrinsic fermion phases cancel in the bilinear.

For a Hermitian fermion bilinear, an antiunitary operator implementing CPT conjugates numerical coefficients and exchanges particle and antiparticle fields at $-x$. Reordering the fermions then gives $\Theta :\overline{\psi }\Gamma \psi (x):{\Theta }^{-1}=:\overline{\psi }{\gamma }_5\Gamma {\gamma }_5\psi (-x):$ when ${\Gamma }^{†}={\gamma }^0\Gamma {\gamma }^0$. Thus scalar, pseudoscalar and tensor bilinears are even, while vector and axial-vector bilinears are odd. Each derivative contributes another minus sign. The graded reordering is essential; treating Dirac fields as commuting quantities gives the wrong transformation.

Each of $D$ embedding bosons contributes one to the chiral central charge, and each real worldsheet Majorana fermion contributes one half. The reparameterization bc system contributes $-26$, and the superconformal ghosts contribute $+11$. Vanishing total worldsheet Weyl anomaly therefore requires $3D/2-26+11=0$, or $D=10$. This differs from the critical dimension 26 of the bosonic string, which has no worldsheet fermions or superconformal ghosts.