Foundations

A positive-definite local energy difference can prove Lyapunov stability of an isolated equilibrium even when its quadratic Hessian is degenerate.

A finite-outcome POVM is a family of positive operators E_a whose sum is the identity.

An instrument associates outcome a with a completely positive trace-nonincreasing map ℐ_a.

A quantum channel Φ(ρ)=Σ_kK_kρK_k† with Σ_kK_k†K_k=I remains positive when extended by an identity map on an arbitrary ancilla.

A time-homogeneous completely positive trace-preserving semigroup has a GKSL generator: a Hamiltonian commutator plus dissipators with nonnegative rates.

For stationary zero-mean Gaussian phase noise f(t), a qubit coherence factor is exp[−½∫₀ᵀ∫₀ᵀK(t−t′)dt dt′], where K is the covariance of angular-frequency noise.

For a uniform dimensionless scalar order parameter m with symmetry m↦−m, an effective free-energy density f=f₀+a m²/2+b m⁴/4 has minima m=0 for a>0 and m²=−a/b for a<0 when b>0.

Connected order-parameter correlations define a correlation length ξ.

Coarse-graining changes couplings while preserving long-distance observables after rescaling.

In the local London model, ∇×j_s=−n_s q_s²B/m_s.

Ginzburg–Landau describes superconductivity through a complex order parameter, symmetry and a gradient/field-dependent free energy, yielding a coherence length and electromagnetic response.

For the local single-component GL model, the SI free-energy density includes α|ψ|²+β|ψ|⁴/2, a covariant gradient cost and B²/(2μ₀).

The homogeneous spin-singlet mean-field pairing model gives quasiparticles E_k=√(ξ_k²+|Δ_k|²), whose thermal populations enter a self-consistent gap equation.

Bardeen, Cooper and Schrieffer's 1957 paper proposed a phonon-mediated attractive interaction and a correlated paired ground state, connecting an excitation gap to thermodynamic and electromagnetic behaviour.

In correlated superconductors, spin-fluctuation-mediated pairing, electron–phonon contributions and other electronic correlations must be tested against gap symmetry, spectroscopy and thermodynamics for the particular material.

The asymptotically flat, spherical vacuum exterior has ds²=−(1−r_s/r)c²dt²+(1−r_s/r)⁻¹dr²+r²dΩ², r_s=2GM/c².

Two static clocks outside a Schwarzschild source compare proper rates through √(1−r_s/r).

Weak far-zone gravitational waves change proper separations through transverse-traceless strain.

For a differentiable local field Lagrangian, a continuous global symmetry that changes it by a total divergence yields an on-shell conserved Noether current.

LIGO's 2016 report of the 14 September 2015 event combined a coherent transient in two detectors with noise-background tests and waveform comparison.

Quantizing a real free scalar field replaces its amplitudes by operators with [φ(t,x),π(t,y)]=iδ³(x−y) in ħ=c=1 units.

The interaction-picture time-ordered expansion organizes scattering amplitudes in couplings; loops introduce regulated momentum integrals.

The strong coupling α_s depends on renormalization scale and scheme.

A precision Standard Model fit compares predicted electroweak observables with correlated measurements, jointly propagating input-parameter, systematic and missing-order uncertainties.

The Muon g−2 Collaboration's June 2026 detailed report describes the final positive-muon anomaly measurement at 127 ppb precision, including precession frequency, calibrated magnetic field and beam-dynamics corrections.

Data-driven dispersion calculations and lattice QCD estimate the same hadronic contribution through different inputs and systematics.

A single-band Hubbard model combines hopping t_ij with on-site repulsion U n_i↑n_i↓.

Single-site dynamical mean-field theory maps an interacting lattice problem to a self-consistent impurity problem with a frequency-dependent local self-energy.

For a gapped noninteracting two-dimensional band system, the occupied-band Berry curvature integrates to an integer C=(2π)⁻¹∫_BZTrF_xy d²k.

Spinful time-reversal symmetry with T²=−1 permits a Z₂ band distinction even when total Chern number vanishes.