Foundations

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.

An ideal spinless one-dimensional p-wave pairing chain has a gapped topological phase with boundary Majorana modes for nonzero pairing and |μ|<2|t| in the uniform nearest-neighbour model.

Will, Cochran and collaborators reported signatures of non-equilibrium topological order in a periodically driven quantum-processor experiment published 10 September 2025.

Liu, Sau and Das Sarma's 2018 nanowire calculations show how trivial Andreev bound states can generate local zero-bias peaks resembling topological Majorana signals.

The Boltzmann equation evolves a dilute one-particle distribution with streaming, force and collision terms.

ETH proposes that suitable local-observable matrix elements in many chaotic energy eigenstates have thermal structure, allowing narrow-energy isolated preparations to approach thermal local expectations.

For perturbation H′(t)=−f(t)B, the retarded response of A is χ_AB(t)=(i/ħ)θ(t)⟨[A(t),B(0)]⟩ in an equilibrium reference state.

Expanding a diffeomorphism-invariant gravitational action in curvature and derivatives organizes low-energy quantum corrections without claiming a complete short-distance theory.

String theory organizes extended-object states and includes a massless spin-two excitation in appropriate constructions; loop approaches quantize geometric variables with spin-network/holonomy methods; asymptotic safety seeks a suitable ultraviolet renormalization-group fixed point.

The 2017 Marletto–Vedral proposal links entanglement between two masses to nonclassical mediation under explicit subsystem and locality assumptions.

For constant-density incompressible flow, averaging u=U+u′ adds the Reynolds stress R_ij=⟨u′_iu′_j⟩ to the mean momentum equation.

DNS aims to resolve the dynamically relevant viscous scales of a specified flow; LES filters resolved scales and models subfilter stress.

A nonlinear polarization drives new frequency components.

In a lossless scalar narrowband Kerr envelope convention, i∂_zA=−D∂²_tA+δ|A|²A represents second-order dispersion and self-phase modulation.

Yee FDTD staggers electric/magnetic fields in space and time to approximate Maxwell curl evolution.

Nichols' worked compressible mixing-layer comparison specifies two streams, geometry, initial transients and statistical sampling before comparing RANS, LES and hybrid predictions with measurements downstream.

Displacement compares endpoints; distance measures the travelled path.

Closed formulas require constant acceleration over the interval.

Newtonian force balances need an inertial frame or explicit inertial terms.

A constant speed can still have an acceleration.

Mass is in kilograms; gravitational weight is a force.

Contact geometry determines the force direction.

List forces on the selected body once.