If an odd graded derivation $Q$ preserves the action and integration measure, then $⟨QX⟩=0$ whenever boundary contributions and anomalies vanish. This applies to finite-dimensional zero-dimensional supersymmetric field theories as well as suitably regulated field integrals.

A symmetry-preserving deformation can leave protected observables unchanged while concentrating their integral on a smaller critical locus. Supersymmetric Ward identities justify the deformation; convergence and boundary terms must be controlled.

For a free massless pseudoclassical spinning particle, let $p_m$ be the canonical momentum and ${\psi }^m$ odd coordinates in a Grassmann algebra. Choose the Dirac bracket conventionwith all $p_m$ commuting with the ${\psi }^n$ and with each other. The odd supercharge $\mathcal{Q}$ then obeysThe first equality follows immediately by contracting the elementary Dirac bracket with $p_m{p}_n$. It expresses the closure of two odd symmetry transformations into an even worldline translation.

In units $\hslash =1$, canonical quantization replaces the graded Dirac bracket by the corresponding graded commutator multiplied by $i$. Thus $\{{\hat{\psi }}^m,{\hat{\psi }}^n\}={\eta }^{mn}$ andConsequently the physical-state constraint $\hat{\mathcal{Q}}|\Psi ⟩=0$ implies $\hat{\mathcal{H}}|\Psi ⟩=0$, the massless mass-shell condition. Specifying these bracket and unit conventions fixes the normalization of the worldline supersymmetry algebra.

A supersymmetric action is invariant under supersymmetry up to boundary terms. For ordinary four-dimensional chiral superfields, a full superspace integration of a real Kähler potential and a chiral superspace integration of a superpotential provide the two-derivative ungauged action.

String theories with worldsheet fermionic degrees of freedom and the corresponding supersymmetric consistency conditions. In the RNS string description, consistent supersymmetric GSO projections produce spacetime supersymmetry and remove the NS tachyon. The flat critical dimension is ten. Worldsheet supersymmetry alone does not guarantee spacetime supersymmetry without a suitable projection.

A string theory combining a bosonic left-moving system with a supersymmetric right-moving system. Ten target coordinates leave sixteen left-moving internal degrees of freedom for gauge currents. Modular consistency allows the familiar gauge algebras ${\mathfrak{e}}_8\oplus {\mathfrak{e}}_8$ and $\mathfrak{so}(32)$. Suitable compactification can produce spacetime supersymmetry and chiral gauge matter.

A string construction combining worldsheet orientation reversal with a suitable spacetime involution. It projects the spectrum and introduces fixed-locus sources whose charges must be treated together with branes and fluxes. In compactifications, such projections can reduce supersymmetry and modify the gauge and matter spectrum. Charge cancellation is part of consistency.

An eleven-dimensional nonperturbative framework suggested by superstring dualities, with eleven-dimensional supergravity as its low-energy limit. Type IIA at coupling $g_s$ has an interpretation on a circle of radius $R_{11}=g_s\sqrt{{\alpha }^{\prime }}$, so an additional dimension becomes visible at strong coupling. Perturbative string descriptions represent particular limits of this framework.

A statistical Hamiltonian assigns an energy to a microscopic configuration. Its Boltzmann weight defines the canonical ensemble, and summing or integrating that weight gives a partition function. A reduced Hamiltonian absorbs inverse temperature into its coefficients, so the weight is $e^{-H_{\mathrm{reduced}}}$. An Ising model uses discrete Ising spin configurations, while Landau-Ginzburg theory uses a coarse scalar field.

Kinetic theory describes many-particle systems through a distribution on phase space, retaining both position and velocity. In the number-density convention, $f(t,\mathbf{x},\mathbf{v})d^{3}x{d}^{3}v$ is the expected number of particles in a small position–velocity region. For identical particles of mass $m$, its integrals give the number density $n$ and bulk velocity $\mathbf{u}$:This $f$ is not normalized to one: its integral over the whole phase space gives the total expected particle number. Transport moves particles along trajectories, while collision or diffusion terms change the distribution. The Boltzmann equation, Hamiltonian Liouville equation and Fokker-Planck equation describe different physical assumptions, rather than interchangeable collision laws.

An ideal gas is a motivating example: the pressure on a wall comes from microscopic momentum transfers, even when the bulk velocity is zero. The Maxwell-Boltzmann velocity distribution separates this random molecular motion from bulk motion; see ideal gas pressure from wall collisions. The attractive feature of kinetic theory is that a distribution can describe two opposing populations where one average velocity hides almost everything. It also tells us when averaging into a continuum approximation loses essential information: compare the intermolecular mean free path with the apparatus scale using the Knudsen number. ^[ref]

Superconductivity is a phase of matter that supports steady electric current with zero bulk electrical resistivity and exhibits the Meissner effect: equilibrium screening of a sufficiently weak, static magnetic field from the interior. It occurs below a material-dependent superconducting transition temperature, provided the electric current and applied magnetic field are also sufficiently small. ^[ref] Bulk expulsion means that a sample much thicker than its London penetration depth has an almost field-free interior; the magnetic field need not vanish at its surface. Conventional BCS theory explains superconductivity through correlated electrons forming Cooper pairs. ^[ref]

The two observations answer different questions. A resistance-versus-temperature experiment for superconductivity asks whether a steady electric current requires a measurable electric potential difference. A Field-cooled Meissner experiment asks whether a magnetic field already present in the material is expelled when it enters superconductivity. An ideal perfect conductor with zero electrical resistivity can preserve that pre-existing magnetic field; zero electrical resistance alone therefore does not establish the Meissner effect.

The usefulness follows directly from Joule heating, $P=I^{2}R$: maintaining a steady electric current through a segment with zero electrical resistance produces no resistive heating in that segment. This makes persistent electric currents attractive for producing steady magnetic fields. The beauty is that one phase has both a transport property and an equilibrium magnetic response, and the second cannot be inferred just by making an ordinary electrical resistance very small. The London equations encode this extra magnetic response. ^[ref]

Hyperbolic motion in special relativity is rectilinear motion with constant proper acceleration. A representative positive-$x$ world line is $x=c{t}_0\cosh (\tau /t_0)$, $t=t_0\sinh (\tau /t_0)$, with proper acceleration $c/t_0$. This describes a hyperbola in spacetime, not a hyperbolic Kepler orbit.

The outgoing angular spectrum propagates prescribed scalar boundary data into a half-space. For $e^{-i\omega t}$ time dependence, $\beta (q)=\sqrt{k^2-q^2}$ has nonnegative real part on propagating modes and nonnegative imaginary part on evanescent modes. Its trace at the boundary is $g$, and each mode solves the Helmholtz equation. The spectral representation can be interpreted through distributions for nondecaying plane-wave data.

Taylor expansion of the Dirichlet boundary condition gives the first-order boundary trace $-h{\partial }_z{\psi }^{[0]}$ at the mean plane. The outgoing angular spectrum then propagates it. A zero-mean surface gives zero mean for this linear rough correction, while the complete scattered field still contains its nonzero flat reflection.

For normal plane-wave incidence, the second-order Dirichlet boundary condition gives ${\psi }_s^{[2]}=-h{\partial }_z{\psi }_s^{[1]}$ at the mean plane, since ${\partial }_z^2{\psi }^{[0]}$ vanishes there. The Dirichlet-to-Neumann map for a Helmholtz half-space then yields $2kh\mathcal{B}h$. This nonlocal expression is not generally $2k^2{h}^2$; that local limit needs roughness varying slowly compared with the wavelength.

For normal incidence on a Dirichlet rough surface, the scattered trace on the moving boundary is exactly $-e^{-ikh(x)}$. Its trace on the mean plane is instead $-1+2ikh+2kh\mathcal{B}h+\cdots$. The two traces are related by a normal Taylor expansion and must not be identified. Their ensemble averages can therefore have different quadratic coefficients without inconsistency.

For a real zero-mean stationary height process, its power spectrum of surface height is the Fourier transform of the height covariance function. It is even and nonnegative, and ${\sigma }^2=(2\pi {)}^{-1}\int S_h(q)dq$. More general stationary processes have a spectral measure rather than a density. Weighted spectral moments control the existence of boundary derivatives and second-order mean scattering corrections.

Through second order in height, a normally incident acoustic plane wave has the displayed mean reflected amplitude for a Dirichlet boundary. The propagating part of the power spectrum of surface height contributes a positive real correction to the negative flat reflection coefficient, while the evanescent part contributes an imaginary correction. The mean linear rough correction is zero.

The rigid-lid approximation suppresses free surface displacement in a fluid layer while retaining a variable pressure that enforces the mass constraint. At constant depth it gives horizontally nondivergent motion. In a one-layer quasi-geostrophic approximation, removing free-surface stretching eliminates the $-\psi /R_D^2$ term from shallow-water quasi-geostrophic potential vorticity. This is also the small-horizontal-scale limit compared with the Rossby deformation radius, when that term is negligible rather than exactly absent.

For linear surface gravity waves with wavelength much larger than the depth, ${\omega }^2=gh{\kappa }^2$. It follows from ${\omega }^2=g\kappa \tanh (\kappa h)$ when $\kappa h\ll 1$, and the wave speed is $\sqrt{gh}$.